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 ) ) )