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Metamath Proof Explorer

Theorem unop

Description: Basic inner product property of a unitary operator. (Contributed by NM, 22-Jan-2006) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 elunop ⊢ ( 𝑇 ∈ UniOp ↔ ( 𝑇 : ℋ –onto→ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( ( 𝑇 ‘ 𝑥 ) ·ih ( 𝑇 ‘ 𝑦 ) ) = ( 𝑥 ·ih 𝑦 ) ) )
2 1 simprbi ⊢ ( 𝑇 ∈ UniOp → ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( ( 𝑇 ‘ 𝑥 ) ·ih ( 𝑇 ‘ 𝑦 ) ) = ( 𝑥 ·ih 𝑦 ) )
3 2 3ad2ant1 ⊢ ( ( 𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( ( 𝑇 ‘ 𝑥 ) ·ih ( 𝑇 ‘ 𝑦 ) ) = ( 𝑥 ·ih 𝑦 ) )
4 fveq2 ⊢ ( 𝑥 = 𝐴 → ( 𝑇 ‘ 𝑥 ) = ( 𝑇 ‘ 𝐴 ) )
5 4 oveq1d ⊢ ( 𝑥 = 𝐴 → ( ( 𝑇 ‘ 𝑥 ) ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑇 ‘ 𝐴 ) ·ih ( 𝑇 ‘ 𝑦 ) ) )
6 oveq1 ⊢ ( 𝑥 = 𝐴 → ( 𝑥 ·ih 𝑦 ) = ( 𝐴 ·ih 𝑦 ) )
7 5 6 eqeq12d ⊢ ( 𝑥 = 𝐴 → ( ( ( 𝑇 ‘ 𝑥 ) ·ih ( 𝑇 ‘ 𝑦 ) ) = ( 𝑥 ·ih 𝑦 ) ↔ ( ( 𝑇 ‘ 𝐴 ) ·ih ( 𝑇 ‘ 𝑦 ) ) = ( 𝐴 ·ih 𝑦 ) ) )
8 fveq2 ⊢ ( 𝑦 = 𝐵 → ( 𝑇 ‘ 𝑦 ) = ( 𝑇 ‘ 𝐵 ) )
9 8 oveq2d ⊢ ( 𝑦 = 𝐵 → ( ( 𝑇 ‘ 𝐴 ) ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑇 ‘ 𝐴 ) ·ih ( 𝑇 ‘ 𝐵 ) ) )
10 oveq2 ⊢ ( 𝑦 = 𝐵 → ( 𝐴 ·ih 𝑦 ) = ( 𝐴 ·ih 𝐵 ) )
11 9 10 eqeq12d ⊢ ( 𝑦 = 𝐵 → ( ( ( 𝑇 ‘ 𝐴 ) ·ih ( 𝑇 ‘ 𝑦 ) ) = ( 𝐴 ·ih 𝑦 ) ↔ ( ( 𝑇 ‘ 𝐴 ) ·ih ( 𝑇 ‘ 𝐵 ) ) = ( 𝐴 ·ih 𝐵 ) ) )
12 7 11 rspc2v ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( ( 𝑇 ‘ 𝑥 ) ·ih ( 𝑇 ‘ 𝑦 ) ) = ( 𝑥 ·ih 𝑦 ) → ( ( 𝑇 ‘ 𝐴 ) ·ih ( 𝑇 ‘ 𝐵 ) ) = ( 𝐴 ·ih 𝐵 ) ) )
13 12 3adant1 ⊢ ( ( 𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( ( 𝑇 ‘ 𝑥 ) ·ih ( 𝑇 ‘ 𝑦 ) ) = ( 𝑥 ·ih 𝑦 ) → ( ( 𝑇 ‘ 𝐴 ) ·ih ( 𝑇 ‘ 𝐵 ) ) = ( 𝐴 ·ih 𝐵 ) ) )
14 3 13 mpd ⊢ ( ( 𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝑇 ‘ 𝐴 ) ·ih ( 𝑇 ‘ 𝐵 ) ) = ( 𝐴 ·ih 𝐵 ) )