{"authority_effect":"NONE","cards":[{"assumptions":"Real normed vector spaces E and F; s is open and connected; t is open; f is a surjective isometry from s to t.","card_ref":{"record_type":"LFH_DECLARATION_CARD","revision":1,"sha256":"ffd0c21c4c389feaa7abe9e5ca64bc9546fb5736d9768da63c049de11648ba84","uid":"lfh:lfh-declaration-card:sha256:30b5b915102151510645ff9e55aef46c99b259d2cb09540049f2b73b0eb07373"},"citations":[{"label":"_private.MathlibAnnex.Analysis.Normed.Affine.IsometryExtension.OpenConnected.0.MathlibAnnex.IsometryEquiv.glue_local_affine_isometry","source_locator":{"end_column":44,"end_line":118,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/OpenConnected.lean","source_sha256":"235562fa4eb1df95ac0320f53756178e6bf8a0739e3a536a6f3afd3069b8ec91","start_column":1,"start_line":74},"text":"The proof or construction of Mankiewicz extension on open connected domains uses the project declaration “Gluing locally defined affine isometries on a connected open set” at the indicated step."},{"label":"MathlibAnnex.IsometryEquiv.exists_affineExtension_eqOn_ball","target_card_ref":{"record_type":"LFH_DECLARATION_CARD","revision":1,"sha256":"57980774b7bb9d703dad9b73dc930e03203d182af996346c7408eb373b1e0322","uid":"lfh:lfh-declaration-card:sha256:c2f8d7f019c50fb6f9c50efe6fa6ba6cb03392611bc794c16ab04a68a88630e3"},"text":"The proof or construction of Mankiewicz extension on open connected domains uses the project declaration “Local affine-isometry chart on a smaller ball” at the indicated step."}],"conclusion":"There is a unique ambient real affine isometry equivalence whose restriction to s is f.","definition_explanation":null,"display_title":"Mankiewicz extension on open connected domains","name":"MathlibAnnex.IsometryEquiv.existsUnique_affineExtension","natural_language_statement":"Let f be a surjective isometry from an open connected subset s of a real normed space onto an open subset t of another real normed space. Then there exists a unique ambient real affine isometry equivalence A agreeing with f on all of s. Target connectedness is not a separate assumption.","one_sentence_role":"Glues local affine-isometry charts to obtain the principal open-domain extension theorem.","proof_steps":[{"step_id":"step-1","text":"For each x∈s, choose positive source and target ball radii from openness and take their minimum R."},{"step_id":"step-2","text":"Apply the local affine-chart theorem to obtain an ambient affine isometry agreeing with f on ball(x,R/8)."},{"step_id":"step-3","text":"Feed these pointwise local charts into the connected gluing lemma."},{"step_id":"step-4","text":"The gluing lemma returns one affine isometry agreeing with f everywhere on s and proves uniqueness from agreement on the nonempty open source."}],"proof_summary":"At each source point, openness of s and t supplies matching small ambient balls. The local chart theorem gives an affine isometry agreeing with f on a smaller ball around that point. These local charts agree on overlaps and therefore form a locally constant chart-valued map; connectedness of s makes it constant. The resulting global chart is the unique extension.","source_locator":{"end_column":37,"end_line":147,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/OpenConnected.lean","source_sha256":"235562fa4eb1df95ac0320f53756178e6bf8a0739e3a536a6f3afd3069b8ec91","start_column":1,"start_line":120},"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/OpenConnected.lean","source_signature":{"end_byte":4956,"extraction_profile":"PARSED_COMMAND_TOP_LEVEL_DECLARATION_HEADER_V1","sha256":"b22c3cacae7bebc98f7faec2113392af6797bc65b527361b314ff3d1fcfe07eb","source_command_range":{"end_column":37,"end_line":147,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/OpenConnected.lean","source_sha256":"235562fa4eb1df95ac0320f53756178e6bf8a0739e3a536a6f3afd3069b8ec91","start_column":1,"start_line":120},"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/OpenConnected.lean","source_sha256":"235562fa4eb1df95ac0320f53756178e6bf8a0739e3a536a6f3afd3069b8ec91","start_byte":4737,"text":"theorem existsUnique_affineExtension\n    {s : Set E} {t : Set F} (f : s ≃ᵢ t)\n    (hs : IsOpen s) (hsc : IsConnected s) (ht : IsOpen t) :\n    ∃! A : E ≃ᵃⁱ[ℝ] F,\n      ∀ x : s, A (x : E) = ((f x : t) : F)"},"stable_card_id":"30b5b915102151510645ff9e55aef46c99b259d2cb09540049f2b73b0eb07373"},{"assumptions":"Real normed vector spaces; f is an isometry equivalence between open balls with radii r > 0 and R > 0.","card_ref":{"record_type":"LFH_DECLARATION_CARD","revision":1,"sha256":"0010c8ccb8965671842243a51487d560ec2424c0e83c8ce493dd7a161cdd859c","uid":"lfh:lfh-declaration-card:sha256:d53fe8e4dd14b54afcbe429368e93867012aae1050cc51eff076031144a34aed"},"citations":[{"label":"MathlibAnnex.IsometryEquiv.existsUnique_affineExtension","target_card_ref":{"record_type":"LFH_DECLARATION_CARD","revision":1,"sha256":"ffd0c21c4c389feaa7abe9e5ca64bc9546fb5736d9768da63c049de11648ba84","uid":"lfh:lfh-declaration-card:sha256:30b5b915102151510645ff9e55aef46c99b259d2cb09540049f2b73b0eb07373"},"text":"The proof or construction of Affine extension from open balls uses the project declaration “Mankiewicz extension on open connected domains” at the indicated step."}],"conclusion":"f extends uniquely to an ambient real affine isometry equivalence. The two positive radii need not be assumed equal.","definition_explanation":null,"display_title":"Affine extension from open balls","name":"MathlibAnnex.IsometryEquiv.existsUnique_affineExtension_ball","natural_language_statement":"Let f be a surjective isometry from the open ball ball(c,r) onto the open ball ball(d,R) in real normed spaces, where r>0 and R>0; the two radii need not be equal. Then there is a unique ambient real affine isometry equivalence A such that A(x)=f(x) for every x in the source ball.","one_sentence_role":"Derives the open-ball form of the Mankiewicz extension theorem from the open-connected-domain theorem.","proof_steps":[{"step_id":"step-1","text":"Use positivity of R to record connectedness of the target ball; this fact is available but is not an additional hypothesis of the invoked extension theorem."