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Metamath Proof Explorer

Theorem cphipeq0

Description: The inner product of a vector with itself is zero iff the vector is zero. Part of Definition 3.1-1 of Kreyszig p. 129. Complex version of ipeq0 . (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 cphipeq0 ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → ( ( 𝐴 , 𝐴 ) = 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 cphclm ⊢ ( 𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod )
5 eqid ⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
6 5 clm0 ⊢ ( 𝑊 ∈ ℂMod → 0 = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) )
7 4 6 syl ⊢ ( 𝑊 ∈ ℂPreHil → 0 = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) )
8 7 adantr ⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → 0 = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) )
9 8 eqeq2d ⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → ( ( 𝐴 , 𝐴 ) = 0 ↔ ( 𝐴 , 𝐴 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) )
10 cphphl ⊢ ( 𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil )
11 eqid ⊢ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) )
12 5 1 2 11 3 ipeq0 ⊢ ( ( 𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ) → ( ( 𝐴 , 𝐴 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ↔ 𝐴 = 0 ) )
13 10 12 sylan ⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → ( ( 𝐴 , 𝐴 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ↔ 𝐴 = 0 ) )
14 9 13 bitrd ⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → ( ( 𝐴 , 𝐴 ) = 0 ↔ 𝐴 = 0 ) )