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  "assumptions": "Let A be a nonzero unital complex C*-algebra, H a separable complex Hilbert space, and π a unital 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.Representation.IsSingletonIrreducibleModelAmongNonUnital",
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      "text": "Exact formal dependency; inspect the linked Card and exact source."
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  "name": "MathlibAnnex.Analysis.CStarAlgebra.Representation.finiteDimensional_algebra_of_singleton_amongNonUnital",
  "natural_language_statement": "Let A be a nonzero unital complex C*-algebra, H a separable complex Hilbert space, and π a unital representation of A on H that is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A. Then A is finite-dimensional over ℂ; finite dimension of A is the conclusion, not an initial hypothesis.",
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      "text": "Exact Lean statement: ∀ {A : Type u} [inst : CStarAlgebra A] [inst_1 : PartialOrder A] [StarOrderedRing A] {H : Type v} [inst_3 : NormedAddCommGroup H] [inst_4 : InnerProductSpace ℂ H] [inst_5 : CompleteSpace H] [Nontrivial A] [TopologicalSpace.SeparableSpace H] (pi : MathlibAnnex.Analysis.CStarAlgebra.Representation A H), pi.IsSingletonIrreducibleModelAmongNonUnital → FiniteDimensional ℂ A"
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      "text": "FiniteDimensional ℂ A."
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      "text": "The source converts the quantifier and applies the unital algebra-finiteness theorem. The underlying argument first makes H finite-dimensional and π injective, then embeds A into the finite-dimensional bounded-operator space on H."
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  ],
  "proof_summary": "The source converts the quantifier and applies the unital algebra-finiteness theorem. The underlying argument first makes H finite-dimensional and π injective, then embeds A into the finite-dimensional bounded-operator space on H.",
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    "text": "theorem finiteDimensional_algebra_of_singleton_amongNonUnital\n    [Nontrivial A] [TopologicalSpace.SeparableSpace H]\n    (pi : Representation A H)\n    (hsingle : IsSingletonIrreducibleModelAmongNonUnital.{u, v, u} pi) :\n    FiniteDimensional ℂ A"
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