},{"step_id":"step-2","text":"Apply the open-connected extension theorem with the openness of both balls and the connectedness of the source ball."},{"step_id":"step-3","text":"The existence, pointwise agreement, and uniqueness clauses are exactly those returned by that theorem."}],"proof_summary":"Both balls are open, and the source ball is connected because its radius is positive. These facts put the isometry directly within the scope of the open-connected extension theorem.","source_locator":{"end_column":52,"end_line":33,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Ball.lean","source_sha256":"81913268d8f7f5d2e3ee5097edf683fee537074f09480b15fbe3d4cce2f1d369","start_column":1,"start_line":24},"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Ball.lean","source_signature":{"end_byte":940,"extraction_profile":"PARSED_COMMAND_TOP_LEVEL_DECLARATION_HEADER_V1","sha256":"1147496eebcfc5d5c278e699b2b7dce593b0aac44eb32a8acbbbe7e59be546ad","source_command_range":{"end_column":52,"end_line":33,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Ball.lean","source_sha256":"81913268d8f7f5d2e3ee5097edf683fee537074f09480b15fbe3d4cce2f1d369","start_column":1,"start_line":24},"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Ball.lean","source_sha256":"81913268d8f7f5d2e3ee5097edf683fee537074f09480b15fbe3d4cce2f1d369","start_byte":712,"text":"theorem existsUnique_affineExtension_ball\n    {c : E} {d : F} {r R : ℝ} (f : ball c r ≃ᵢ ball d R)\n    (hr : 0 < r) (hR : 0 < R) :\n    ∃! A : E ≃ᵃⁱ[ℝ] F,\n      ∀ x : ball c r, A (x : E) = ((f x : ball d R) : F)"},"stable_card_id":"d53fe8e4dd14b54afcbe429368e93867012aae1050cc51eff076031144a34aed"},{"assumptions":"Real normed vector spaces; f is an isometry equivalence between closed balls with the same radius r > 0.","card_ref":{"record_type":"LFH_DECLARATION_CARD","revision":1,"sha256":"17f27f1cddaac7817c4669469925411815ac3e0475b586fd47b2e04f5d14e0d3","uid":"lfh:lfh-declaration-card:sha256:92f93cdb5b6da9a1e1a1837b4e8470509909095a54e82328e641ff0b3c842995"},"citations":[{"label":"MathlibAnnex.IsometryEquiv.map_center_of_reflectionInvariant","target_card_ref":{"record_type":"LFH_DECLARATION_CARD","revision":1,"sha256":"a97a1976b14207fc263907ee8fef3f37bda326fe215a2ac5516026cc01017806","uid":"lfh:lfh-declaration-card:sha256:078cf36ae52c92c7dc2f32abc038c608c14c167df35f6f1db0514b019f459281"},"text":"The proof or construction of Affine extension from equal-radius closed balls uses the project declaration “Center transport under bounded point-reflection symmetry” at the indicated step."},{"label":"_private.MathlibAnnex.Analysis.Normed.Affine.IsometryExtension.Ball.0.MathlibAnnex.IsometryEquiv.pointReflection_mapsTo_closedBall","source_locator":{"end_column":11,"end_line":39,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Ball.lean","source_sha256":"81913268d8f7f5d2e3ee5097edf683fee537074f09480b15fbe3d4cce2f1d369","start_column":1,"start_line":35},"text":"The proof or construction of Affine extension from equal-radius closed balls uses the project declaration “Closed balls are invariant under center reflection” at the indicated step."},{"label":"MathlibAnnex.IsometryEquiv.existsUnique_affineExtension_of_convex","target_card_ref":{"record_type":"LFH_DECLARATION_CARD","revision":1,"sha256":"7dc5136edabd6e33db77b7a9f9a2122488e5b12c078056cb01de6e0ab95ce426","uid":"lfh:lfh-declaration-card:sha256:45c4db7ff375613be920f48ef4a7427063daaa3084fa98c5dd90c595522e7428"},"text":"The proof or construction of Affine extension from equal-radius closed balls uses the project declaration “Convex-set extension via ambient interiors” at the indicated step."}],"conclusion":"f extends uniquely to an ambient real affine isometry equivalence.","definition_explanation":null,"display_title":"Affine extension from equal-radius closed balls","name":"MathlibAnnex.IsometryEquiv.existsUnique_affineExtension_closedBall","natural_language_statement":"Let f be a surjective isometry between the closed balls closedBall(c,r) and closedBall(d,r) of the same radius r>0 in real normed spaces. Then f has a unique ambient real affine isometry-equivalence extension agreeing with f on the whole source closed ball.","one_sentence_role":"Extends an isometry of closed balls by first recovering the centers and then passing through the convex-interior extension theorem.","proof_steps":[{"step_id":"step-1","text":"Regard c and d as points of their respective closed balls."},{"step_id":"step-2","text":"Use boundedness and point-reflection invariance of closed balls to prove that f maps the source center to the target center."},{"step_id":"step-3","text":"For each source point, rewrite interior membership as a strict distance-to-center inequality and transport that distance through f using the center identity."},{"step_id":"step-4","text":"Show that the source closed ball has nonempty ambient interior because r is positive."},{"step_id":"step-5","text":"Apply the convex-set extension theorem to the closed ball, the nonempty interior, and the proved interior-membership equivalence."}],"proof_summary":"The center-transport theorem shows that f sends c to d because both closed balls are bounded and invariant under reflection about their centers. Distance preservation then identifies membership in the two ambient interiors, which are the corresponding open balls. The convex extension theorem completes the proof.","source_locator":{"end_column":42,"end_line":78,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Ball.lean","source_sha256":"81913268d8f7f5d2e3ee5097edf683fee537074f09480b15fbe3d4cce2f1d369","start_column":1,"start_line":41},"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Ball.lean","source_signature":{"end_byte":1735,"extraction_profile":"PARSED_COMMAND_TOP_LEVEL_DECLARATION_HEADER_V1","sha256":"16f64003b8ad024edaaecebead56b269fe82f5468d9fc6b8a517072dbdc89ceb","source_command_range":{"end_column":42,"end_line":78,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Ball.lean","source_sha256":"81913268d8f7f5d2e3ee5097edf683fee537074f09480b15fbe3d4cce2f1d369","start_column":1,"start_line":41},"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Ball.lean","source_sha256":"81913268d8f7f5d2e3ee5097edf683fee537074f09480b15fbe3d4cce2f1d369","start_byte":1484,"text":"theorem