Theorem xmstps 15116

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

Assertion

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

Proof of Theorem xmstps

Step	Hyp	Ref	Expression
1		eqid 2229	. . 3 |- ( TopOpen ` M ) = ( TopOpen ` M )
2		eqid 2229	. . 3 |- ( Base ` M ) = ( Base ` M )
3		eqid 2229	. . 3 |- ( ( dist ` M ) |` ( ( Base ` M ) X. ( Base ` M ) ) ) = ( ( dist ` M ) |` ( ( Base ` M ) X. ( Base ` M ) ) )
4	1, 2, 3	isxms 15110	. 2 |- ( M e. *MetSp <-> ( M e. TopSp /\ ( TopOpen ` M ) = ( MetOpen ` ( ( dist ` M ) |` ( ( Base ` M ) X. ( Base ` M ) ) ) ) ) )
5	4	simplbi 274	1 |- ( M e. *MetSp -> M e. TopSp )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   X. cxp 4714   |` cres 4718   ` cfv 5314   Basecbs 13018   distcds 13105   TopOpenctopn 13259   MetOpencmopn 14490   TopSpctps 14689   *MetSpcxms 14995

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-rab 2517  df-v 2801  df-un 3201  df-in 3203  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-xp 4722  df-res 4728  df-iota 5274  df-fv 5322  df-xms 14998

This theorem is referenced by:  mstps  15118