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  "assumptions": "Let A be a nonzero unital complex C*-algebra, H a complex Hilbert space, and π a representation of A on H that is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A. Assume additionally that π is injective. No separability assumption on H is required.",
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      "label": "MathlibAnnex.Analysis.CStarAlgebra.isIrreducible_pureState_gnsStarAlgHom",
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  "name": "MathlibAnnex.Analysis.CStarAlgebra.Representation.isSimpleCStarAlgebra_of_singleton_of_injective",
  "natural_language_statement": "Let A be a nonzero unital complex C*-algebra, H a complex Hilbert space, and π a representation of A on H that is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A. Assume additionally that π is injective. No separability assumption on H is required. A is simple in the exact closed-two-sided-ideal sense IsSimpleCStarAlgebra A.",
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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] (pi : MathlibAnnex.Analysis.CStarAlgebra.Representation A H), pi.IsSingletonIrreducibleModel → Function.Injective ⇑pi → MathlibAnnex.Analysis.CStarAlgebra.IsSimpleCStarAlgebra A"
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      "text": "IsSingletonIrreducibleModel π ∧ Injective π → IsSimpleCStarAlgebra A."
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      "text": "For a closed two-sided ideal I, the source splits I=⊤ from a proper I. In the proper case an irreducible GNS representation annihilating I belongs to the unique class represented by π; the intertwiner makes π vanish on each x∈I, and injectivity forces x=0, so I=⊥."
    }
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  "proof_summary": "For a closed two-sided ideal I, the source splits I=⊤ from a proper I. In the proper case an irreducible GNS representation annihilating I belongs to the unique class represented by π; the intertwiner makes π vanish on each x∈I, and injectivity forces x=0, so I=⊥.",
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    "text": "theorem isSimpleCStarAlgebra_of_singleton_of_injective [Nontrivial A]\n    (pi : Representation A H)\n    (hsingle : IsSingletonIrreducibleModel.{u, v, u} pi)\n    (hinj : Function.Injective pi) : IsSimpleCStarAlgebra A"
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