Theorem cmsms 25488

Description: A complete metric space is a metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)

Assertion

Ref	Expression
cmsms	⊢ (𝐺 ∈ CMetSp → 𝐺 ∈ MetSp)

Proof of Theorem cmsms

Step	Hyp	Ref	Expression
1		eqid 2763	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2763	. . 3 ⊢ ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) = ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))
3	1, 2	iscms 25485	. 2 ⊢ (𝐺 ∈ CMetSp ↔ (𝐺 ∈ MetSp ∧ ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) ∈ (CMet‘(Base‘𝐺))))
4	3	simplbi 501	1 ⊢ (𝐺 ∈ CMetSp → 𝐺 ∈ MetSp)

Syntax hints:   → wi 4   ∈ wcel 2143   × cxp 5661   ↾ cres 5665  ‘cfv 6538  Basecbs 17270  distcds 17320  MetSpcms 24456  CMetccmet 25394  CMetSpccms 25472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5270

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-xp 5669  df-res 5675  df-iota 6494  df-fv 6546  df-cms 25475

This theorem is referenced by:  cmsss  25491  cmetcusp1  25493  rlmbn  25501  cmscsscms  25513  rrhcn  34368  dya2icoseg2  34649  sitgclbn  34714