Theorem cphipcj

Description: Conjugate of an inner product in a subcomplex pre-Hilbert space. Complex version of ipcj . (Contributed by Mario Carneiro, 16-Oct-2015)

	Ref	Expression
Hypotheses	cphipcj.h	⊢ , = ( ·𝑖 ‘ 𝑊 )
	cphipcj.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
Assertion	cphipcj	⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ∗ ‘ ( 𝐴 , 𝐵 ) ) = ( 𝐵 , 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		cphipcj.h	⊢ , = ( ·𝑖 ‘ 𝑊 )
2		cphipcj.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
3		cphclm	⊢ ( 𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod )
4		eqid	⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
5	4	clmcj	⊢ ( 𝑊 ∈ ℂMod → ∗ = ( *𝑟 ‘ ( Scalar ‘ 𝑊 ) ) )
6	3 5	syl	⊢ ( 𝑊 ∈ ℂPreHil → ∗ = ( *𝑟 ‘ ( Scalar ‘ 𝑊 ) ) )
7	6	3ad2ant1	⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ∗ = ( *𝑟 ‘ ( Scalar ‘ 𝑊 ) ) )
8	7	fveq1d	⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ∗ ‘ ( 𝐴 , 𝐵 ) ) = ( ( *𝑟 ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 𝐴 , 𝐵 ) ) )
9		cphphl	⊢ ( 𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil )
10		eqid	⊢ ( *𝑟 ‘ ( Scalar ‘ 𝑊 ) ) = ( *𝑟 ‘ ( Scalar ‘ 𝑊 ) )
11	4 1 2 10	ipcj	⊢ ( ( 𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ( *𝑟 ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 𝐴 , 𝐵 ) ) = ( 𝐵 , 𝐴 ) )
12	9 11	syl3an1	⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ( *𝑟 ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 𝐴 , 𝐵 ) ) = ( 𝐵 , 𝐴 ) )
13	8 12	eqtrd	⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ∗ ‘ ( 𝐴 , 𝐵 ) ) = ( 𝐵 , 𝐴 ) )