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Theorem xmstps 22477
Description: A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
xmstps (𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp)

Proof of Theorem xmstps
StepHypRef Expression
1 eqid 2770 . . 3 (TopOpen‘𝑀) = (TopOpen‘𝑀)
2 eqid 2770 . . 3 (Base‘𝑀) = (Base‘𝑀)
3 eqid 2770 . . 3 ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
41, 2, 3isxms 22471 . 2 (𝑀 ∈ ∞MetSp ↔ (𝑀 ∈ TopSp ∧ (TopOpen‘𝑀) = (MetOpen‘((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))))))
54simplbi 479 1 (𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2144   × cxp 5247  cres 5251  cfv 6031  Basecbs 16063  distcds 16157  TopOpenctopn 16289  MetOpencmopn 19950  TopSpctps 20956  ∞MetSpcxme 22341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-xp 5255  df-res 5261  df-iota 5994  df-fv 6039  df-xms 22344
This theorem is referenced by:  mstps  22479  ressxms  22549  prdsxmslem2  22553  tmsxpsmopn  22561  minveclem4a  23419  rrhcn  30375  rrhf  30376  rrexttps  30384  sitmcl  30747
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