Theorem cphnmfval

Description: The value of the norm in a subcomplex pre-Hilbert space is the square root of the inner product of a vector with itself. (Contributed by Mario Carneiro, 7-Oct-2015)

	Ref	Expression
Hypotheses	nmsq.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	nmsq.h	⊢ , = ( ·𝑖 ‘ 𝑊 )
	nmsq.n	⊢ 𝑁 = ( norm ‘ 𝑊 )
Assertion	cphnmfval	⊢ ( 𝑊 ∈ ℂPreHil → 𝑁 = ( 𝑥 ∈ 𝑉 ↦ ( √ ‘ ( 𝑥 , 𝑥 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		nmsq.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		nmsq.h	⊢ , = ( ·𝑖 ‘ 𝑊 )
3		nmsq.n	⊢ 𝑁 = ( 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 ) ) ∧ 𝑁 = ( 𝑥 ∈ 𝑉 ↦ ( √ ‘ ( 𝑥 , 𝑥 ) ) ) ) )
7	6	simp3bi	⊢ ( 𝑊 ∈ ℂPreHil → 𝑁 = ( 𝑥 ∈ 𝑉 ↦ ( √ ‘ ( 𝑥 , 𝑥 ) ) ) )