Theorem cphnm

Description: The square of the norm is the norm of an inner product in a subcomplex pre-Hilbert space. (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	cphnm	⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → ( 𝑁 ‘ 𝐴 ) = ( √ ‘ ( 𝐴 , 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		nmsq.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		nmsq.h	⊢ , = ( ·𝑖 ‘ 𝑊 )
3		nmsq.n	⊢ 𝑁 = ( norm ‘ 𝑊 )
4	1 2 3	cphnmfval	⊢ ( 𝑊 ∈ ℂPreHil → 𝑁 = ( 𝑥 ∈ 𝑉 ↦ ( √ ‘ ( 𝑥 , 𝑥 ) ) ) )
5	4	fveq1d	⊢ ( 𝑊 ∈ ℂPreHil → ( 𝑁 ‘ 𝐴 ) = ( ( 𝑥 ∈ 𝑉 ↦ ( √ ‘ ( 𝑥 , 𝑥 ) ) ) ‘ 𝐴 ) )
6		oveq12	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑥 = 𝐴 ) → ( 𝑥 , 𝑥 ) = ( 𝐴 , 𝐴 ) )
7	6	anidms	⊢ ( 𝑥 = 𝐴 → ( 𝑥 , 𝑥 ) = ( 𝐴 , 𝐴 ) )
8	7	fveq2d	⊢ ( 𝑥 = 𝐴 → ( √ ‘ ( 𝑥 , 𝑥 ) ) = ( √ ‘ ( 𝐴 , 𝐴 ) ) )
9		eqid	⊢ ( 𝑥 ∈ 𝑉 ↦ ( √ ‘ ( 𝑥 , 𝑥 ) ) ) = ( 𝑥 ∈ 𝑉 ↦ ( √ ‘ ( 𝑥 , 𝑥 ) ) )
10		fvex	⊢ ( √ ‘ ( 𝐴 , 𝐴 ) ) ∈ V
11	8 9 10	fvmpt	⊢ ( 𝐴 ∈ 𝑉 → ( ( 𝑥 ∈ 𝑉 ↦ ( √ ‘ ( 𝑥 , 𝑥 ) ) ) ‘ 𝐴 ) = ( √ ‘ ( 𝐴 , 𝐴 ) ) )
12	5 11	sylan9eq	⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → ( 𝑁 ‘ 𝐴 ) = ( √ ‘ ( 𝐴 , 𝐴 ) ) )