Theorem bncmet 25674

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 25671	. 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 25673	. 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 1569   ∈ wcel 2144   × cxp 5648   ↾ cres 5652  ‘cfv 6539  Basecbs 17393  distcds 17443  CMetccmet 25581  CMetSpccms 25659  Bancbn 25660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5656  df-res 5662  df-iota 6494  df-fv 6547  df-cms 25662  df-bn 25663

This theorem is used by: (None)