Theorem xmstps 14973

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 2206	. . 3 |- ( TopOpen ` M ) = ( TopOpen ` M )
2		eqid 2206	. . 3 |- ( Base ` M ) = ( Base ` M )
3		eqid 2206	. . 3 |- ( ( dist ` M ) |` ( ( Base ` M ) X. ( Base ` M ) ) ) = ( ( dist ` M ) |` ( ( Base ` M ) X. ( Base ` M ) ) )
4	1, 2, 3	isxms 14967	. 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 1373   e. wcel 2177   X. cxp 4677   |` cres 4681   ` cfv 5276   Basecbs 12876   distcds 12962   TopOpenctopn 13116   MetOpencmopn 14347   TopSpctps 14546   *MetSpcxms 14852

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491  df-rab 2494  df-v 2775  df-un 3171  df-in 3173  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-br 4048  df-opab 4110  df-xp 4685  df-res 4691  df-iota 5237  df-fv 5284  df-xms 14855

This theorem is referenced by:  mstps  14975