This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem cphorthcom

Description: Orthogonality (meaning inner product is 0) is commutative. Complex version of iporthcom . (Contributed by Mario Carneiro, 16-Oct-2015)

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

Proof

Step Hyp Ref Expression
1 cphipcj.h ⊢ , = ( ·𝑖 ‘ 𝑊 )
2 cphipcj.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
3 cphphl ⊢ ( 𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil )
4 eqid ⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
5 eqid ⊢ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) )
6 4 1 2 5 iporthcom ⊢ ( ( 𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ( 𝐴 , 𝐵 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ↔ ( 𝐵 , 𝐴 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) )
7 3 6 syl3an1 ⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ( 𝐴 , 𝐵 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ↔ ( 𝐵 , 𝐴 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) )
8 cphclm ⊢ ( 𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod )
9 4 clm0 ⊢ ( 𝑊 ∈ ℂMod → 0 = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) )
10 8 9 syl ⊢ ( 𝑊 ∈ ℂPreHil → 0 = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) )
11 10 3ad2ant1 ⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → 0 = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) )
12 11 eqeq2d ⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ( 𝐴 , 𝐵 ) = 0 ↔ ( 𝐴 , 𝐵 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) )
13 11 eqeq2d ⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ( 𝐵 , 𝐴 ) = 0 ↔ ( 𝐵 , 𝐴 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) )
14 7 12 13 3bitr4d ⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ( 𝐴 , 𝐵 ) = 0 ↔ ( 𝐵 , 𝐴 ) = 0 ) )