Theorem bncmet 25314

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 25311	. 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 25313	. 2 ⊢ (𝑀 ∈ CMetSp → 𝐷 ∈ (CMet‘𝑋))
5	1, 4	syl 17	1 ⊢ (𝑀 ∈ Ban → 𝐷 ∈ (CMet‘𝑋))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   × cxp 5629   ↾ cres 5633  ‘cfv 6498  Basecbs 17179  distcds 17229  CMetccmet 25221  CMetSpccms 25299  Bancbn 25300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-xp 5637  df-res 5643  df-iota 6454  df-fv 6506  df-cms 25302  df-bn 25303

This theorem is referenced by: (None)