Theorem xmstopn 15042

Description: The topology component of an extended metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)

Hypotheses

Ref	Expression
isms.j	|- J = ( TopOpen ` K )
isms.x	|- X = ( Base ` K )
isms.d	|- D = ( ( dist ` K ) |` ( X X. X ) )

Assertion

Ref	Expression
xmstopn	|- ( K e. *MetSp -> J = ( MetOpen ` D ) )

Proof of Theorem xmstopn

Step	Hyp	Ref	Expression
1		isms.j	. . 3 |- J = ( TopOpen ` K )
2		isms.x	. . 3 |- X = ( Base ` K )
3		isms.d	. . 3 |- D = ( ( dist ` K ) |` ( X X. X ) )
4	1, 2, 3	isxms 15038	. 2 |- ( K e. *MetSp <-> ( K e. TopSp /\ J = ( MetOpen ` D ) ) )
5	4	simprbi 275	1 |- ( K e. *MetSp -> J = ( MetOpen ` D ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2178   X. cxp 4691   |` cres 4695   ` cfv 5290   Basecbs 12947   distcds 13033   TopOpenctopn 13187   MetOpencmopn 14418   TopSpctps 14617   *MetSpcxms 14923

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 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-rab 2495  df-v 2778  df-un 3178  df-in 3180  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-xp 4699  df-res 4705  df-iota 5251  df-fv 5298  df-xms 14926

This theorem is referenced by: (None)