Metamath Proof Explorer

Theorem cnvunop

Description: The inverse (converse) of a unitary operator in Hilbert space is unitary. Theorem in AkhiezerGlazman p. 72. (Contributed by NM, 22-Jan-2006) (New usage is discouraged.)

Ref Expression
Assertion cnvunop ( 𝑇 ∈ UniOp → ◡ 𝑇 ∈ UniOp )

Proof

Step Hyp Ref Expression
1 unopf1o ⊢ ( 𝑇 ∈ UniOp → 𝑇 : ℋ –1-1-onto→ ℋ )
2 f1ocnv ⊢ ( 𝑇 : ℋ –1-1-onto→ ℋ → ◡ 𝑇 : ℋ –1-1-onto→ ℋ )
3 f1ofo ⊢ ( ◡ 𝑇 : ℋ –1-1-onto→ ℋ → ◡ 𝑇 : ℋ –onto→ ℋ )
4 2 3 syl ⊢ ( 𝑇 : ℋ –1-1-onto→ ℋ → ◡ 𝑇 : ℋ –onto→ ℋ )
5 1 4 syl ⊢ ( 𝑇 ∈ UniOp → ◡ 𝑇 : ℋ –onto→ ℋ )
6 simpl ⊢ ( ( 𝑇 ∈ UniOp ∧ ( 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ) ) → 𝑇 ∈ UniOp )
7 fof ⊢ ( ◡ 𝑇 : ℋ –onto→ ℋ → ◡ 𝑇 : ℋ ⟶ ℋ )
8 5 7 syl ⊢ ( 𝑇 ∈ UniOp → ◡ 𝑇 : ℋ ⟶ ℋ )
9 8 ffvelrnda ⊢ ( ( 𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ) → ( ◡ 𝑇 ‘ 𝑥 ) ∈ ℋ )
10 9 adantrr ⊢ ( ( 𝑇 ∈ UniOp ∧ ( 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ) ) → ( ◡ 𝑇 ‘ 𝑥 ) ∈ ℋ )
11 8 ffvelrnda ⊢ ( ( 𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ ) → ( ◡ 𝑇 ‘ 𝑦 ) ∈ ℋ )
12 11 adantrl ⊢ ( ( 𝑇 ∈ UniOp ∧ ( 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ) ) → ( ◡ 𝑇 ‘ 𝑦 ) ∈ ℋ )
13 unop ⊢ ( ( 𝑇 ∈ UniOp ∧ ( ◡ 𝑇 ‘ 𝑥 ) ∈ ℋ ∧ ( ◡ 𝑇 ‘ 𝑦 ) ∈ ℋ ) → ( ( 𝑇 ‘ ( ◡ 𝑇 ‘ 𝑥 ) ) ·ih ( 𝑇 ‘ ( ◡ 𝑇 ‘ 𝑦 ) ) ) = ( ( ◡ 𝑇 ‘ 𝑥 ) ·ih ( ◡ 𝑇 ‘ 𝑦 ) ) )
14 6 10 12 13 syl3anc ⊢ ( ( 𝑇 ∈ UniOp ∧ ( 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ) ) → ( ( 𝑇 ‘ ( ◡ 𝑇 ‘ 𝑥 ) ) ·ih ( 𝑇 ‘ ( ◡ 𝑇 ‘ 𝑦 ) ) ) = ( ( ◡ 𝑇 ‘ 𝑥 ) ·ih ( ◡ 𝑇 ‘ 𝑦 ) ) )
15 f1ocnvfv2 ⊢ ( ( 𝑇 : ℋ –1-1-onto→ ℋ ∧ 𝑥 ∈ ℋ ) → ( 𝑇 ‘ ( ◡ 𝑇 ‘ 𝑥 ) ) = 𝑥 )
16 15 adantrr ⊢ ( ( 𝑇 : ℋ –1-1-onto→ ℋ ∧ ( 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ) ) → ( 𝑇 ‘ ( ◡ 𝑇 ‘ 𝑥 ) ) = 𝑥 )
17 f1ocnvfv2 ⊢ ( ( 𝑇 : ℋ –1-1-onto→ ℋ ∧ 𝑦 ∈ ℋ ) → ( 𝑇 ‘ ( ◡ 𝑇 ‘ 𝑦 ) ) = 𝑦 )
18 17 adantrl ⊢ ( ( 𝑇 : ℋ –1-1-onto→ ℋ ∧ ( 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ) ) → ( 𝑇 ‘ ( ◡ 𝑇 ‘ 𝑦 ) ) = 𝑦 )
19 16 18 oveq12d ⊢ ( ( 𝑇 : ℋ –1-1-onto→ ℋ ∧ ( 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ) ) → ( ( 𝑇 ‘ ( ◡ 𝑇 ‘ 𝑥 ) ) ·ih ( 𝑇 ‘ ( ◡ 𝑇 ‘ 𝑦 ) ) ) = ( 𝑥 ·ih 𝑦 ) )
20 1 19 sylan ⊢ ( ( 𝑇 ∈ UniOp ∧ ( 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ) ) → ( ( 𝑇 ‘ ( ◡ 𝑇 ‘ 𝑥 ) ) ·ih ( 𝑇 ‘ ( ◡ 𝑇 ‘ 𝑦 ) ) ) = ( 𝑥 ·ih 𝑦 ) )
21 14 20 eqtr3d ⊢ ( ( 𝑇 ∈ UniOp ∧ ( 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ) ) → ( ( ◡ 𝑇 ‘ 𝑥 ) ·ih ( ◡ 𝑇 ‘ 𝑦 ) ) = ( 𝑥 ·ih 𝑦 ) )
22 21 ralrimivva ⊢ ( 𝑇 ∈ UniOp → ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( ( ◡ 𝑇 ‘ 𝑥 ) ·ih ( ◡ 𝑇 ‘ 𝑦 ) ) = ( 𝑥 ·ih 𝑦 ) )
23 elunop ⊢ ( ◡ 𝑇 ∈ UniOp ↔ ( ◡ 𝑇 : ℋ –onto→ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( ( ◡ 𝑇 ‘ 𝑥 ) ·ih ( ◡ 𝑇 ‘ 𝑦 ) ) = ( 𝑥 ·ih 𝑦 ) ) )
24 5 22 23 sylanbrc ⊢ ( 𝑇 ∈ UniOp → ◡ 𝑇 ∈ UniOp )