Theorem cmscmet 25314

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 25313	. 2 ⊢ (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMet‘𝑋)))
4	3	simprbi 497	1 ⊢ (𝑀 ∈ CMetSp → 𝐷 ∈ (CMet‘𝑋))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   × cxp 5630   ↾ cres 5634  ‘cfv 6500  Basecbs 17148  distcds 17198  MetSpcms 24274  CMetccmet 25222  CMetSpccms 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 2709  ax-nul 5253

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 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5638  df-res 5644  df-iota 6456  df-fv 6508  df-cms 25303

This theorem is referenced by:  bncmet  25315  cmsss  25319  cmetcusp1  25321  cmscsscms  25341  minveclem3a  25395