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Metamath Proof Explorer

Theorem ishmo

Description: The predicate "is a hermitian operator." (Contributed by NM, 26-Jan-2008) (New usage is discouraged.)

Ref Expression
Hypotheses hmoval.8 ⊢ 𝐻 = ( HmOp ‘ 𝑈 )
hmoval.9 ⊢ 𝐴 = ( 𝑈 adj 𝑈 )
Assertion ishmo ( 𝑈 ∈ NrmCVec → ( 𝑇 ∈ 𝐻 ↔ ( 𝑇 ∈ dom 𝐴 ∧ ( 𝐴 ‘ 𝑇 ) = 𝑇 ) ) )

Proof

Step Hyp Ref Expression
1 hmoval.8 ⊢ 𝐻 = ( HmOp ‘ 𝑈 )
2 hmoval.9 ⊢ 𝐴 = ( 𝑈 adj 𝑈 )
3 1 2 hmoval ⊢ ( 𝑈 ∈ NrmCVec → 𝐻 = { 𝑡 ∈ dom 𝐴 ∣ ( 𝐴 ‘ 𝑡 ) = 𝑡 } )
4 3 eleq2d ⊢ ( 𝑈 ∈ NrmCVec → ( 𝑇 ∈ 𝐻 ↔ 𝑇 ∈ { 𝑡 ∈ dom 𝐴 ∣ ( 𝐴 ‘ 𝑡 ) = 𝑡 } ) )
5 fveq2 ⊢ ( 𝑡 = 𝑇 → ( 𝐴 ‘ 𝑡 ) = ( 𝐴 ‘ 𝑇 ) )
6 id ⊢ ( 𝑡 = 𝑇 → 𝑡 = 𝑇 )
7 5 6 eqeq12d ⊢ ( 𝑡 = 𝑇 → ( ( 𝐴 ‘ 𝑡 ) = 𝑡 ↔ ( 𝐴 ‘ 𝑇 ) = 𝑇 ) )
8 7 elrab ⊢ ( 𝑇 ∈ { 𝑡 ∈ dom 𝐴 ∣ ( 𝐴 ‘ 𝑡 ) = 𝑡 } ↔ ( 𝑇 ∈ dom 𝐴 ∧ ( 𝐴 ‘ 𝑇 ) = 𝑇 ) )
9 4 8 bitrdi ⊢ ( 𝑈 ∈ NrmCVec → ( 𝑇 ∈ 𝐻 ↔ ( 𝑇 ∈ dom 𝐴 ∧ ( 𝐴 ‘ 𝑇 ) = 𝑇 ) ) )