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Theorem cmsms 23878
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
StepHypRef Expression
1 eqid 2818 . . 3 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2818 . . 3 ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) = ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))
31, 2iscms 23875 . 2 (𝐺 ∈ CMetSp ↔ (𝐺 ∈ MetSp ∧ ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) ∈ (CMet‘(Base‘𝐺))))
43simplbi 498 1 (𝐺 ∈ CMetSp → 𝐺 ∈ MetSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105   × cxp 5546  cres 5550  cfv 6348  Basecbs 16471  distcds 16562  MetSpcms 22855  CMetccmet 23784  CMetSpccms 23862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-res 5560  df-iota 6307  df-fv 6356  df-cms 23865
This theorem is referenced by:  cmsss  23881  cmetcusp1  23883  rlmbn  23891  cmscsscms  23903  rrhcn  31137  dya2icoseg2  31435  sitgclbn  31500
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