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Theorem indistgp 17799
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 443 . 2  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  G  e.  Grp )
2 simpr 447 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  J  =  { (/) ,  B }
)
3 distgp.1 . . . . . 6  |-  B  =  ( Base `  G
)
4 fvex 5555 . . . . . 6  |-  ( Base `  G )  e.  _V
53, 4eqeltri 2366 . . . . 5  |-  B  e. 
_V
6 indistopon 16754 . . . . 5  |-  ( B  e.  _V  ->  { (/) ,  B }  e.  (TopOn `  B ) )
75, 6ax-mp 8 . . . 4  |-  { (/) ,  B }  e.  (TopOn `  B )
82, 7syl6eqel 2384 . . 3  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  J  e.  (TopOn `  B )
)
9 distgp.2 . . . 4  |-  J  =  ( TopOpen `  G )
103, 9istps 16690 . . 3  |-  ( G  e.  TopSp 
<->  J  e.  (TopOn `  B ) )
118, 10sylibr 203 . 2  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  G  e.  TopSp )
12 eqid 2296 . . . . . 6  |-  ( -g `  G )  =  (
-g `  G )
133, 12grpsubf 14561 . . . . 5  |-  ( G  e.  Grp  ->  ( -g `  G ) : ( B  X.  B
) --> B )
1413adantr 451 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  ( -g `  G ) : ( B  X.  B
) --> B )
155, 5xpex 4817 . . . . 5  |-  ( B  X.  B )  e. 
_V
165, 15elmap 6812 . . . 4  |-  ( (
-g `  G )  e.  ( B  ^m  ( B  X.  B ) )  <-> 
( -g `  G ) : ( B  X.  B ) --> B )
1714, 16sylibr 203 . . 3  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  ( -g `  G )  e.  ( B  ^m  ( B  X.  B ) ) )
182oveq2d 5890 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  (
( J  tX  J
)  Cn  J )  =  ( ( J 
tX  J )  Cn 
{ (/) ,  B }
) )
19 txtopon 17302 . . . . . 6  |-  ( ( J  e.  (TopOn `  B )  /\  J  e.  (TopOn `  B )
)  ->  ( J  tX  J )  e.  (TopOn `  ( B  X.  B
) ) )
208, 8, 19syl2anc 642 . . . . 5  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  ( J  tX  J )  e.  (TopOn `  ( B  X.  B ) ) )
21 cnindis 17036 . . . . 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 643 . . . 4  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  (
( J  tX  J
)  Cn  { (/) ,  B } )  =  ( B  ^m  ( B  X.  B ) ) )
2318, 22eqtrd 2328 . . 3  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  (
( J  tX  J
)  Cn  J )  =  ( B  ^m  ( B  X.  B
) ) )
2417, 23eleqtrrd 2373 . 2  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  ( -g `  G )  e.  ( ( J  tX  J )  Cn  J
) )
259, 12istgp2 17790 . 2  |-  ( G  e.  TopGrp 
<->  ( G  e.  Grp  /\  G  e.  TopSp  /\  ( -g `  G )  e.  ( ( J  tX  J )  Cn  J
) ) )
261, 11, 24, 25syl3anbrc 1136 1  |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B } )  ->  G  e.  TopGrp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   {cpr 3654    X. cxp 4703   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   Basecbs 13164   TopOpenctopn 13342   Grpcgrp 14378   -gcsg 14381  TopOnctopon 16648   TopSpctps 16650    Cn ccn 16970    tX ctx 17271   TopGrpctgp 17770
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-map 6790  df-topgen 13360  df-0g 13420  df-mnd 14383  df-plusf 14384  df-grp 14505  df-minusg 14506  df-sbg 14507  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cn 16973  df-cnp 16974  df-tx 17273  df-tmd 17771  df-tgp 17772
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