Theorem nmsq

Description: The square of the norm is the norm of an inner product in a subcomplex pre-Hilbert space. Equation I4 of Ponnusamy p. 362. (Contributed by NM, 1-Feb-2007) (Revised by Mario Carneiro, 7-Oct-2015)

	Ref	Expression
Hypotheses	nmsq.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	nmsq.h	⊢ , = ( ·𝑖 ‘ 𝑊 )
	nmsq.n	⊢ 𝑁 = ( norm ‘ 𝑊 )
Assertion	nmsq	⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → ( ( 𝑁 ‘ 𝐴 ) ↑ 2 ) = ( 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	cphnm	⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → ( 𝑁 ‘ 𝐴 ) = ( √ ‘ ( 𝐴 , 𝐴 ) ) )
5	4	oveq1d	⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → ( ( 𝑁 ‘ 𝐴 ) ↑ 2 ) = ( ( √ ‘ ( 𝐴 , 𝐴 ) ) ↑ 2 ) )
6	1 2	cphipcl	⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ) → ( 𝐴 , 𝐴 ) ∈ ℂ )
7	6	3anidm23	⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → ( 𝐴 , 𝐴 ) ∈ ℂ )
8	7	sqsqrtd	⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → ( ( √ ‘ ( 𝐴 , 𝐴 ) ) ↑ 2 ) = ( 𝐴 , 𝐴 ) )
9	5 8	eqtrd	⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → ( ( 𝑁 ‘ 𝐴 ) ↑ 2 ) = ( 𝐴 , 𝐴 ) )