Theorem tcphbas

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

	Ref	Expression
Hypotheses	tcphval.n	⊢ 𝐺 = ( toℂPreHil ‘ 𝑊 )
	tcphbas.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
Assertion	tcphbas	⊢ 𝑉 = ( Base ‘ 𝐺 )

Proof

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