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  "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.",
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      "label": "_private.MathlibAnnex.Analysis.Normed.Affine.Reflection.0.MathlibAnnex.IsometryEquiv.apply_center_eq_of_isBounded_of_mapsTo_pointReflection",
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      "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."
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      "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."
    }
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  "conclusion": "f maps the source reflection center to the target reflection center.",
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  "display_title": "Center transport under bounded point-reflection symmetry",
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  "name": "MathlibAnnex.IsometryEquiv.map_center_of_mapsTo_pointReflection",
  "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.",
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    "text": "theorem map_center_of_mapsTo_pointReflection\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⟩"
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