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Metamath Proof Explorer

Theorem unopadj

Description: The inverse (converse) of a unitary operator is its adjoint. Equation 2 of AkhiezerGlazman p. 72. (Contributed by NM, 22-Jan-2006) (New usage is discouraged.)

Ref Expression
Assertion unopadj ( ( 𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝑇 ‘ 𝐴 ) ·ih 𝐵 ) = ( 𝐴 ·ih ( ◡ 𝑇 ‘ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 unopf1o ⊢ ( 𝑇 ∈ UniOp → 𝑇 : ℋ –1-1-onto→ ℋ )
2 f1ocnvfv2 ⊢ ( ( 𝑇 : ℋ –1-1-onto→ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( ◡ 𝑇 ‘ 𝐵 ) ) = 𝐵 )
3 1 2 sylan ⊢ ( ( 𝑇 ∈ UniOp ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( ◡ 𝑇 ‘ 𝐵 ) ) = 𝐵 )
4 3 3adant2 ⊢ ( ( 𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( ◡ 𝑇 ‘ 𝐵 ) ) = 𝐵 )
5 4 oveq2d ⊢ ( ( 𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝑇 ‘ 𝐴 ) ·ih ( 𝑇 ‘ ( ◡ 𝑇 ‘ 𝐵 ) ) ) = ( ( 𝑇 ‘ 𝐴 ) ·ih 𝐵 ) )
6 f1ocnv ⊢ ( 𝑇 : ℋ –1-1-onto→ ℋ → ◡ 𝑇 : ℋ –1-1-onto→ ℋ )
7 f1of ⊢ ( ◡ 𝑇 : ℋ –1-1-onto→ ℋ → ◡ 𝑇 : ℋ ⟶ ℋ )
8 1 6 7 3syl ⊢ ( 𝑇 ∈ UniOp → ◡ 𝑇 : ℋ ⟶ ℋ )
9 8 ffvelcdmda ⊢ ( ( 𝑇 ∈ UniOp ∧ 𝐵 ∈ ℋ ) → ( ◡ 𝑇 ‘ 𝐵 ) ∈ ℋ )
10 9 3adant2 ⊢ ( ( 𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ◡ 𝑇 ‘ 𝐵 ) ∈ ℋ )
11 unop ⊢ ( ( 𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ ( ◡ 𝑇 ‘ 𝐵 ) ∈ ℋ ) → ( ( 𝑇 ‘ 𝐴 ) ·ih ( 𝑇 ‘ ( ◡ 𝑇 ‘ 𝐵 ) ) ) = ( 𝐴 ·ih ( ◡ 𝑇 ‘ 𝐵 ) ) )
12 10 11 syld3an3 ⊢ ( ( 𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝑇 ‘ 𝐴 ) ·ih ( 𝑇 ‘ ( ◡ 𝑇 ‘ 𝐵 ) ) ) = ( 𝐴 ·ih ( ◡ 𝑇 ‘ 𝐵 ) ) )
13 5 12 eqtr3d ⊢ ( ( 𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝑇 ‘ 𝐴 ) ·ih 𝐵 ) = ( 𝐴 ·ih ( ◡ 𝑇 ‘ 𝐵 ) ) )