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  "assumptions": "Let A be a nonzero complex C*-algebra, not assumed unital; let H be a separable complex Hilbert space; and let π be a representation of A on H that is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A.",
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      "label": "MathlibAnnex.Analysis.CStarAlgebra.NonUnitalCStarRepresentation.exists_nonzero_projection_rankOne_map",
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  "conclusion": "For every pair x,y∈H, including zero vectors, there exists a∈A with π(a)=rankOne ℂ x y; this operator has rank at most one.",
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  "definition_equation": "∀ x y : H, ∃ a : A, π a = rankOne ℂ x y.",
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  "display_title": "Preimages for all operators of the form rankOne",
  "exact_elaborated_lean_type": "∀ {A : Type u} [inst : NonUnitalCStarAlgebra A] {H : Type v} [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℂ H] [inst_3 : CompleteSpace H] [Nontrivial A] [inst_5 : PartialOrder A] [StarOrderedRing A] [TopologicalSpace.SeparableSpace H] (pi : MathlibAnnex.Analysis.CStarAlgebra.NonUnitalCStarRepresentation A H), pi.IsSingletonIrreducibleModel → ∀ (x y : H), ∃ a, pi a = ((InnerProductSpace.rankOne ℂ) x) y",
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  "name": "MathlibAnnex.Analysis.CStarAlgebra.NonUnitalCStarRepresentation.exists_preimage_rankOne_of_singleton",
  "natural_language_statement": "Let A be a nonzero complex C*-algebra, not assumed unital; let H be a separable complex Hilbert space; and let π be a representation of A on H that is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A. For every pair x,y∈H, including zero vectors, there exists a∈A with π(a)=rankOne ℂ x y; this operator has rank at most one.",
  "one_sentence_role": "Provides preimages for every operator of the form rankOne ℂ x y, the rank-at-most-one generators used to obtain all compact operators.",
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  "proof_steps": [
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      "step_id": "formal_type",
      "text": "Exact Lean statement: ∀ {A : Type u} [inst : NonUnitalCStarAlgebra A] {H : Type v} [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℂ H] [inst_3 : CompleteSpace H] [Nontrivial A] [inst_5 : PartialOrder A] [StarOrderedRing A] [TopologicalSpace.SeparableSpace H] (pi : MathlibAnnex.Analysis.CStarAlgebra.NonUnitalCStarRepresentation A H), pi.IsSingletonIrreducibleModel → ∀ (x y : H), ∃ a, pi a = ((InnerProductSpace.rankOne ℂ) x) y"
    },
    {
      "step_id": "source_equation",
      "text": "∀ x y : H, ∃ a : A, π a = rankOne ℂ x y."
    },
    {
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      "text": "The source obtains one nonzero rank-one projection in π(A), uses the unique-class hypothesis for π together with injectivity, and invokes the rank-one-preimage theorem to move from that projection to every x and y, including zero vectors."
    }
  ],
  "proof_summary": "The source obtains one nonzero rank-one projection in π(A), uses the unique-class hypothesis for π together with injectivity, and invokes the rank-one-preimage theorem to move from that projection to every x and y, including zero vectors.",
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    "text": "theorem exists_preimage_rankOne_of_singleton [Nontrivial A]\n    [PartialOrder A] [StarOrderedRing A]\n    [TopologicalSpace.SeparableSpace H]\n    (pi : NonUnitalCStarRepresentation A H)\n    (hsingle : IsSingletonIrreducibleModel.{u, v, u} pi) :\n    ∀ x y : H, ∃ a : A, pi a = InnerProductSpace.rankOne ℂ x y"
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