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Theorem distgp 18134
Description: Any group equipped with the discrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.)
Hypotheses
Ref Expression
distgp.1  |-  B  =  ( Base `  G
)
distgp.2  |-  J  =  ( TopOpen `  G )
Assertion
Ref Expression
distgp  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  G  e.  TopGrp )

Proof of Theorem distgp
StepHypRef Expression
1 simpl 445 . 2  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  G  e.  Grp )
2 simpr 449 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  J  =  ~P B )
3 distgp.1 . . . . . 6  |-  B  =  ( Base `  G
)
4 fvex 5745 . . . . . 6  |-  ( Base `  G )  e.  _V
53, 4eqeltri 2508 . . . . 5  |-  B  e. 
_V
6 distopon 17066 . . . . 5  |-  ( B  e.  _V  ->  ~P B  e.  (TopOn `  B
) )
75, 6ax-mp 5 . . . 4  |-  ~P B  e.  (TopOn `  B )
82, 7syl6eqel 2526 . . 3  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  J  e.  (TopOn `  B ) )
9 distgp.2 . . . 4  |-  J  =  ( TopOpen `  G )
103, 9istps 17006 . . 3  |-  ( G  e.  TopSp 
<->  J  e.  (TopOn `  B ) )
118, 10sylibr 205 . 2  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  G  e.  TopSp
)
12 eqid 2438 . . . . . 6  |-  ( -g `  G )  =  (
-g `  G )
133, 12grpsubf 14873 . . . . 5  |-  ( G  e.  Grp  ->  ( -g `  G ) : ( B  X.  B
) --> B )
1413adantr 453 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( -g `  G ) : ( B  X.  B ) --> B )
155, 5xpex 4993 . . . . 5  |-  ( B  X.  B )  e. 
_V
165, 15elmap 7045 . . . 4  |-  ( (
-g `  G )  e.  ( B  ^m  ( B  X.  B ) )  <-> 
( -g `  G ) : ( B  X.  B ) --> B )
1714, 16sylibr 205 . . 3  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( -g `  G )  e.  ( B  ^m  ( B  X.  B ) ) )
182, 2oveq12d 6102 . . . . . 6  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( J  tX  J )  =  ( ~P B  tX  ~P B ) )
19 txdis 17669 . . . . . . 7  |-  ( ( B  e.  _V  /\  B  e.  _V )  ->  ( ~P B  tX  ~P B )  =  ~P ( B  X.  B
) )
205, 5, 19mp2an 655 . . . . . 6  |-  ( ~P B  tX  ~P B
)  =  ~P ( B  X.  B )
2118, 20syl6eq 2486 . . . . 5  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( J  tX  J )  =  ~P ( B  X.  B
) )
2221oveq1d 6099 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( ( J  tX  J )  Cn  J )  =  ( ~P ( B  X.  B )  Cn  J
) )
23 cndis 17360 . . . . 5  |-  ( ( ( B  X.  B
)  e.  _V  /\  J  e.  (TopOn `  B
) )  ->  ( ~P ( B  X.  B
)  Cn  J )  =  ( B  ^m  ( B  X.  B
) ) )
2415, 8, 23sylancr 646 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( ~P ( B  X.  B
)  Cn  J )  =  ( B  ^m  ( B  X.  B
) ) )
2522, 24eqtrd 2470 . . 3  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( ( J  tX  J )  Cn  J )  =  ( B  ^m  ( B  X.  B ) ) )
2617, 25eleqtrrd 2515 . 2  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( -g `  G )  e.  ( ( J  tX  J
)  Cn  J ) )
279, 12istgp2 18126 . 2  |-  ( G  e.  TopGrp 
<->  ( G  e.  Grp  /\  G  e.  TopSp  /\  ( -g `  G )  e.  ( ( J  tX  J )  Cn  J
) ) )
281, 11, 26, 27syl3anbrc 1139 1  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  G  e.  TopGrp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   ~Pcpw 3801    X. cxp 4879   -->wf 5453   ` cfv 5457  (class class class)co 6084    ^m cmap 7021   Basecbs 13474   TopOpenctopn 13654   Grpcgrp 14690   -gcsg 14693  TopOnctopon 16964   TopSpctps 16966    Cn ccn 17293    tX ctx 17597   TopGrpctgp 18106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-map 7023  df-topgen 13672  df-0g 13732  df-mnd 14695  df-plusf 14696  df-grp 14817  df-minusg 14818  df-sbg 14819  df-top 16968  df-bases 16970  df-topon 16971  df-topsp 16972  df-cn 17296  df-cnp 17297  df-tx 17599  df-tmd 18107  df-tgp 18108
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