Theorem tcphds

Description: The distance of a pre-Hilbert space augmented with norm. (Contributed by Thierry Arnoux, 30-Jun-2019)

	Ref	Expression
Hypotheses	tcphval.n	⊢ 𝐺 = ( toℂPreHil ‘ 𝑊 )
	tcphds.n	⊢ 𝑁 = ( norm ‘ 𝐺 )
	tcphds.m	⊢ − = ( -g ‘ 𝑊 )
Assertion	tcphds	⊢ ( 𝑊 ∈ Grp → ( 𝑁 ∘ − ) = ( dist ‘ 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		tcphval.n	⊢ 𝐺 = ( toℂPreHil ‘ 𝑊 )
2		tcphds.n	⊢ 𝑁 = ( norm ‘ 𝐺 )
3		tcphds.m	⊢ − = ( -g ‘ 𝑊 )
4		eqid	⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
5		eqid	⊢ ( ·𝑖 ‘ 𝑊 ) = ( ·𝑖 ‘ 𝑊 )
6	1 2 4 5	tchnmfval	⊢ ( 𝑊 ∈ Grp → 𝑁 = ( 𝑥 ∈ ( Base ‘ 𝑊 ) ↦ ( √ ‘ ( 𝑥 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) ) ) )
7	6	coeq1d	⊢ ( 𝑊 ∈ Grp → ( 𝑁 ∘ − ) = ( ( 𝑥 ∈ ( Base ‘ 𝑊 ) ↦ ( √ ‘ ( 𝑥 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) ) ) ∘ − ) )
8	4	tcphex	⊢ ( 𝑥 ∈ ( Base ‘ 𝑊 ) ↦ ( √ ‘ ( 𝑥 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) ) ) ∈ V
9	1 4 5	tcphval	⊢ 𝐺 = ( 𝑊 toNrmGrp ( 𝑥 ∈ ( Base ‘ 𝑊 ) ↦ ( √ ‘ ( 𝑥 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) ) ) )
10	9 3	tngds	⊢ ( ( 𝑥 ∈ ( Base ‘ 𝑊 ) ↦ ( √ ‘ ( 𝑥 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) ) ) ∈ V → ( ( 𝑥 ∈ ( Base ‘ 𝑊 ) ↦ ( √ ‘ ( 𝑥 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) ) ) ∘ − ) = ( dist ‘ 𝐺 ) )
11	8 10	ax-mp	⊢ ( ( 𝑥 ∈ ( Base ‘ 𝑊 ) ↦ ( √ ‘ ( 𝑥 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) ) ) ∘ − ) = ( dist ‘ 𝐺 )
12	7 11	eqtrdi	⊢ ( 𝑊 ∈ Grp → ( 𝑁 ∘ − ) = ( dist ‘ 𝐺 ) )