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Metamath Proof Explorer

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 )