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Theorem distgp 19669
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 457 . 2  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  G  e.  Grp )
2 simpr 461 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  J  =  ~P B )
3 distgp.1 . . . . . 6  |-  B  =  ( Base `  G
)
4 fvex 5700 . . . . . 6  |-  ( Base `  G )  e.  _V
53, 4eqeltri 2512 . . . . 5  |-  B  e. 
_V
6 distopon 18600 . . . . 5  |-  ( B  e.  _V  ->  ~P B  e.  (TopOn `  B
) )
75, 6ax-mp 5 . . . 4  |-  ~P B  e.  (TopOn `  B )
82, 7syl6eqel 2530 . . 3  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  J  e.  (TopOn `  B ) )
9 distgp.2 . . . 4  |-  J  =  ( TopOpen `  G )
103, 9istps 18540 . . 3  |-  ( G  e.  TopSp 
<->  J  e.  (TopOn `  B ) )
118, 10sylibr 212 . 2  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  G  e.  TopSp
)
12 eqid 2442 . . . . . 6  |-  ( -g `  G )  =  (
-g `  G )
133, 12grpsubf 15604 . . . . 5  |-  ( G  e.  Grp  ->  ( -g `  G ) : ( B  X.  B
) --> B )
1413adantr 465 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( -g `  G ) : ( B  X.  B ) --> B )
155, 5xpex 6507 . . . . 5  |-  ( B  X.  B )  e. 
_V
165, 15elmap 7240 . . . 4  |-  ( (
-g `  G )  e.  ( B  ^m  ( B  X.  B ) )  <-> 
( -g `  G ) : ( B  X.  B ) --> B )
1714, 16sylibr 212 . . 3  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( -g `  G )  e.  ( B  ^m  ( B  X.  B ) ) )
182, 2oveq12d 6108 . . . . . 6  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( J  tX  J )  =  ( ~P B  tX  ~P B ) )
19 txdis 19204 . . . . . . 7  |-  ( ( B  e.  _V  /\  B  e.  _V )  ->  ( ~P B  tX  ~P B )  =  ~P ( B  X.  B
) )
205, 5, 19mp2an 672 . . . . . 6  |-  ( ~P B  tX  ~P B
)  =  ~P ( B  X.  B )
2118, 20syl6eq 2490 . . . . 5  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( J  tX  J )  =  ~P ( B  X.  B
) )
2221oveq1d 6105 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( ( J  tX  J )  Cn  J )  =  ( ~P ( B  X.  B )  Cn  J
) )
23 cndis 18894 . . . . 5  |-  ( ( ( B  X.  B
)  e.  _V  /\  J  e.  (TopOn `  B
) )  ->  ( ~P ( B  X.  B
)  Cn  J )  =  ( B  ^m  ( B  X.  B
) ) )
2415, 8, 23sylancr 663 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( ~P ( B  X.  B
)  Cn  J )  =  ( B  ^m  ( B  X.  B
) ) )
2522, 24eqtrd 2474 . . 3  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( ( J  tX  J )  Cn  J )  =  ( B  ^m  ( B  X.  B ) ) )
2617, 25eleqtrrd 2519 . 2  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  ( -g `  G )  e.  ( ( J  tX  J
)  Cn  J ) )
279, 12istgp2 19661 . 2  |-  ( G  e.  TopGrp 
<->  ( G  e.  Grp  /\  G  e.  TopSp  /\  ( -g `  G )  e.  ( ( J  tX  J )  Cn  J
) ) )
281, 11, 26, 27syl3anbrc 1172 1  |-  ( ( G  e.  Grp  /\  J  =  ~P B
)  ->  G  e.  TopGrp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2971   ~Pcpw 3859    X. cxp 4837   -->wf 5413   ` cfv 5417  (class class class)co 6090    ^m cmap 7213   Basecbs 14173   TopOpenctopn 14359   Grpcgrp 15409   -gcsg 15412  TopOnctopon 18498   TopSpctps 18500    Cn ccn 18827    tX ctx 19132   TopGrpctgp 19641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-map 7215  df-0g 14379  df-topgen 14381  df-mnd 15414  df-plusf 15415  df-grp 15544  df-minusg 15545  df-sbg 15546  df-top 18502  df-bases 18504  df-topon 18505  df-topsp 18506  df-cn 18830  df-cnp 18831  df-tx 19134  df-tmd 19642  df-tgp 19643
This theorem is referenced by: (None)
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