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Metamath Proof Explorer

Theorem cphphl

Description: A subcomplex pre-Hilbert space is a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015)

Ref Expression
Assertion cphphl ( 𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil )

Proof

Step Hyp Ref Expression
1 eqid ⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
2 eqid ⊢ ( ·𝑖 ‘ 𝑊 ) = ( ·𝑖 ‘ 𝑊 )
3 eqid ⊢ ( norm ‘ 𝑊 ) = ( norm ‘ 𝑊 )
4 eqid ⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
5 eqid ⊢ ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) )
6 1 2 3 4 5 iscph ⊢ ( 𝑊 ∈ ℂPreHil ↔ ( ( 𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ ( Scalar ‘ 𝑊 ) = ( ℂfld ↾s ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ) ∧ ( √ “ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∩ ( 0 [,) +∞ ) ) ) ⊆ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( norm ‘ 𝑊 ) = ( 𝑥 ∈ ( Base ‘ 𝑊 ) ↦ ( √ ‘ ( 𝑥 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) ) ) ) )
7 6 simp1bi ⊢ ( 𝑊 ∈ ℂPreHil → ( 𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ ( Scalar ‘ 𝑊 ) = ( ℂfld ↾s ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ) )
8 7 simp1d ⊢ ( 𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil )