Theorem xmstps 13057

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 2165	. . 3 |- ( TopOpen ` M ) = ( TopOpen ` M )
2		eqid 2165	. . 3 |- ( Base ` M ) = ( Base ` M )
3		eqid 2165	. . 3 |- ( ( dist ` M ) |` ( ( Base ` M ) X. ( Base ` M ) ) ) = ( ( dist ` M ) |` ( ( Base ` M ) X. ( Base ` M ) ) )
4	1, 2, 3	isxms 13051	. 2 |- ( M e. *MetSp <-> ( M e. TopSp /\ ( TopOpen ` M ) = ( MetOpen ` ( ( dist ` M ) |` ( ( Base ` M ) X. ( Base ` M ) ) ) ) ) )
5	4	simplbi 272	1 |- ( M e. *MetSp -> M e. TopSp )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   X. cxp 4601   |` cres 4605   ` cfv 5187   Basecbs 12390   distcds 12461   TopOpenctopn 12552   MetOpencmopn 12585   TopSpctps 12628   *MetSpcxms 12936

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-rex 2449  df-rab 2452  df-v 2727  df-un 3119  df-in 3121  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-br 3982  df-opab 4043  df-xp 4609  df-res 4615  df-iota 5152  df-fv 5195  df-xms 12939

This theorem is referenced by:  mstps  13059