Theorem bncmet 23944

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 23941	. 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 23943	. 2 ⊢ (𝑀 ∈ CMetSp → 𝐷 ∈ (CMet‘𝑋))
5	1, 4	syl 17	1 ⊢ (𝑀 ∈ Ban → 𝐷 ∈ (CMet‘𝑋))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110   × cxp 5547   ↾ cres 5551  ‘cfv 6349  Basecbs 16477  distcds 16568  CMetccmet 23851  CMetSpccms 23929  Bancbn 23930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-nul 5202

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-xp 5555  df-res 5561  df-iota 6308  df-fv 6357  df-cms 23932  df-bn 23933

This theorem is referenced by: (None)