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Theorem indistgp 18131
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 445 . 2  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  G  e.  Grp )
2 simpr 449 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  J  =  { (/) ,  B }
)
3 distgp.1 . . . . . 6  |-  B  =  ( Base `  G
)
4 fvex 5743 . . . . . 6  |-  ( Base `  G )  e.  _V
53, 4eqeltri 2507 . . . . 5  |-  B  e. 
_V
6 indistopon 17066 . . . . 5  |-  ( B  e.  _V  ->  { (/) ,  B }  e.  (TopOn `  B ) )
75, 6ax-mp 8 . . . 4  |-  { (/) ,  B }  e.  (TopOn `  B )
82, 7syl6eqel 2525 . . 3  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  J  e.  (TopOn `  B )
)
9 distgp.2 . . . 4  |-  J  =  ( TopOpen `  G )
103, 9istps 17002 . . 3  |-  ( G  e.  TopSp 
<->  J  e.  (TopOn `  B ) )
118, 10sylibr 205 . 2  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  G  e.  TopSp )
12 eqid 2437 . . . . . 6  |-  ( -g `  G )  =  (
-g `  G )
133, 12grpsubf 14869 . . . . 5  |-  ( G  e.  Grp  ->  ( -g `  G ) : ( B  X.  B
) --> B )
1413adantr 453 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  ( -g `  G ) : ( B  X.  B
) --> B )
155, 5xpex 4991 . . . . 5  |-  ( B  X.  B )  e. 
_V
165, 15elmap 7043 . . . 4  |-  ( (
-g `  G )  e.  ( B  ^m  ( B  X.  B ) )  <-> 
( -g `  G ) : ( B  X.  B ) --> B )
1714, 16sylibr 205 . . 3  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  ( -g `  G )  e.  ( B  ^m  ( B  X.  B ) ) )
182oveq2d 6098 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  (
( J  tX  J
)  Cn  J )  =  ( ( J 
tX  J )  Cn 
{ (/) ,  B }
) )
19 txtopon 17624 . . . . . 6  |-  ( ( J  e.  (TopOn `  B )  /\  J  e.  (TopOn `  B )
)  ->  ( J  tX  J )  e.  (TopOn `  ( B  X.  B
) ) )
208, 8, 19syl2anc 644 . . . . 5  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  ( J  tX  J )  e.  (TopOn `  ( B  X.  B ) ) )
21 cnindis 17357 . . . . 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 645 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  (
( J  tX  J
)  Cn  { (/) ,  B } )  =  ( B  ^m  ( B  X.  B ) ) )
2318, 22eqtrd 2469 . . 3  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  (
( J  tX  J
)  Cn  J )  =  ( B  ^m  ( B  X.  B
) ) )
2417, 23eleqtrrd 2514 . 2  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  ( -g `  G )  e.  ( ( J  tX  J )  Cn  J
) )
259, 12istgp2 18122 . 2  |-  ( G  e.  TopGrp 
<->  ( G  e.  Grp  /\  G  e.  TopSp  /\  ( -g `  G )  e.  ( ( J  tX  J )  Cn  J
) ) )
261, 11, 24, 25syl3anbrc 1139 1  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  G  e.  TopGrp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2957   (/)c0 3629   {cpr 3816    X. cxp 4877   -->wf 5451   ` cfv 5455  (class class class)co 6082    ^m cmap 7019   Basecbs 13470   TopOpenctopn 13650   Grpcgrp 14686   -gcsg 14689  TopOnctopon 16960   TopSpctps 16962    Cn ccn 17289    tX ctx 17593   TopGrpctgp 18102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-map 7021  df-topgen 13668  df-0g 13728  df-mnd 14691  df-plusf 14692  df-grp 14813  df-minusg 14814  df-sbg 14815  df-top 16964  df-bases 16966  df-topon 16967  df-topsp 16968  df-cn 17292  df-cnp 17293  df-tx 17595  df-tmd 18103  df-tgp 18104
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