Theorem cmscmet 25338

Description: The induced metric on a complete normed group is complete. (Contributed by Mario Carneiro, 15-Oct-2015.)

Hypotheses

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

Assertion

Ref	Expression
cmscmet	⊢ (𝑀 ∈ CMetSp → 𝐷 ∈ (CMet‘𝑋))

Proof of Theorem cmscmet

Step	Hyp	Ref	Expression
1		iscms.1	. . 3 ⊢ 𝑋 = (Base‘𝑀)
2		iscms.2	. . 3 ⊢ 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))
3	1, 2	iscms 25337	. 2 ⊢ (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMet‘𝑋)))
4	3	simprbi 498	1 ⊢ (𝑀 ∈ CMetSp → 𝐷 ∈ (CMet‘𝑋))

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119   × cxp 5623   ↾ cres 5627  ‘cfv 6492  Basecbs 17177  distcds 17227  MetSpcms 24308  CMetccmet 25246  CMetSpccms 25324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-nul 5235

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-xp 5631  df-res 5637  df-iota 6448  df-fv 6500  df-cms 25327

This theorem is referenced by:  bncmet  25339  cmsss  25343  cmetcusp1  25345  cmscsscms  25365  minveclem3a  25419