Theorem cphdi

Description: Distributive law for inner product (left-distributivity). Complex version of ipdi . (Contributed by Mario Carneiro, 16-Oct-2015)

	Ref	Expression
Hypotheses	cphipcj.h	⊢ , = ( ·𝑖 ‘ 𝑊 )
	cphipcj.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	cphdir.P	⊢ + = ( +g ‘ 𝑊 )
Assertion	cphdi	⊢ ( ( 𝑊 ∈ ℂPreHil ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → ( 𝐴 , ( 𝐵 + 𝐶 ) ) = ( ( 𝐴 , 𝐵 ) + ( 𝐴 , 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		cphipcj.h	⊢ , = ( ·𝑖 ‘ 𝑊 )
2		cphipcj.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
3		cphdir.P	⊢ + = ( +g ‘ 𝑊 )
4		cphphl	⊢ ( 𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil )
5		eqid	⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
6		eqid	⊢ ( +g ‘ ( Scalar ‘ 𝑊 ) ) = ( +g ‘ ( Scalar ‘ 𝑊 ) )
7	5 1 2 3 6	ipdi	⊢ ( ( 𝑊 ∈ PreHil ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → ( 𝐴 , ( 𝐵 + 𝐶 ) ) = ( ( 𝐴 , 𝐵 ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐴 , 𝐶 ) ) )
8	4 7	sylan	⊢ ( ( 𝑊 ∈ ℂPreHil ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → ( 𝐴 , ( 𝐵 + 𝐶 ) ) = ( ( 𝐴 , 𝐵 ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐴 , 𝐶 ) ) )
9		cphclm	⊢ ( 𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod )
10	5	clmadd	⊢ ( 𝑊 ∈ ℂMod → + = ( +g ‘ ( Scalar ‘ 𝑊 ) ) )
11	9 10	syl	⊢ ( 𝑊 ∈ ℂPreHil → + = ( +g ‘ ( Scalar ‘ 𝑊 ) ) )
12	11	adantr	⊢ ( ( 𝑊 ∈ ℂPreHil ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → + = ( +g ‘ ( Scalar ‘ 𝑊 ) ) )
13	12	oveqd	⊢ ( ( 𝑊 ∈ ℂPreHil ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → ( ( 𝐴 , 𝐵 ) + ( 𝐴 , 𝐶 ) ) = ( ( 𝐴 , 𝐵 ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐴 , 𝐶 ) ) )
14	8 13	eqtr4d	⊢ ( ( 𝑊 ∈ ℂPreHil ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → ( 𝐴 , ( 𝐵 + 𝐶 ) ) = ( ( 𝐴 , 𝐵 ) + ( 𝐴 , 𝐶 ) ) )