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Theorem tgplacthmeo 17802
Description: The left group action of element  A in a topological group  G is a homeomorphism from the group to itself. (Contributed by Mario Carneiro, 14-Aug-2015.)
Hypotheses
Ref Expression
tgplacthmeo.1  |-  F  =  ( x  e.  X  |->  ( A  .+  x
) )
tgplacthmeo.2  |-  X  =  ( Base `  G
)
tgplacthmeo.3  |-  .+  =  ( +g  `  G )
tgplacthmeo.4  |-  J  =  ( TopOpen `  G )
Assertion
Ref Expression
tgplacthmeo  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  F  e.  ( J  Homeo  J ) )
Distinct variable groups:    x, A    x, G    x, J    x,  .+    x, X
Allowed substitution hint:    F( x)

Proof of Theorem tgplacthmeo
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 tgptmd 17778 . . 3  |-  ( G  e.  TopGrp  ->  G  e. TopMnd )
2 tgplacthmeo.1 . . . 4  |-  F  =  ( x  e.  X  |->  ( A  .+  x
) )
3 tgplacthmeo.2 . . . 4  |-  X  =  ( Base `  G
)
4 tgplacthmeo.3 . . . 4  |-  .+  =  ( +g  `  G )
5 tgplacthmeo.4 . . . 4  |-  J  =  ( TopOpen `  G )
62, 3, 4, 5tmdlactcn 17801 . . 3  |-  ( ( G  e. TopMnd  /\  A  e.  X )  ->  F  e.  ( J  Cn  J
) )
71, 6sylan 457 . 2  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  F  e.  ( J  Cn  J
) )
8 tgpgrp 17777 . . . . . 6  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
9 eqid 2296 . . . . . . 7  |-  ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) )  =  ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) )
10 eqid 2296 . . . . . . 7  |-  ( inv g `  G )  =  ( inv g `  G )
119, 3, 4, 10grplactcnv 14580 . . . . . 6  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) ) `  A ) : X -1-1-onto-> X  /\  `' ( ( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x
) ) ) `  A )  =  ( ( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x
) ) ) `  ( ( inv g `  G ) `  A
) ) ) )
128, 11sylan 457 . . . . 5  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( ( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x ) ) ) `
 A ) : X -1-1-onto-> X  /\  `' ( ( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x
) ) ) `  A )  =  ( ( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x
) ) ) `  ( ( inv g `  G ) `  A
) ) ) )
1312simprd 449 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  `' ( ( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x ) ) ) `
 A )  =  ( ( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x ) ) ) `
 ( ( inv g `  G ) `
 A ) ) )
149, 3grplactfval 14578 . . . . . . 7  |-  ( A  e.  X  ->  (
( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x
) ) ) `  A )  =  ( x  e.  X  |->  ( A  .+  x ) ) )
1514adantl 452 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x
) ) ) `  A )  =  ( x  e.  X  |->  ( A  .+  x ) ) )
1615, 2syl6eqr 2346 . . . . 5  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x
) ) ) `  A )  =  F )
1716cnveqd 4873 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  `' ( ( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x ) ) ) `
 A )  =  `' F )
183, 10grpinvcl 14543 . . . . . 6  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( inv g `  G ) `  A
)  e.  X )
198, 18sylan 457 . . . . 5  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( inv g `  G ) `  A
)  e.  X )
209, 3grplactfval 14578 . . . . 5  |-  ( ( ( inv g `  G ) `  A
)  e.  X  -> 
( ( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x ) ) ) `
 ( ( inv g `  G ) `
 A ) )  =  ( x  e.  X  |->  ( ( ( inv g `  G
) `  A )  .+  x ) ) )
2119, 20syl 15 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x
) ) ) `  ( ( inv g `  G ) `  A
) )  =  ( x  e.  X  |->  ( ( ( inv g `  G ) `  A
)  .+  x )
) )
2213, 17, 213eqtr3d 2336 . . 3  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  `' F  =  ( x  e.  X  |->  ( ( ( inv g `  G ) `  A
)  .+  x )
) )
23 eqid 2296 . . . . . 6  |-  ( x  e.  X  |->  ( ( ( inv g `  G ) `  A
)  .+  x )
)  =  ( x  e.  X  |->  ( ( ( inv g `  G ) `  A
)  .+  x )
)
2423, 3, 4, 5tmdlactcn 17801 . . . . 5  |-  ( ( G  e. TopMnd  /\  (
( inv g `  G ) `  A
)  e.  X )  ->  ( x  e.  X  |->  ( ( ( inv g `  G
) `  A )  .+  x ) )  e.  ( J  Cn  J
) )
251, 24sylan 457 . . . 4  |-  ( ( G  e.  TopGrp  /\  (
( inv g `  G ) `  A
)  e.  X )  ->  ( x  e.  X  |->  ( ( ( inv g `  G
) `  A )  .+  x ) )  e.  ( J  Cn  J
) )
2619, 25syldan 456 . . 3  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
x  e.  X  |->  ( ( ( inv g `  G ) `  A
)  .+  x )
)  e.  ( J  Cn  J ) )
2722, 26eqeltrd 2370 . 2  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  `' F  e.  ( J  Cn  J ) )
28 ishmeo 17466 . 2  |-  ( F  e.  ( J  Homeo  J )  <->  ( F  e.  ( J  Cn  J
)  /\  `' F  e.  ( J  Cn  J
) ) )
297, 27, 28sylanbrc 645 1  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  F  e.  ( J  Homeo  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    e. cmpt 4093   `'ccnv 4704   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   TopOpenctopn 13342   Grpcgrp 14378   inv gcminusg 14379    Cn ccn 16970    Homeo chmeo 17460  TopMndctmd 17769   TopGrpctgp 17770
This theorem is referenced by:  subgntr  17805  opnsubg  17806  cldsubg  17809  tgpconcompeqg  17810  tgpconcomp  17811  snclseqg  17814  divstgpopn  17818  tsmsxplem1  17851
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-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cn 16973  df-cnp 16974  df-tx 17273  df-hmeo 17462  df-tmd 17771  df-tgp 17772
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