existsUnique_affineExtension_closedBall\n    {c : E} {d : F} {r : ℝ} (f : closedBall c r ≃ᵢ closedBall d r)\n    (hr : 0 < r) :\n    ∃! A : E ≃ᵃⁱ[ℝ] F,\n      ∀ x : closedBall c r,\n        A (x : E) = ((f x : closedBall d r) : F)"},"stable_card_id":"92f93cdb5b6da9a1e1a1837b4e8470509909095a54e82328e641ff0b3c842995"},{"assumptions":"Real normed vector spaces; the source s is convex with nonempty interior; f is an isometry equivalence s to t; x is interior to s exactly when f(x) is interior to t.","card_ref":{"record_type":"LFH_DECLARATION_CARD","revision":1,"sha256":"7dc5136edabd6e33db77b7a9f9a2122488e5b12c078056cb01de6e0ab95ce426","uid":"lfh:lfh-declaration-card:sha256:45c4db7ff375613be920f48ef4a7427063daaa3084fa98c5dd90c595522e7428"},"citations":[{"label":"MathlibAnnex.IsometryEquiv.existsUnique_affineExtension","target_card_ref":{"record_type":"LFH_DECLARATION_CARD","revision":1,"sha256":"ffd0c21c4c389feaa7abe9e5ca64bc9546fb5736d9768da63c049de11648ba84","uid":"lfh:lfh-declaration-card:sha256:30b5b915102151510645ff9e55aef46c99b259d2cb09540049f2b73b0eb07373"},"text":"The proof or construction of Convex-set extension via ambient interiors uses the project declaration “Mankiewicz extension on open connected domains” at the indicated step."},{"label":"_private.MathlibAnnex.Analysis.Normed.Affine.IsometryExtension.Convex.0.MathlibAnnex.IsometryEquiv.interiorRestriction","source_locator":{"end_column":53,"end_line":49,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Convex.lean","source_sha256":"f6cab253fd6b2054d08c87bab759859c8d2bf89a27a6bb4ae14eff4801601f4c","start_column":1,"start_line":24},"text":"The proof or construction of Convex-set extension via ambient interiors uses the project declaration “Restriction of an isometry to ambient interiors” at the indicated step."}],"conclusion":"f extends uniquely to an ambient real affine isometry equivalence. No additional convexity assumption on t is stated.","definition_explanation":null,"display_title":"Convex-set extension via ambient interiors","name":"MathlibAnnex.IsometryEquiv.existsUnique_affineExtension_of_convex","natural_language_statement":"Let f be a surjective isometry from a convex subset s of a real normed space onto a subset t. Assume the ambient interior of s is nonempty and, for every x in s, x lies in interior(s) if and only if f(x) lies in interior(t). Then f extends uniquely to an ambient real affine isometry equivalence. No separate convexity assumption on t is required by this statement.","one_sentence_role":"Transfers the open-connected extension theorem to convex sets whose ambient interiors are matched exactly by the isometry.","proof_steps":[{"step_id":"step-1","text":"Construct an isometry equivalence between interior(s) and interior(t) from the bidirectional interior-membership hypothesis."},{"step_id":"step-2","text":"Apply the open-connected extension theorem to the two open interiors, using connectedness of the source interior obtained from convexity of s."},{"step_id":"step-3","text":"Identify the subtype of points of s lying in interior(s) and prove it is dense in s via the convex closure-of-interior theorem."},{"step_id":"step-4","text":"Use continuity of the affine extension and of f to extend their equality from the dense interior subtype to every point of s."},{"step_id":"step-5","text":"For uniqueness, restrict any competing ambient affine isometry to the source interior and invoke the uniqueness clause from the open-connected theorem."}],"proof_summary":"Restrict f to an isometry between the two ambient interiors. Convexity and nonempty interior make the source interior connected, so the open-connected theorem yields an affine extension on the interiors. The interior of a convex set with nonempty interior is dense in the set; continuity therefore extends agreement to all of s. Uniqueness is inherited by restriction to the interior.","source_locator":{"end_column":10,"end_line":95,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Convex.lean","source_sha256":"f6cab253fd6b2054d08c87bab759859c8d2bf89a27a6bb4ae14eff4801601f4c","start_column":1,"start_line":51},"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Convex.lean","source_signature":{"end_byte":2246,"extraction_profile":"PARSED_COMMAND_TOP_LEVEL_DECLARATION_HEADER_V1","sha256":"2285ef6e2982a0c593448cf1b20e65f6b3bbfaebdb960c90657435910b0b39de","source_command_range":{"end_column":10,"end_line":95,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Convex.lean","source_sha256":"f6cab253fd6b2054d08c87bab759859c8d2bf89a27a6bb4ae14eff4801601f4c","start_column":1,"start_line":51},"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Convex.lean","source_sha256":"f6cab253fd6b2054d08c87bab759859c8d2bf89a27a6bb4ae14eff4801601f4c","start_byte":1932,"text":"theorem existsUnique_affineExtension_of_convex\n    {s : Set E} {t : Set F} (f : s ≃ᵢ t)\n    (hs : Convex ℝ s) (hsint : (interior s).Nonempty)\n    (hfi : ∀ x : s,\n      (x : E) ∈ interior s ↔ ((f x : t) : F) ∈ interior t) :\n    ∃! A : E ≃ᵃⁱ[ℝ] F,\n      ∀ x : s, A (x : E) = ((f x : t) : F)"},"stable_card_id":"45c4db7ff375613be920f48ef4a7427063daaa3084fa98c5dd90c595522e7428"},{"assumptions":"Real normed vector spaces; f is an isometry equivalence s to t; R > 0; radius-R balls about c and f(c) are contained in s and t.","card_ref":{"record_type":"LFH_DECLARATION_CARD","revision":1,"sha256":"57980774b7bb9d703dad9b73dc930e03203d182af996346c7408eb373b1e0322","uid":"lfh:lfh-declaration-card:sha256:c2f8d7f019c50fb6f9c50efe6fa6ba6cb03392611bc794c16ab04a68a88630e3"},"citations":[{"label":"_private.MathlibAnnex.Analysis.Normed.Affine.IsometryExtension.Local.0.MathlibAnnex.IsometryEquiv.midpointLinearIsometry","source_locator":{"end_column":25,"end_line":573,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_column":1,"start_line":568},"text":"The proof or construction of Local affine-isometry chart on a smaller ball uses the project declaration “Linear isometry induced by midpoint and norm preservation” at the indicated step."