Theorem msxms 22259

Description: A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
msxms	|- ( M e. MetSp -> M e. *MetSp )

Proof of Theorem msxms

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( TopOpen ` M ) = ( TopOpen ` M )
2		eqid 2622	. . 3 |- ( Base ` M ) = ( Base ` M )
3		eqid 2622	. . 3 |- ( ( dist ` M ) |` ( ( Base ` M ) X. ( Base ` M ) ) ) = ( ( dist ` M ) |` ( ( Base ` M ) X. ( Base ` M ) ) )
4	1, 2, 3	isms 22254	. 2 |- ( M e. MetSp <-> ( M e. *MetSp /\ ( ( dist ` M ) |` ( ( Base ` M ) X. ( Base ` M ) ) ) e. ( Met ` ( Base ` M ) ) ) )
5	4	simplbi 476	1 |- ( M e. MetSp -> M e. *MetSp )

Syntax hints:   -> wi 4   e. wcel 1990   X. cxp 5112   |` cres 5116   ` cfv 5888   Basecbs 15857   distcds 15950   TopOpenctopn 16082   Metcme 19732   *MetSpcxme 22122   MetSpcmt 22123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-res 5126  df-iota 5851  df-fv 5896  df-ms 22126

This theorem is referenced by:  mstps  22260  imasf1oms  22295  ressms  22331  prdsms  22336  ngpxms  22405  ngptgp  22440  nlmvscnlem2  22489  nlmvscn  22491  nrginvrcn  22496  nghmcn  22549  cnfldxms  22580  nmhmcn  22920  ipcnlem2  23043  ipcn  23045  nglmle  23100  cmetcusp1  23149  dya2icoseg2  30340