Theorem cphip0l

Description: Inner product with a zero first argument. Part of proof of Theorem 6.44 of Ponnusamy p. 361. Complex version of ip0l . (Contributed by Mario Carneiro, 16-Oct-2015)

	Ref	Expression
Hypotheses	cphipcj.h	⊢ , = ( ·𝑖 ‘ 𝑊 )
	cphipcj.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	cphip0l.z	⊢ 0 = ( 0g ‘ 𝑊 )
Assertion	cphip0l	⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → ( 0 , 𝐴 ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		cphipcj.h	⊢ , = ( ·𝑖 ‘ 𝑊 )
2		cphipcj.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
3		cphip0l.z	⊢ 0 = ( 0g ‘ 𝑊 )
4		cphphl	⊢ ( 𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil )
5		eqid	⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
6		eqid	⊢ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) )
7	5 1 2 6 3	ip0l	⊢ ( ( 𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ) → ( 0 , 𝐴 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) )
8	4 7	sylan	⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → ( 0 , 𝐴 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) )
9		cphclm	⊢ ( 𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod )
10	5	clm0	⊢ ( 𝑊 ∈ ℂMod → 0 = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) )
11	9 10	syl	⊢ ( 𝑊 ∈ ℂPreHil → 0 = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) )
12	11	adantr	⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → 0 = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) )
13	8 12	eqtr4d	⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → ( 0 , 𝐴 ) = 0 )