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Metamath Proof Explorer

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 ) ) )