Theorem cphtcphnm

Description: The norm of a norm-augmented subcomplex pre-Hilbert space is the same as the original norm on it. (Contributed by Mario Carneiro, 11-Oct-2015)

	Ref	Expression
Hypotheses	tcphval.n	⊢ 𝐺 = ( toℂPreHil ‘ 𝑊 )
	cphtcphnm.n	⊢ 𝑁 = ( norm ‘ 𝑊 )
Assertion	cphtcphnm	⊢ ( 𝑊 ∈ ℂPreHil → 𝑁 = ( norm ‘ 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		tcphval.n	⊢ 𝐺 = ( toℂPreHil ‘ 𝑊 )
2		cphtcphnm.n	⊢ 𝑁 = ( norm ‘ 𝑊 )
3		eqid	⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
4		eqid	⊢ ( ·𝑖 ‘ 𝑊 ) = ( ·𝑖 ‘ 𝑊 )
5	3 4 2	cphnmfval	⊢ ( 𝑊 ∈ ℂPreHil → 𝑁 = ( 𝑥 ∈ ( Base ‘ 𝑊 ) ↦ ( √ ‘ ( 𝑥 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) ) ) )
6		cphlmod	⊢ ( 𝑊 ∈ ℂPreHil → 𝑊 ∈ LMod )
7		lmodgrp	⊢ ( 𝑊 ∈ LMod → 𝑊 ∈ Grp )
8		eqid	⊢ ( norm ‘ 𝐺 ) = ( norm ‘ 𝐺 )
9	1 8 3 4	tchnmfval	⊢ ( 𝑊 ∈ Grp → ( norm ‘ 𝐺 ) = ( 𝑥 ∈ ( Base ‘ 𝑊 ) ↦ ( √ ‘ ( 𝑥 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) ) ) )
10	6 7 9	3syl	⊢ ( 𝑊 ∈ ℂPreHil → ( norm ‘ 𝐺 ) = ( 𝑥 ∈ ( Base ‘ 𝑊 ) ↦ ( √ ‘ ( 𝑥 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) ) ) )
11	5 10	eqtr4d	⊢ ( 𝑊 ∈ ℂPreHil → 𝑁 = ( norm ‘ 𝐺 ) )