Theorem tcphnmval

Description: The norm of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015)

	Ref	Expression
Hypotheses	tcphval.n	⊢ 𝐺 = ( toℂPreHil ‘ 𝑊 )
	tcphnmval.n	⊢ 𝑁 = ( norm ‘ 𝐺 )
	tcphnmval.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	tcphnmval.h	⊢ , = ( ·𝑖 ‘ 𝑊 )
Assertion	tcphnmval	⊢ ( ( 𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ 𝑋 ) = ( √ ‘ ( 𝑋 , 𝑋 ) ) )

Proof

Step	Hyp	Ref	Expression
1		tcphval.n	⊢ 𝐺 = ( toℂPreHil ‘ 𝑊 )
2		tcphnmval.n	⊢ 𝑁 = ( norm ‘ 𝐺 )
3		tcphnmval.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
4		tcphnmval.h	⊢ , = ( ·𝑖 ‘ 𝑊 )
5	1 2 3 4	tchnmfval	⊢ ( 𝑊 ∈ Grp → 𝑁 = ( 𝑥 ∈ 𝑉 ↦ ( √ ‘ ( 𝑥 , 𝑥 ) ) ) )
6	5	fveq1d	⊢ ( 𝑊 ∈ Grp → ( 𝑁 ‘ 𝑋 ) = ( ( 𝑥 ∈ 𝑉 ↦ ( √ ‘ ( 𝑥 , 𝑥 ) ) ) ‘ 𝑋 ) )
7		oveq12	⊢ ( ( 𝑥 = 𝑋 ∧ 𝑥 = 𝑋 ) → ( 𝑥 , 𝑥 ) = ( 𝑋 , 𝑋 ) )
8	7	anidms	⊢ ( 𝑥 = 𝑋 → ( 𝑥 , 𝑥 ) = ( 𝑋 , 𝑋 ) )
9	8	fveq2d	⊢ ( 𝑥 = 𝑋 → ( √ ‘ ( 𝑥 , 𝑥 ) ) = ( √ ‘ ( 𝑋 , 𝑋 ) ) )
10		eqid	⊢ ( 𝑥 ∈ 𝑉 ↦ ( √ ‘ ( 𝑥 , 𝑥 ) ) ) = ( 𝑥 ∈ 𝑉 ↦ ( √ ‘ ( 𝑥 , 𝑥 ) ) )
11		fvex	⊢ ( √ ‘ ( 𝑋 , 𝑋 ) ) ∈ V
12	9 10 11	fvmpt	⊢ ( 𝑋 ∈ 𝑉 → ( ( 𝑥 ∈ 𝑉 ↦ ( √ ‘ ( 𝑥 , 𝑥 ) ) ) ‘ 𝑋 ) = ( √ ‘ ( 𝑋 , 𝑋 ) ) )
13	6 12	sylan9eq	⊢ ( ( 𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ 𝑋 ) = ( √ ‘ ( 𝑋 , 𝑋 ) ) )