Theorem cmscmet 25399

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 25398	. 2 ⊢ (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMet‘𝑋)))
4	3	simprbi 496	1 ⊢ (𝑀 ∈ CMetSp → 𝐷 ∈ (CMet‘𝑋))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108   × cxp 5698   ↾ cres 5702  ‘cfv 6573  Basecbs 17258  distcds 17320  MetSpcms 24349  CMetccmet 25307  CMetSpccms 25385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-res 5712  df-iota 6525  df-fv 6581  df-cms 25388

This theorem is referenced by:  bncmet  25400  cmsss  25404  cmetcusp1  25406  cmscsscms  25426  minveclem3a  25480