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 → ( 𝑇𝐻 ↔ ( 𝑇 ∈ dom 𝐴 ∧ ( 𝐴𝑇 ) = 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 ∣ ( 𝐴𝑡 ) = 𝑡 } )
4 3 eleq2d ( 𝑈 ∈ NrmCVec → ( 𝑇𝐻𝑇 ∈ { 𝑡 ∈ dom 𝐴 ∣ ( 𝐴𝑡 ) = 𝑡 } ) )
5 fveq2 ( 𝑡 = 𝑇 → ( 𝐴𝑡 ) = ( 𝐴𝑇 ) )
6 id ( 𝑡 = 𝑇𝑡 = 𝑇 )
7 5 6 eqeq12d ( 𝑡 = 𝑇 → ( ( 𝐴𝑡 ) = 𝑡 ↔ ( 𝐴𝑇 ) = 𝑇 ) )
8 7 elrab ( 𝑇 ∈ { 𝑡 ∈ dom 𝐴 ∣ ( 𝐴𝑡 ) = 𝑡 } ↔ ( 𝑇 ∈ dom 𝐴 ∧ ( 𝐴𝑇 ) = 𝑇 ) )
9 4 8 bitrdi ( 𝑈 ∈ NrmCVec → ( 𝑇𝐻 ↔ ( 𝑇 ∈ dom 𝐴 ∧ ( 𝐴𝑇 ) = 𝑇 ) ) )