},{"label":"_private.MathlibAnnex.Analysis.Normed.Affine.IsometryExtension.Local.0.MathlibAnnex.IsometryEquiv.radialMap_midpoint_of_norm_lt","source_locator":{"end_column":9,"end_line":693,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_column":1,"start_line":632},"text":"The proof or construction of Local affine-isometry chart on a smaller ball uses the project declaration “Local midpoint preservation of the radial map” at the indicated step."},{"label":"_private.MathlibAnnex.Analysis.Normed.Affine.IsometryExtension.Local.0.MathlibAnnex.IsometryEquiv.map_midpoint_of_local_of_smul_pos","source_locator":{"end_column":67,"end_line":549,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_column":1,"start_line":519},"text":"The proof or construction of Local affine-isometry chart on a smaller ball uses the project declaration “Positive homogeneity globalizes local midpoint preservation” at the indicated step."},{"label":"_private.MathlibAnnex.Analysis.Normed.Affine.IsometryExtension.Local.0.MathlibAnnex.IsometryEquiv.radialMap_smul_of_pos","source_locator":{"end_column":7,"end_line":452,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_column":1,"start_line":430},"text":"The proof or construction of Local affine-isometry chart on a smaller ball uses the project declaration “Positive homogeneity of the radial map” at the indicated step."},{"label":"_private.MathlibAnnex.Analysis.Normed.Affine.IsometryExtension.Local.0.MathlibAnnex.IsometryEquiv.radialMap","source_locator":{"end_column":84,"end_line":409,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_column":1,"start_line":402},"text":"The proof or construction of Local affine-isometry chart on a smaller ball uses the project declaration “Radial extension map” at the indicated step."},{"label":"_private.MathlibAnnex.Analysis.Normed.Affine.IsometryExtension.Local.0.MathlibAnnex.IsometryEquiv.radialMap_eq_sub_of_norm_lt","source_locator":{"end_column":9,"end_line":630,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_column":1,"start_line":578},"text":"The proof or construction of Local affine-isometry chart on a smaller ball uses the project declaration “The radial map agrees locally with translated f” at the indicated step."},{"label":"_private.MathlibAnnex.Analysis.Normed.Affine.IsometryExtension.Local.0.MathlibAnnex.IsometryEquiv.norm_radialMap","source_locator":{"end_column":13,"end_line":517,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_column":1,"start_line":501},"text":"The proof or construction of Local affine-isometry chart on a smaller ball uses the project declaration “The radial map preserves norms” at the indicated step."}],"conclusion":"An ambient affine isometry equivalence agrees with f on the ball about c of radius R/8.","definition_explanation":null,"display_title":"Local affine-isometry chart on a smaller ball","name":"MathlibAnnex.IsometryEquiv.exists_affineExtension_eqOn_ball","natural_language_statement":"Suppose a set isometry f:s≃ᵢt is defined on subsets containing the ambient balls ball(c,R) and ball(f(c),R), with R>0. Then there is an ambient real affine isometry equivalence A that agrees with f at every source point lying in the smaller ball ball(c,R/8).","one_sentence_role":"Constructs the local ambient affine chart used in the gluing proof for open connected domains.","proof_steps":[{"step_id":"step-1","text":"Set the working radius to r=R/8 and construct the radial map T from f."},{"step_id":"step-2","text":"Establish T(0)=0, preservation of norms, positive homogeneity, and local midpoint preservation; rescaling upgrades midpoint preservation to all vectors."},{"step_id":"step-3","text":"Package T first as an additive homomorphism and then as a real linear isometry."},{"step_id":"step-4","text":"Show the range of the linear isometry contains ball(0,r) by pulling small target vectors back through f and applying the local radial formula."},{"step_id":"step-5","text":"A submodule with nonempty interior is the whole space, so the linear isometry is surjective and becomes a linear isometry equivalence."},{"step_id":"step-6","text":"Translate the linear equivalence from c to f(c), and use the local radial formula once more to prove agreement with f on ball(c,R/8)."}],"proof_summary":"A radial extension centered at c is built from values of f on the sphere of radius R/8. Local midpoint preservation and positive homogeneity make the radial map globally midpoint-preserving; norm preservation then turns it into a real linear isometry. Its range contains an open ball, so it is surjective. Translating this linear equivalence produces the required affine chart, and the local radial formula proves agreement with f.","source_locator":{"end_column":7,"end_line":786,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_column":1,"start_line":695},"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_signature":{"end_byte":29470,"extraction_profile":"PARSED_COMMAND_TOP_LEVEL_DECLARATION_HEADER_V1","sha256":"b59a4f9d6b1f1b1cac0bb748eb24d5c8ee702cce7674e19be0b4182356798fdb","source_command_range":{"end_column":7,"end_line":786,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_column":1,"start_line":695},"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_byte":29158,"text":"theorem exists_affineExtension_eqOn_ball\n    {s : Set E} {t : Set F} (f : s ≃ᵢ t) (c : s) {R : ℝ} (hR : 0 < R)\n    (hsball : ball (c : E) R ⊆ s)\n    (htball : ball ((f c : t) : F) R ⊆ t) :\n    ∃ A : E ≃ᵃⁱ[ℝ] F, ∀ x : s,\n      (x : E) ∈ ball (c : E) (R / 8) → A (x : E) = ((f x : t) : F)"},"stable_card_id":"c2f8d7f019c50fb6f9c50efe6fa6ba6cb03392611bc794c16ab04a68a88630e3"},{"assumptions":"Real normed affine spaces; source and target sets contain their centers and are invariant under reflection through those centers; the source is bounded; f is an isometry equivalence.","card_ref":{"record_type":"LFH_DECLARATION_CARD","revision":1,"sha256":"a97a1976b14207fc263907ee8fef3f37bda326fe215a2ac5516026cc01017806","uid":"lfh:lfh-declaration-card:sha256:078cf36ae52c92c7dc2f32abc038c608c14c167df35f6f1db0514b019f459281"},"citations":[{"label":"_private.MathlibAnnex.Analysis.Normed.Affine.Reflection.0.MathlibAnnex.IsometryEquiv.fix_center_of_bounded_pointReflection_invariant","source_locator":{"end_column":43,"end_line":72,"source_path":"MathlibAnnex/Analysis/Normed/Affine/Reflection.lean","source_sha256":"f289f13a3b7459b738604ee35278417298da58619c6daa14e3491944fc851982","start_column":1,"start_line":38},"text":"The proof or construction of Center transport under bounded point-reflection symmetry uses the project declaration “Every self-isometry fixes the center of a bounded reflection-invariant set” at the indicated step."