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Theorem msxms 13252
Description: A metric space is an extended metric space. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
msxms (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)

Proof of Theorem msxms
StepHypRef Expression
1 eqid 2170 . . 3 (TopOpen‘𝑀) = (TopOpen‘𝑀)
2 eqid 2170 . . 3 (Base‘𝑀) = (Base‘𝑀)
3 eqid 2170 . . 3 ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
41, 2, 3isms 13247 . 2 (𝑀 ∈ MetSp ↔ (𝑀 ∈ ∞MetSp ∧ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) ∈ (Met‘(Base‘𝑀))))
54simplbi 272 1 (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2141   × cxp 4609  cres 4613  cfv 5198  Basecbs 12416  distcds 12489  TopOpenctopn 12580  Metcmet 12775  ∞MetSpcxms 13130  MetSpcms 13131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-rab 2457  df-v 2732  df-un 3125  df-in 3127  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-res 4623  df-iota 5160  df-fv 5206  df-ms 13134
This theorem is referenced by:  mstps  13253
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