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Theorem distgp 17995
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 443 . 2  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  G  e.  Grp )
2 simpr 447 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  J  =  ~P B )
3 distgp.1 . . . . . 6  |-  B  =  ( Base `  G
)
4 fvex 5646 . . . . . 6  |-  ( Base `  G )  e.  _V
53, 4eqeltri 2436 . . . . 5  |-  B  e. 
_V
6 distopon 16951 . . . . 5  |-  ( B  e.  _V  ->  ~P B  e.  (TopOn `  B
) )
75, 6ax-mp 8 . . . 4  |-  ~P B  e.  (TopOn `  B )
82, 7syl6eqel 2454 . . 3  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  J  e.  (TopOn `  B ) )
9 distgp.2 . . . 4  |-  J  =  ( TopOpen `  G )
103, 9istps 16891 . . 3  |-  ( G  e.  TopSp 
<->  J  e.  (TopOn `  B ) )
118, 10sylibr 203 . 2  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  G  e.  TopSp
)
12 eqid 2366 . . . . . 6  |-  ( -g `  G )  =  (
-g `  G )
133, 12grpsubf 14755 . . . . 5  |-  ( G  e.  Grp  ->  ( -g `  G ) : ( B  X.  B
) --> B )
1413adantr 451 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( -g `  G ) : ( B  X.  B ) --> B )
155, 5xpex 4904 . . . . 5  |-  ( B  X.  B )  e. 
_V
165, 15elmap 6939 . . . 4  |-  ( (
-g `  G )  e.  ( B  ^m  ( B  X.  B ) )  <-> 
( -g `  G ) : ( B  X.  B ) --> B )
1714, 16sylibr 203 . . 3  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( -g `  G )  e.  ( B  ^m  ( B  X.  B ) ) )
182, 2oveq12d 5999 . . . . . 6  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( J  tX  J )  =  ( ~P B  tX  ~P B ) )
19 txdis 17543 . . . . . . 7  |-  ( ( B  e.  _V  /\  B  e.  _V )  ->  ( ~P B  tX  ~P B )  =  ~P ( B  X.  B
) )
205, 5, 19mp2an 653 . . . . . 6  |-  ( ~P B  tX  ~P B
)  =  ~P ( B  X.  B )
2118, 20syl6eq 2414 . . . . 5  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( J  tX  J )  =  ~P ( B  X.  B
) )
2221oveq1d 5996 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( ( J  tX  J )  Cn  J )  =  ( ~P ( B  X.  B )  Cn  J
) )
23 cndis 17236 . . . . 5  |-  ( ( ( B  X.  B
)  e.  _V  /\  J  e.  (TopOn `  B
) )  ->  ( ~P ( B  X.  B
)  Cn  J )  =  ( B  ^m  ( B  X.  B
) ) )
2415, 8, 23sylancr 644 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( ~P ( B  X.  B
)  Cn  J )  =  ( B  ^m  ( B  X.  B
) ) )
2522, 24eqtrd 2398 . . 3  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( ( J  tX  J )  Cn  J )  =  ( B  ^m  ( B  X.  B ) ) )
2617, 25eleqtrrd 2443 . 2  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( -g `  G )  e.  ( ( J  tX  J
)  Cn  J ) )
279, 12istgp2 17987 . 2  |-  ( G  e.  TopGrp 
<->  ( G  e.  Grp  /\  G  e.  TopSp  /\  ( -g `  G )  e.  ( ( J  tX  J )  Cn  J
) ) )
281, 11, 26, 27syl3anbrc 1137 1  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  G  e.  TopGrp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   _Vcvv 2873   ~Pcpw 3714    X. cxp 4790   -->wf 5354   ` cfv 5358  (class class class)co 5981    ^m cmap 6915   Basecbs 13356   TopOpenctopn 13536   Grpcgrp 14572   -gcsg 14575  TopOnctopon 16849   TopSpctps 16851    Cn ccn 17171    tX ctx 17472   TopGrpctgp 17967
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-map 6917  df-topgen 13554  df-0g 13614  df-mnd 14577  df-plusf 14578  df-grp 14699  df-minusg 14700  df-sbg 14701  df-top 16853  df-bases 16855  df-topon 16856  df-topsp 16857  df-cn 17174  df-cnp 17175  df-tx 17474  df-tmd 17968  df-tgp 17969
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