Theorem tcphvsca

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

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

Proof

Step	Hyp	Ref	Expression
1		tcphval.n	⊢ 𝐺 = ( toℂPreHil ‘ 𝑊 )
2		tcphvsca.s	⊢ · = ( ·𝑠 ‘ 𝑊 )
3		eqid	⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
4	3	tcphex	⊢ ( 𝑥 ∈ ( Base ‘ 𝑊 ) ↦ ( √ ‘ ( 𝑥 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) ) ) ∈ V
5		eqid	⊢ ( ·𝑖 ‘ 𝑊 ) = ( ·𝑖 ‘ 𝑊 )
6	1 3 5	tcphval	⊢ 𝐺 = ( 𝑊 toNrmGrp ( 𝑥 ∈ ( Base ‘ 𝑊 ) ↦ ( √ ‘ ( 𝑥 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) ) ) )
7	6 2	tngvsca	⊢ ( ( 𝑥 ∈ ( Base ‘ 𝑊 ) ↦ ( √ ‘ ( 𝑥 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) ) ) ∈ V → · = ( ·𝑠 ‘ 𝐺 ) )
8	4 7	ax-mp	⊢ · = ( ·𝑠 ‘ 𝐺 )