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Metamath Proof Explorer

Theorem cphipcj

Description: Conjugate of an inner product in a subcomplex pre-Hilbert space. Complex version of ipcj . (Contributed by Mario Carneiro, 16-Oct-2015)

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

Proof

Step Hyp Ref Expression
1 cphipcj.h ⊢ , = ( ·𝑖 ‘ 𝑊 )
2 cphipcj.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
3 cphclm ⊢ ( 𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod )
4 eqid ⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
5 4 clmcj ⊢ ( 𝑊 ∈ ℂMod → ∗ = ( *𝑟 ‘ ( Scalar ‘ 𝑊 ) ) )
6 3 5 syl ⊢ ( 𝑊 ∈ ℂPreHil → ∗ = ( *𝑟 ‘ ( Scalar ‘ 𝑊 ) ) )
7 6 3ad2ant1 ⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ∗ = ( *𝑟 ‘ ( Scalar ‘ 𝑊 ) ) )
8 7 fveq1d ⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ∗ ‘ ( 𝐴 , 𝐵 ) ) = ( ( *𝑟 ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 𝐴 , 𝐵 ) ) )
9 cphphl ⊢ ( 𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil )
10 eqid ⊢ ( *𝑟 ‘ ( Scalar ‘ 𝑊 ) ) = ( *𝑟 ‘ ( Scalar ‘ 𝑊 ) )
11 4 1 2 10 ipcj ⊢ ( ( 𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ( *𝑟 ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 𝐴 , 𝐵 ) ) = ( 𝐵 , 𝐴 ) )
12 9 11 syl3an1 ⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ( *𝑟 ‘ ( Scalar ‘ 𝑊 ) ) ‘ ( 𝐴 , 𝐵 ) ) = ( 𝐵 , 𝐴 ) )
13 8 12 eqtrd ⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ∗ ‘ ( 𝐴 , 𝐵 ) ) = ( 𝐵 , 𝐴 ) )