Theorem xmstopn 15479

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 15475	. 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 1402   e. wcel 2209   X. cxp 4767   |` cres 4771   ` cfv 5372   Basecbs 13330   distcds 13417   TopOpenctopn 13571   MetOpencmopn 14850   TopSpctps 15054   *MetSpcxms 15360

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-in 3226  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-xp 4775  df-res 4781  df-iota 5332  df-fv 5380  df-xms 15363

This theorem is referenced by: (None)