Theorem cmscmet 25262

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 25261	. 2 ⊢ (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMet‘𝑋)))
4	3	simprbi 496	1 ⊢ (𝑀 ∈ CMetSp → 𝐷 ∈ (CMet‘𝑋))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   × cxp 5621   ↾ cres 5625  ‘cfv 6486  Basecbs 17138  distcds 17188  MetSpcms 24222  CMetccmet 25170  CMetSpccms 25248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5248

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-rab 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-xp 5629  df-res 5635  df-iota 6442  df-fv 6494  df-cms 25251

This theorem is referenced by:  bncmet  25263  cmsss  25267  cmetcusp1  25269  cmscsscms  25289  minveclem3a  25343