Theorem bncmet 25543

Description: The induced metric on Banach space is complete. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 15-Oct-2015.)

Hypotheses

Ref	Expression
iscms.1	⊢ 𝑋 = (Base‘𝑀)
iscms.2	⊢ 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))

Assertion

Ref	Expression
bncmet	⊢ (𝑀 ∈ Ban → 𝐷 ∈ (CMet‘𝑋))

Proof of Theorem bncmet

Step	Hyp	Ref	Expression
1		bncms 25540	. 2 ⊢ (𝑀 ∈ Ban → 𝑀 ∈ CMetSp)
2		iscms.1	. . 3 ⊢ 𝑋 = (Base‘𝑀)
3		iscms.2	. . 3 ⊢ 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))
4	2, 3	cmscmet 25542	. 2 ⊢ (𝑀 ∈ CMetSp → 𝐷 ∈ (CMet‘𝑋))
5	1, 4	syl 18	1 ⊢ (𝑀 ∈ Ban → 𝐷 ∈ (CMet‘𝑋))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146   × cxp 5664   ↾ cres 5668  ‘cfv 6543  Basecbs 17294  distcds 17344  CMetccmet 25450  CMetSpccms 25528  Bancbn 25529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-nul 5274

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-xp 5672  df-res 5678  df-iota 6499  df-fv 6551  df-cms 25531  df-bn 25532

This theorem is used by: (None)