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Metamath Proof Explorer

Theorem cphip0r

Description: Inner product with a zero second argument. Complex version of ip0r . (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 cphip0r ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → ( 𝐴 , 0 ) = 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 ip0r ⊢ ( ( 𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ) → ( 𝐴 , 0 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) )
8 4 7 sylan ⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → ( 𝐴 , 0 ) = ( 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 ) → ( 𝐴 , 0 ) = 0 )