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Metamath Proof Explorer

Theorem cphipcl

Description: An inner product is a member of the complex numbers. (Contributed by Mario Carneiro, 13-Oct-2015)

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

Proof

Step Hyp Ref Expression
1 nmsq.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
2 nmsq.h ⊢ , = ( ·𝑖 ‘ 𝑊 )
3 eqid ⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
4 eqid ⊢ ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) )
5 3 4 cphsubrg ⊢ ( 𝑊 ∈ ℂPreHil → ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∈ ( SubRing ‘ ℂfld ) )
6 cnfldbas ⊢ ℂ = ( Base ‘ ℂfld )
7 6 subrgss ⊢ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∈ ( SubRing ‘ ℂfld ) → ( Base ‘ ( Scalar ‘ 𝑊 ) ) ⊆ ℂ )
8 5 7 syl ⊢ ( 𝑊 ∈ ℂPreHil → ( Base ‘ ( Scalar ‘ 𝑊 ) ) ⊆ ℂ )
9 8 3ad2ant1 ⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( Base ‘ ( Scalar ‘ 𝑊 ) ) ⊆ ℂ )
10 cphphl ⊢ ( 𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil )
11 3 2 1 4 ipcl ⊢ ( ( 𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( 𝐴 , 𝐵 ) ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
12 10 11 syl3an1 ⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( 𝐴 , 𝐵 ) ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
13 9 12 sseldd ⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( 𝐴 , 𝐵 ) ∈ ℂ )