Theorem xmstopn 13249

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 13245	. 2 |- ( K e. *MetSp <-> ( K e. TopSp /\ J = ( MetOpen ` D ) ) )
5	4	simprbi 273	1 |- ( K e. *MetSp -> J = ( MetOpen ` D ) )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   X. cxp 4609   |` cres 4613   ` cfv 5198   Basecbs 12416   distcds 12489   TopOpenctopn 12580   MetOpencmopn 12779   TopSpctps 12822   *MetSpcxms 13130

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-rab 2457  df-v 2732  df-un 3125  df-in 3127  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-res 4623  df-iota 5160  df-fv 5206  df-xms 13133

This theorem is referenced by: (None)