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Theorem indistgp 19671
Description: Any group equipped with the indiscrete 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
indistgp  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  G  e.  TopGrp )

Proof of Theorem indistgp
StepHypRef Expression
1 simpl 457 . 2  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  G  e.  Grp )
2 simpr 461 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  J  =  { (/) ,  B }
)
3 distgp.1 . . . . . 6  |-  B  =  ( Base `  G
)
4 fvex 5701 . . . . . 6  |-  ( Base `  G )  e.  _V
53, 4eqeltri 2513 . . . . 5  |-  B  e. 
_V
6 indistopon 18605 . . . . 5  |-  ( B  e.  _V  ->  { (/) ,  B }  e.  (TopOn `  B ) )
75, 6ax-mp 5 . . . 4  |-  { (/) ,  B }  e.  (TopOn `  B )
82, 7syl6eqel 2531 . . 3  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  J  e.  (TopOn `  B )
)
9 distgp.2 . . . 4  |-  J  =  ( TopOpen `  G )
103, 9istps 18541 . . 3  |-  ( G  e.  TopSp 
<->  J  e.  (TopOn `  B ) )
118, 10sylibr 212 . 2  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  G  e.  TopSp )
12 eqid 2443 . . . . . 6  |-  ( -g `  G )  =  (
-g `  G )
133, 12grpsubf 15605 . . . . 5  |-  ( G  e.  Grp  ->  ( -g `  G ) : ( B  X.  B
) --> B )
1413adantr 465 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  ( -g `  G ) : ( B  X.  B
) --> B )
155, 5xpex 6508 . . . . 5  |-  ( B  X.  B )  e. 
_V
165, 15elmap 7241 . . . 4  |-  ( (
-g `  G )  e.  ( B  ^m  ( B  X.  B ) )  <-> 
( -g `  G ) : ( B  X.  B ) --> B )
1714, 16sylibr 212 . . 3  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  ( -g `  G )  e.  ( B  ^m  ( B  X.  B ) ) )
182oveq2d 6107 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  (
( J  tX  J
)  Cn  J )  =  ( ( J 
tX  J )  Cn 
{ (/) ,  B }
) )
19 txtopon 19164 . . . . . 6  |-  ( ( J  e.  (TopOn `  B )  /\  J  e.  (TopOn `  B )
)  ->  ( J  tX  J )  e.  (TopOn `  ( B  X.  B
) ) )
208, 8, 19syl2anc 661 . . . . 5  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  ( J  tX  J )  e.  (TopOn `  ( B  X.  B ) ) )
21 cnindis 18896 . . . . 5  |-  ( ( ( J  tX  J
)  e.  (TopOn `  ( B  X.  B
) )  /\  B  e.  _V )  ->  (
( J  tX  J
)  Cn  { (/) ,  B } )  =  ( B  ^m  ( B  X.  B ) ) )
2220, 5, 21sylancl 662 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  (
( J  tX  J
)  Cn  { (/) ,  B } )  =  ( B  ^m  ( B  X.  B ) ) )
2318, 22eqtrd 2475 . . 3  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  (
( J  tX  J
)  Cn  J )  =  ( B  ^m  ( B  X.  B
) ) )
2417, 23eleqtrrd 2520 . 2  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  ( -g `  G )  e.  ( ( J  tX  J )  Cn  J
) )
259, 12istgp2 19662 . 2  |-  ( G  e.  TopGrp 
<->  ( G  e.  Grp  /\  G  e.  TopSp  /\  ( -g `  G )  e.  ( ( J  tX  J )  Cn  J
) ) )
261, 11, 24, 25syl3anbrc 1172 1  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  G  e.  TopGrp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2972   (/)c0 3637   {cpr 3879    X. cxp 4838   -->wf 5414   ` cfv 5418  (class class class)co 6091    ^m cmap 7214   Basecbs 14174   TopOpenctopn 14360   Grpcgrp 15410   -gcsg 15413  TopOnctopon 18499   TopSpctps 18501    Cn ccn 18828    tX ctx 19133   TopGrpctgp 19642
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-map 7216  df-0g 14380  df-topgen 14382  df-mnd 15415  df-plusf 15416  df-grp 15545  df-minusg 15546  df-sbg 15547  df-top 18503  df-bases 18505  df-topon 18506  df-topsp 18507  df-cn 18831  df-cnp 18832  df-tx 19135  df-tmd 19643  df-tgp 19644
This theorem is referenced by: (None)
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