Theorem hmop

Description: Basic inner product property of a Hermitian operator. (Contributed by NM, 19-Mar-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	hmop	⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 ·ih ( 𝑇 ‘ 𝐵 ) ) = ( ( 𝑇 ‘ 𝐴 ) ·ih 𝐵 ) )

Proof

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