Theorem ipnm

Description: Norm expressed in terms of inner product. (Contributed by NM, 11-Sep-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ipid.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
	ipid.6	⊢ 𝑁 = ( normCV ‘ 𝑈 )
	ipid.7	⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 )
Assertion	ipnm	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( 𝑁 ‘ 𝐴 ) = ( √ ‘ ( 𝐴 𝑃 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ipid.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		ipid.6	⊢ 𝑁 = ( normCV ‘ 𝑈 )
3		ipid.7	⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 )
4	1 2 3	ipidsq	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝑃 𝐴 ) = ( ( 𝑁 ‘ 𝐴 ) ↑ 2 ) )
5	4	fveq2d	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( √ ‘ ( 𝐴 𝑃 𝐴 ) ) = ( √ ‘ ( ( 𝑁 ‘ 𝐴 ) ↑ 2 ) ) )
6	1 2	nvcl	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( 𝑁 ‘ 𝐴 ) ∈ ℝ )
7	1 2	nvge0	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → 0 ≤ ( 𝑁 ‘ 𝐴 ) )
8	6 7	sqrtsqd	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( √ ‘ ( ( 𝑁 ‘ 𝐴 ) ↑ 2 ) ) = ( 𝑁 ‘ 𝐴 ) )
9	5 8	eqtr2d	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( 𝑁 ‘ 𝐴 ) = ( √ ‘ ( 𝐴 𝑃 𝐴 ) ) )