},{"label":"_private.MathlibAnnex.Analysis.Normed.Affine.Reflection.0.MathlibAnnex.IsometryEquiv.restrictedPointReflection","source_locator":{"end_column":53,"end_line":36,"source_path":"MathlibAnnex/Analysis/Normed/Affine/Reflection.lean","source_sha256":"f289f13a3b7459b738604ee35278417298da58619c6daa14e3491944fc851982","start_column":1,"start_line":23},"text":"The proof or construction of Center transport under bounded point-reflection symmetry uses the project declaration “Point reflection restricted to an invariant subset” at the indicated step."}],"conclusion":"f maps the source reflection center to the target reflection center.","definition_explanation":null,"display_title":"Center transport under bounded point-reflection symmetry","name":"MathlibAnnex.IsometryEquiv.map_center_of_reflectionInvariant","natural_language_statement":"Let f:s≃ᵢt be an isometry equivalence between subsets of real normed affine spaces. Suppose c∈s and d∈t, the source set s is bounded, and s and t are invariant under point reflection about c and d respectively. Then f(c)=d as subtype points.","one_sentence_role":"Identifies the distinguished centers of two bounded reflection-invariant subsets under an isometry equivalence.","proof_steps":[{"step_id":"step-1","text":"Restrict target point reflection about d to an isometry equivalence of t."},{"step_id":"step-2","text":"Conjugate this restricted reflection by f to form a self-isometry g of s."},{"step_id":"step-3","text":"Apply the bounded reflection-center fixed-point lemma to g at c."},{"step_id":"step-4","text":"Transport the resulting equality through f to show that restricted reflection fixes f(c)."},{"step_id":"step-5","text":"Forget the subtype and apply the characterization of fixed points of point reflection to conclude f(c)=d."}],"proof_summary":"Conjugate the target reflection about d by f to obtain a self-isometry g of s. Every self-isometry of the bounded source reflection space fixes c. Translating that fixed-point equation back through f shows that f(c) is fixed by reflection about d, hence equals d.","source_locator":{"end_column":44,"end_line":91,"source_path":"MathlibAnnex/Analysis/Normed/Affine/Reflection.lean","source_sha256":"f289f13a3b7459b738604ee35278417298da58619c6daa14e3491944fc851982","start_column":1,"start_line":74},"source_path":"MathlibAnnex/Analysis/Normed/Affine/Reflection.lean","source_signature":{"end_byte":3494,"extraction_profile":"PARSED_COMMAND_TOP_LEVEL_DECLARATION_HEADER_V1","sha256":"3b3145eab7c31e36860891735f2e6254540af0d8716168b7b1d7379ca2beac9a","source_command_range":{"end_column":44,"end_line":91,"source_path":"MathlibAnnex/Analysis/Normed/Affine/Reflection.lean","source_sha256":"f289f13a3b7459b738604ee35278417298da58619c6daa14e3491944fc851982","start_column":1,"start_line":74},"source_path":"MathlibAnnex/Analysis/Normed/Affine/Reflection.lean","source_sha256":"f289f13a3b7459b738604ee35278417298da58619c6daa14e3491944fc851982","start_byte":3199,"text":"theorem map_center_of_reflectionInvariant\n    {s : Set P} {t : Set Q} {c : P} {d : Q} (f : s ≃ᵢ t)\n    (hc : c ∈ s) (hd : d ∈ t) (hs : IsBounded s)\n    (hsreflect : MapsTo (pointReflection ℝ c) s s)\n    (htreflect : MapsTo (pointReflection ℝ d) t t) :\n    f ⟨c, hc⟩ = ⟨d, hd⟩"},"stable_card_id":"078cf36ae52c92c7dc2f32abc038c608c14c167df35f6f1db0514b019f459281"},{"assumptions":"Real normed vector spaces; radius-R balls about c and f(c) lie in the source and target; R > 0; x,y lie in the radius-R/4 source ball; a lies in [0,1].","card_ref":{"record_type":"LFH_DECLARATION_CARD","revision":1,"sha256":"4efc62d8a56c8dfa3372c385987153dc2be7eb7c3e7d7266df28cec985605db0","uid":"lfh:lfh-declaration-card:sha256:0cfc0bfe5c6e6e9641207ba94e1890139d7e6a138ab41f3ea638d101cd0bfb04"},"citations":[{"label":"_private.MathlibAnnex.Analysis.Normed.Affine.IsometryExtension.Local.0.MathlibAnnex.IsometryEquiv.map_lineMap_of_map_midpoint","source_locator":{"end_column":25,"end_line":351,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_column":1,"start_line":194},"text":"The proof or construction of Affine-segment preservation on a quarter ball uses the project declaration “Continuous midpoint maps preserve affine segments” at the indicated step."},{"label":"MathlibAnnex.IsometryEquiv.map_midpoint_of_mem_ball","target_card_ref":{"record_type":"LFH_DECLARATION_CARD","revision":1,"sha256":"81c408d08caad98e784a6f6c6f5aaacbe2736834dee2eb84cce9af3ab14ccf02","uid":"lfh:lfh-declaration-card:sha256:d802849b06652256f387426dabbf331ecbd04b5b96f2c15ca4bdaffbf30d6a87"},"text":"The proof or construction of Affine-segment preservation on a quarter ball uses the project declaration “Midpoint preservation on a quarter ball” at the indicated step."}],"conclusion":"f preserves the affine combination (1-a)x + ay, with the exact subtype membership witnesses supplied in Lean.","definition_explanation":null,"display_title":"Affine-segment preservation on a quarter ball","name":"MathlibAnnex.IsometryEquiv.map_lineMap_of_mem_ball","natural_language_statement":"Under the same ambient-ball hypotheses as the local midpoint theorem, if x and y lie in ball(c,R/4), then for every a in [0,1], f sends the affine point lineMap(x,y,a) to lineMap(f(x),f(y),a).","one_sentence_role":"Upgrades local midpoint preservation to preservation of every affine segment parameter inside the quarter-ball chart.","proof_steps":[{"step_id":"step-1","text":"Define the inclusion of the quarter ball into s and the ambient-valued restriction g of f."