Theorem hmopadj

Description: A Hermitian operator is self-adjoint. (Contributed by NM, 24-Mar-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	hmopadj	⊢ ( 𝑇 ∈ HrmOp → ( adjℎ ‘ 𝑇 ) = 𝑇 )

Proof

Step	Hyp	Ref	Expression
1		hmopf	⊢ ( 𝑇 ∈ HrmOp → 𝑇 : ℋ ⟶ ℋ )
2		hmop	⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ) → ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑦 ) )
3	2	eqcomd	⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ) → ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑦 ) = ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) )
4	3	3expib	⊢ ( 𝑇 ∈ HrmOp → ( ( 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ) → ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑦 ) = ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) ) )
5	4	ralrimivv	⊢ ( 𝑇 ∈ HrmOp → ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑦 ) = ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) )
6		adjeq	⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑦 ) = ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) ) → ( adjℎ ‘ 𝑇 ) = 𝑇 )
7	1 1 5 6	syl3anc	⊢ ( 𝑇 ∈ HrmOp → ( adjℎ ‘ 𝑇 ) = 𝑇 )