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 ) ) )