},{"step_id":"step-2","text":"Use continuity of the inclusion and of the isometry to prove continuity of g."},{"step_id":"step-3","text":"Apply the quarter-ball midpoint theorem to obtain midpoint preservation for every pair of points in the restricted ball."},{"step_id":"step-4","text":"Invoke the continuous convex-set lemma that upgrades midpoint preservation to affine-line-map preservation on [0,1]."},{"step_id":"step-5","text":"Unfold the local definitions to obtain the stated equality for f."}],"proof_summary":"Restrict the domain to the convex quarter ball and view f there as a continuous map into the ambient target space. The preceding local midpoint theorem supplies midpoint preservation on that convex set. A general midpoint-to-line-map lemma then gives preservation of every segment parameter.","source_locator":{"end_column":36,"end_line":386,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_column":1,"start_line":353},"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_signature":{"end_byte":15338,"extraction_profile":"PARSED_COMMAND_TOP_LEVEL_DECLARATION_HEADER_V1","sha256":"957f6f46d627ba94d831620e28a0082444ee749f8adfe7119b2dfeed3e1714e4","source_command_range":{"end_column":36,"end_line":386,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_column":1,"start_line":353},"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_byte":14758,"text":"theorem map_lineMap_of_mem_ball\n    {s : Set E} {t : Set F} (f : s ≃ᵢ t) (c : s) {R : ℝ} (hR : 0 < R)\n    (hsball : ball (c : E) R ⊆ s)\n    (htball : ball ((f c : t) : F) R ⊆ t)\n    (x y : s) (hx : (x : E) ∈ ball (c : E) (R / 4))\n    (hy : (y : E) ∈ ball (c : E) (R / 4))\n    (a : ℝ) (ha : a ∈ Icc (0 : ℝ) 1) :\n    ((f ⟨AffineMap.lineMap (x : E) (y : E) a,\n          hsball (ball_subset_ball (by linarith : R / 4 ≤ R)\n            ((convex_ball (c : E) (R / 4)).lineMap_mem hx hy ha))⟩ : t) : F) =\n      AffineMap.lineMap ((f x : t) : F) ((f y : t) : F) a"},"stable_card_id":"0cfc0bfe5c6e6e9641207ba94e1890139d7e6a138ab41f3ea638d101cd0bfb04"},{"assumptions":"Real normed vector spaces; radius-R balls about c and f(c) lie in the source and target; R > 0; x,y lie in the radius-R/4 source ball.","card_ref":{"record_type":"LFH_DECLARATION_CARD","revision":1,"sha256":"81c408d08caad98e784a6f6c6f5aaacbe2736834dee2eb84cce9af3ab14ccf02","uid":"lfh:lfh-declaration-card:sha256:d802849b06652256f387426dabbf331ecbd04b5b96f2c15ca4bdaffbf30d6a87"},"citations":[{"label":"MathlibAnnex.IsometryEquiv.map_midpoint_of_symmetricLens_subset","target_card_ref":{"record_type":"LFH_DECLARATION_CARD","revision":1,"sha256":"b8c28fe867604d317c6bb50248b4684deb39c56fb6fcd964b2329c93e42d14b6","uid":"lfh:lfh-declaration-card:sha256:e55f926ea1e3391b9a6426100e98fe7349c95254cdc2437a77d973137bea95c8"},"text":"The proof or construction of Midpoint preservation on a quarter ball uses the project declaration “Midpoint preservation under lens containment” at the indicated step."},{"label":"MathlibAnnex.symmetricLens","target_card_ref":{"record_type":"LFH_DECLARATION_CARD","revision":1,"sha256":"8ec9c339586d12c353b6eeece0fed3c279aef6a9933e7bea978a68d85a9c2ca7","uid":"lfh:lfh-declaration-card:sha256:b6295e62b69808246ec83346365d7b0e00269f627c02dcd6ce3d253fb920949a"},"text":"The proof or construction of Midpoint preservation on a quarter ball uses the project declaration “Symmetric lens” at the indicated step."}],"conclusion":"f sends the midpoint of x and y to the midpoint of f(x) and f(y).","definition_explanation":null,"display_title":"Midpoint preservation on a quarter ball","name":"MathlibAnnex.IsometryEquiv.map_midpoint_of_mem_ball","natural_language_statement":"Let f:s≃ᵢt and let c∈s. Assume R>0, ball(c,R)⊆s, and ball(f(c),R)⊆t. If x,y∈s both lie in ball(c,R/4), then f sends their midpoint to the midpoint of f(x) and f(y).","one_sentence_role":"Obtains a uniform local midpoint law from symmetric-lens center rigidity.","proof_steps":[{"step_id":"step-1","text":"Use the triangle inequality through c to bound dist(x,y) by less than R/2."},{"step_id":"step-2","text":"Deduce the half-distance condition needed for a symmetric lens of radius R/2."},{"step_id":"step-3","text":"Show every point of the source lens lies in ball(c,R) by combining its distance to x with the R/4 bound on x."},{"step_id":"step-4","text":"Transport the R/4 bound from x to f(x), then prove the analogous containment of the target lens in ball(f(c),R)."},{"step_id":"step-5","text":"Apply midpoint preservation for a set isometry whose corresponding lenses remain inside the source and target sets."}],"proof_summary":"The two endpoints are less than R/2 apart. Their symmetric lens of radius R/2 lies in the source R-ball, and the corresponding target lens lies in the target R-ball. The subset-local lens theorem therefore applies.","source_locator":{"end_column":79,"end_line":192,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_column":1,"start_line":137},"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_signature":{"end_byte":6691,"extraction_profile":"PARSED_COMMAND_TOP_LEVEL_DECLARATION_HEADER_V1","sha256":"cb472dd816bb08047ca8db0e178d2840a9408e2e3fae224b33828dec78d41605","source_command_range":{"end_column":79,"end_line":192,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_column":1,"start_line":137},"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_byte":5912,"text":"theorem map_midpoint_of_mem_ball\n    {s : Set E} {t : Set F} (f : s ≃ᵢ t) (c : s) {R : ℝ} (hR : 0 < R)\n    (hsball : ball (c : E) R ⊆ s)\n    (htball : ball ((f c : t) : F) R ⊆ t)\n    (x y : s) (hx : (x : E) ∈ ball (c : E) (R / 4))\n    (hy : (y : E) ∈ ball (c : E) (R / 4)) :\n    ((f ⟨midpoint ℝ (x : E) (y : E),\n          hsball (by\n            rw [mem_ball]\n            calc\n              dist (midpoint ℝ (x : E) (y : E)) (c : E) ≤\n                  (dist (x : E) (c : E) + dist (y : E) (c : E)) / 2 := by\n                    simpa [dist_comm] using\n                      dist_midpoint_midpoint_le (x : E) (y : E) (c : E) (c : E)\n              _ < R := by rw [mem_ball] at hx hy; linarith)⟩ : t) : F) =\n      midpoint ℝ ((f x : t) : F) ((f y : t) : F)"},"stable_card_id":"d802849b06652256f387426dabbf331ecbd04b5b96f2c15ca4bdaffbf30d6a87"},{"assumptions":"Real normed vector spaces; an isometry equivalence between symmetric lenses of the same radius r; both half-distances between the respective foci are at most r.","card_ref":{"record_type":"LFH_DECLARATION_CARD","revision":1,"sha256":"21daf8652777c81a08e94a4a7a3eff63eef5bfd09483b15da2b62d60227b0e87","uid":"lfh:lfh-declaration-card:sha256:4a485c9aabdc6f37ae97b47bfba6021a6bedf685894a44120f0290571e459c36"},"citations":[{"label":"_private.MathlibAnnex.Analysis.Normed.Affine.IsometryExtension.Local.0.MathlibAnnex.symmetricLens_reflectionInvariant","source_locator":{"end_column":19,"end_line":62,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_column":1,"start_line":40},"text":"The proof or construction of Midpoint preservation on a symmetric lens uses the project declaration “A symmetric lens is invariant under reflection about its midpoint” at the indicated step."},{"label":"MathlibAnnex.IsometryEquiv.map_center_of_reflectionInvariant","target_card_ref":{"record_type":"LFH_DECLARATION_CARD","revision":1,"sha256":"a97a1976b14207fc263907ee8fef3f37bda326fe215a2ac5516026cc01017806","uid":"lfh:lfh-declaration-card:sha256:078cf36ae52c92c7dc2f32abc038c608c14c167df35f6f1db0514b019f459281"},"text":"The proof or construction of Midpoint preservation on a symmetric lens uses the project declaration “Center transport under bounded point-reflection symmetry” at the indicated step."},{"label":"MathlibAnnex.symmetricLens","target_card_ref":{"record_type":"LFH_DECLARATION_CARD","revision":1,"sha256":"8ec9c339586d12c353b6eeece0fed3c279aef6a9933e7bea978a68d85a9c2ca7","uid":"lfh:lfh-declaration-card:sha256:b6295e62b69808246ec83346365d7b0e00269f627c02dcd6ce3d253fb920949a"},"text":"The proof or construction of Midpoint preservation on a symmetric lens uses the project declaration “Symmetric lens” at the indicated step."},{"label":"_private.MathlibAnnex.Analysis.Normed.Affine.IsometryExtension.Local.0.MathlibAnnex.midpoint_mem_symmetricLens","source_locator":{"end_column":70,"end_line":38,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_column":1,"start_line":34},"text":"The proof or construction of Midpoint preservation on a symmetric lens uses the project declaration “The midpoint belongs to a sufficiently large symmetric lens” at the indicated step."}],"conclusion":"The isometry maps the midpoint of the source foci to the midpoint of the target foci.","definition_explanation":null,"display_title":"Midpoint preservation on a symmetric lens","name":"MathlibAnnex.IsometryEquiv.map_midpoint_of_symmetricLens","natural_language_statement":"Assume the midpoint of x,y belongs to symmetricLens(x,y,r), and likewise the midpoint of x′,y′ belongs to symmetricLens(x′,y′,r), as expressed by the two half-distance inequalities. Any isometry equivalence between these lenses maps midpoint(x,y) to midpoint(x′,y′).","one_sentence_role":"Turns bounded reflection symmetry of equal-radius lenses into midpoint preservation.","proof_steps":[{"step_id":"step-1","text":"Use the half-distance inequalities to place both midpoints in their respective lenses."},{"step_id":"step-2","text":"Bound the source lens by one of its defining closed balls."},{"step_id":"step-3","text":"Use reflection invariance of each lens about the corresponding midpoint."},{"step_id":"step-4","text":"Apply center transport to the lens isometry and erase the subtype wrapper."}],"proof_summary":"A symmetric lens is bounded and invariant under point reflection about the midpoint of its foci. The center-transport theorem therefore sends the source midpoint to the target midpoint.","source_locator":{"end_column":31,"end_line":80,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_column":1,"start_line":66},"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_signature":{"end_byte":2911,"extraction_profile":"PARSED_COMMAND_TOP_LEVEL_DECLARATION_HEADER_V1","sha256":"c1a93ec73de0de2cab77dc0c341e5a1f6344fd0ddf572ea92b1110b642463813","source_command_range":{"end_column":31,"end_line":80,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_column":1,"start_line":66},"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_byte":2584,"text":"theorem map_midpoint_of_symmetricLens\n    {x y : E} {x' y' : F} {r : ℝ}\n    (hxy : 2⁻¹ * dist x y ≤ r) (hxy' : 2⁻¹ * dist x' y' ≤ r)\n    (f : symmetricLens x y r ≃ᵢ symmetricLens x' y' r) :\n    ((f ⟨midpoint ℝ x y, midpoint_mem_symmetricLens hxy⟩ :\n        symmetricLens x' y' r) : F) = midpoint ℝ x' y'"},"stable_card_id":"4a485c9aabdc6f37ae97b47bfba6021a6bedf685894a44120f0290571e459c36"},{"assumptions":"Real normed vector spaces; f is an isometry equivalence s to t; radius-r symmetric lenses about x,y and f(x),f(y) lie in s and t; half the distance from x to y is at most r.","card_ref":{"record_type":"LFH_DECLARATION_CARD","revision":1,"sha256":"b8c28fe867604d317c6bb50248b4684deb39c56fb6fcd964b2329c93e42d14b6","uid":"lfh:lfh-declaration-card:sha256:e55f926ea1e3391b9a6426100e98fe7349c95254cdc2437a77d973137bea95c8"},"citations":[{"label":"MathlibAnnex.IsometryEquiv.map_midpoint_of_symmetricLens","target_card_ref":{"record_type":"LFH_DECLARATION_CARD","revision":1,"sha256":"21daf8652777c81a08e94a4a7a3eff63eef5bfd09483b15da2b62d60227b0e87","uid":"lfh:lfh-declaration-card:sha256:4a485c9aabdc6f37ae97b47bfba6021a6bedf685894a44120f0290571e459c36"},"text":"The proof or construction of Midpoint preservation under lens containment uses the project declaration “Midpoint preservation on a symmetric lens” at the indicated step."},{"label":"_private.MathlibAnnex.Analysis.Normed.Affine.IsometryExtension.Local.0.MathlibAnnex.IsometryEquiv.restrictToSymmetricLens","source_locator":{"end_column":53,"end_line":118,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_column":1,"start_line":82},"text":"The proof or construction of Midpoint preservation under lens containment uses the project declaration “Restriction of a set isometry to corresponding symmetric lenses” at the indicated step."