Theorem tcphip

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

	Ref	Expression
Hypotheses	tcphval.n	⊢ 𝐺 = ( toℂPreHil ‘ 𝑊 )
	tcphip.s	⊢ · = ( ·𝑖 ‘ 𝑊 )
Assertion	tcphip	⊢ · = ( ·𝑖 ‘ 𝐺 )

Proof

Step	Hyp	Ref	Expression
1		tcphval.n	⊢ 𝐺 = ( toℂPreHil ‘ 𝑊 )
2		tcphip.s	⊢ · = ( ·𝑖 ‘ 𝑊 )
3		eqid	⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
4	3	tcphex	⊢ ( 𝑥 ∈ ( Base ‘ 𝑊 ) ↦ ( √ ‘ ( 𝑥 · 𝑥 ) ) ) ∈ V
5	1 3 2	tcphval	⊢ 𝐺 = ( 𝑊 toNrmGrp ( 𝑥 ∈ ( Base ‘ 𝑊 ) ↦ ( √ ‘ ( 𝑥 · 𝑥 ) ) ) )
6	5 2	tngip	⊢ ( ( 𝑥 ∈ ( Base ‘ 𝑊 ) ↦ ( √ ‘ ( 𝑥 · 𝑥 ) ) ) ∈ V → · = ( ·𝑖 ‘ 𝐺 ) )
7	4 6	ax-mp	⊢ · = ( ·𝑖 ‘ 𝐺 )