Theorem cmscmet 25580

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 25579	. 2 ⊢ (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMet‘𝑋)))
4	3	simprbi 503	1 ⊢ (𝑀 ∈ CMetSp → 𝐷 ∈ (CMet‘𝑋))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   × cxp 5657   ↾ cres 5661  ‘cfv 6537  Basecbs 17307  distcds 17357  MetSpcms 24550  CMetccmet 25488  CMetSpccms 25566

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-nul 5267

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-xp 5665  df-res 5671  df-iota 6493  df-fv 6545  df-cms 25569

This theorem is used by:  bncmet  25581  cmsss  25585  cmetcusp1  25587  cmscsscms  25607  minveclem3a  25661