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Mirrors > Home > MPE Home > Th. List > xmstopn | Structured version Visualization version Unicode version |
Description: The topology component of a metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
isms.j | |
isms.x | |
isms.d |
Ref | Expression |
---|---|
xmstopn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isms.j | . . 3 | |
2 | isms.x | . . 3 | |
3 | isms.d | . . 3 | |
4 | 1, 2, 3 | isxms 22252 | . 2 |
5 | 4 | simprbi 480 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wcel 1990 cxp 5112 cres 5116 cfv 5888 cbs 15857 cds 15950 ctopn 16082 cmopn 19736 ctps 20736 cxme 22122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-res 5126 df-iota 5851 df-fv 5896 df-xms 22125 |
This theorem is referenced by: imasf1oxms 22294 ressxms 22330 prdsxmslem2 22334 tmsxpsmopn 22342 xmsusp 22374 cmetcusp1 23149 minveclem4a 23201 minveclem4 23203 qqhcn 30035 rrhcn 30041 rrexthaus 30051 dya2icoseg2 30340 sitmcl 30413 |
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