},{"label":"MathlibAnnex.symmetricLens","target_card_ref":{"record_type":"LFH_DECLARATION_CARD","revision":1,"sha256":"8ec9c339586d12c353b6eeece0fed3c279aef6a9933e7bea978a68d85a9c2ca7","uid":"lfh:lfh-declaration-card:sha256:b6295e62b69808246ec83346365d7b0e00269f627c02dcd6ce3d253fb920949a"},"text":"The proof or construction of Midpoint preservation under lens containment uses the project declaration “Symmetric lens” at the indicated step."},{"label":"_private.MathlibAnnex.Analysis.Normed.Affine.IsometryExtension.Local.0.MathlibAnnex.midpoint_mem_symmetricLens","source_locator":{"end_column":70,"end_line":38,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_column":1,"start_line":34},"text":"The proof or construction of Midpoint preservation under lens containment uses the project declaration “The midpoint belongs to a sufficiently large symmetric lens” at the indicated step."}],"conclusion":"f preserves the midpoint of x and y. The target half-distance bound follows from the isometry.","definition_explanation":null,"display_title":"Midpoint preservation under lens containment","name":"MathlibAnnex.IsometryEquiv.map_midpoint_of_symmetricLens_subset","natural_language_statement":"Let f:s≃ᵢt and x,y∈s. If the symmetric lens of radius r around x,y lies in s, the corresponding lens around f(x),f(y) lies in t, and half the distance between x and y is at most r, then f sends midpoint(x,y) to midpoint(f(x),f(y)).","one_sentence_role":"Applies the symmetric-lens midpoint theorem inside arbitrary source and target subsets.","proof_steps":[{"step_id":"step-1","text":"Construct the induced isometry equivalence between the two symmetric lenses."},{"step_id":"step-2","text":"Use f.dist_eq to identify the distance between the target foci with the distance between the source foci."},{"step_id":"step-3","text":"Transport the half-distance hypothesis to the target lens."},{"step_id":"step-4","text":"Apply midpoint preservation for an isometry between the two lenses."}],"proof_summary":"Restrict f and its inverse to the two contained lenses. Distance preservation transports the half-distance bound to the target foci. The lens-level midpoint theorem applied to this restricted isometry gives the result.","source_locator":{"end_column":53,"end_line":135,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_column":1,"start_line":120},"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_signature":{"end_byte":5433,"extraction_profile":"PARSED_COMMAND_TOP_LEVEL_DECLARATION_HEADER_V1","sha256":"216cc16af7de6004b202dd6391c656086ec4432fa041056f6e718bea48b69a30","source_command_range":{"end_column":53,"end_line":135,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_column":1,"start_line":120},"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_byte":5003,"text":"theorem map_midpoint_of_symmetricLens_subset\n    {s : Set E} {t : Set F} (f : s ≃ᵢ t) (x y : s) (r : ℝ)\n    (hhalf : 2⁻¹ * dist (x : E) (y : E) ≤ r)\n    (hs : symmetricLens (x : E) (y : E) r ⊆ s)\n    (ht : symmetricLens ((f x : t) : F) ((f y : t) : F) r ⊆ t) :\n    ((f ⟨midpoint ℝ (x : E) (y : E),\n          hs (midpoint_mem_symmetricLens hhalf)⟩ : t) : F) =\n      midpoint ℝ ((f x : t) : F) ((f y : t) : F)"},"stable_card_id":"e55f926ea1e3391b9a6426100e98fe7349c95254cdc2437a77d973137bea95c8"},{"assumptions":"A normed additive commutative group E; points x and y; a real radius r.","card_ref":{"record_type":"LFH_DECLARATION_CARD","revision":1,"sha256":"8ec9c339586d12c353b6eeece0fed3c279aef6a9933e7bea978a68d85a9c2ca7","uid":"lfh:lfh-declaration-card:sha256:b6295e62b69808246ec83346365d7b0e00269f627c02dcd6ce3d253fb920949a"},"citations":[],"conclusion":"The symmetric lens is the intersection of the closed radius-r balls centered at x and y. This definition needs no positivity assumption.","definition_explanation":"A point lies in the symmetric lens exactly when its distance from each focus is at most r. The common midpoint is the center of the point-reflection symmetry that exchanges the two foci.","display_title":"Symmetric lens","name":"MathlibAnnex.symmetricLens","natural_language_statement":"`symmetricLens x y r` is the intersection of the two closed balls of radius r centered at x and y.","one_sentence_role":"Names the reflection-symmetric bounded region used to force preservation of a midpoint.","proof_steps":[],"proof_summary":null,"source_locator":{"end_column":34,"end_line":32,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_column":1,"start_line":30},"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_signature":{"end_byte":928,"extraction_profile":"PARSED_COMMAND_TOP_LEVEL_DECLARATION_HEADER_V1","sha256":"20242e503438b61fd3e9b262cd0d1637a4e4539ba160182d31691065fd277cad","source_command_range":{"end_column":34,"end_line":32,"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_column":1,"start_line":30},"source_path":"MathlibAnnex/Analysis/Normed/Affine/IsometryExtension/Local.lean","source_sha256":"3902a71a10200e7d787caca81f0d4cd9f5655b91c7309954793958b754f6fd2e","start_byte":883,"text":"def symmetricLens (x y : E) (r : ℝ) : Set E"},"stable_card_id":"b6295e62b69808246ec83346365d7b0e00269f627c02dcd6ce3d253fb920949a"}],"catalog_current_ref":{"record_type":"LFH_CATALOG_SNAPSHOT","revision":1,"sha256":"c12d369de0cf81b7edffc9c814a9e69207e10ca0a18c8fef34b6b829ca291f51","uid":"lfh:lfh-catalog-snapshot:sha256:70de7b2343dea138d1d9fbaf5271aa779c685875f4173533273ed9746d29e307"},"current_state":{"catalog_current_selection":"OWNER_DIRECTED_PRIVATE_CURRENT_R1","formal_mathlibannex_admission":true,"formal_public_admission":false,"license_status":"PENDING_OWNER_DECISION","project_overview_owner_approved":true,"publication_state":"NOT_PUBLISHED"},"project_view_sha256":"5af5d00bb7df6f66359109cc4d4ef75215ca7ef5042928445c1e3fa4a0ff42b3","schema":"mathlibannex.publication-candidate-projection.v1","source_binding_ref":{"record_type":"LFH_SOURCE_BUILD_BINDING","revision":1,"sha256":"f749b8a9024c48af33beb39b93f88a9714b75bbdbf0402075de531591a43d078","uid":"lw:lfh-source-build-binding:sha256:ae84e64475f37c6f43c0fb49ffe48d70f8df971325b1248ed267a54728e4d28c"}}