Theorem hlipcj

Description: Conjugate law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hlipf.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
	hlipf.7	⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 )
Assertion	hlipcj	⊢ ( ( 𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝑃 𝐵 ) = ( ∗ ‘ ( 𝐵 𝑃 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		hlipf.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		hlipf.7	⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 )
3		hlnv	⊢ ( 𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec )
4	1 2	dipcj	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( ∗ ‘ ( 𝐵 𝑃 𝐴 ) ) = ( 𝐴 𝑃 𝐵 ) )
5	3 4	syl3an1	⊢ ( ( 𝑈 ∈ CHilOLD ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( ∗ ‘ ( 𝐵 𝑃 𝐴 ) ) = ( 𝐴 𝑃 𝐵 ) )
6	5	3com23	⊢ ( ( 𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ∗ ‘ ( 𝐵 𝑃 𝐴 ) ) = ( 𝐴 𝑃 𝐵 ) )
7	6	eqcomd	⊢ ( ( 𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝑃 𝐵 ) = ( ∗ ‘ ( 𝐵 𝑃 𝐴 ) ) )