Theorem tcphtopn

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

	Ref	Expression
Hypotheses	tcphval.n	⊢ 𝐺 = ( toℂPreHil ‘ 𝑊 )
	tcphtopn.d	⊢ 𝐷 = ( dist ‘ 𝐺 )
	tcphtopn.j	⊢ 𝐽 = ( TopOpen ‘ 𝐺 )
Assertion	tcphtopn	⊢ ( 𝑊 ∈ 𝑉 → 𝐽 = ( MetOpen ‘ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		tcphval.n	⊢ 𝐺 = ( toℂPreHil ‘ 𝑊 )
2		tcphtopn.d	⊢ 𝐷 = ( dist ‘ 𝐺 )
3		tcphtopn.j	⊢ 𝐽 = ( TopOpen ‘ 𝐺 )
4		eqid	⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
5	4	tcphex	⊢ ( 𝑥 ∈ ( Base ‘ 𝑊 ) ↦ ( √ ‘ ( 𝑥 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) ) ) ∈ V
6		eqid	⊢ ( ·𝑖 ‘ 𝑊 ) = ( ·𝑖 ‘ 𝑊 )
7	1 4 6	tcphval	⊢ 𝐺 = ( 𝑊 toNrmGrp ( 𝑥 ∈ ( Base ‘ 𝑊 ) ↦ ( √ ‘ ( 𝑥 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) ) ) )
8		eqid	⊢ ( MetOpen ‘ 𝐷 ) = ( MetOpen ‘ 𝐷 )
9	7 2 8	tngtopn	⊢ ( ( 𝑊 ∈ 𝑉 ∧ ( 𝑥 ∈ ( Base ‘ 𝑊 ) ↦ ( √ ‘ ( 𝑥 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) ) ) ∈ V ) → ( MetOpen ‘ 𝐷 ) = ( TopOpen ‘ 𝐺 ) )
10	5 9	mpan2	⊢ ( 𝑊 ∈ 𝑉 → ( MetOpen ‘ 𝐷 ) = ( TopOpen ‘ 𝐺 ) )
11	3 10	eqtr4id	⊢ ( 𝑊 ∈ 𝑉 → 𝐽 = ( MetOpen ‘ 𝐷 ) )