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Theorem tgplacthmeo 20334
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 20310 . . 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 20333 . . 3  |-  ( ( G  e. TopMnd  /\  A  e.  X )  ->  F  e.  ( J  Cn  J
) )
71, 6sylan 471 . 2  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  F  e.  ( J  Cn  J
) )
8 tgpgrp 20309 . . . . . 6  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
9 eqid 2467 . . . . . . 7  |-  ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) )  =  ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) )
10 eqid 2467 . . . . . . 7  |-  ( invg `  G )  =  ( invg `  G )
119, 3, 4, 10grplactcnv 15936 . . . . . 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
) ) ) `  ( ( invg `  G ) `  A
) ) ) )
128, 11sylan 471 . . . . 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
) ) ) `  ( ( invg `  G ) `  A
) ) ) )
1312simprd 463 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  `' ( ( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x ) ) ) `
 A )  =  ( ( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x ) ) ) `
 ( ( invg `  G ) `
 A ) ) )
149, 3grplactfval 15934 . . . . . . 7  |-  ( A  e.  X  ->  (
( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x
) ) ) `  A )  =  ( x  e.  X  |->  ( A  .+  x ) ) )
1514adantl 466 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x
) ) ) `  A )  =  ( x  e.  X  |->  ( A  .+  x ) ) )
1615, 2syl6eqr 2526 . . . . 5  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x
) ) ) `  A )  =  F )
1716cnveqd 5176 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  `' ( ( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x ) ) ) `
 A )  =  `' F )
183, 10grpinvcl 15893 . . . . . 6  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( invg `  G ) `  A
)  e.  X )
198, 18sylan 471 . . . . 5  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( invg `  G ) `  A
)  e.  X )
209, 3grplactfval 15934 . . . . 5  |-  ( ( ( invg `  G ) `  A
)  e.  X  -> 
( ( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x ) ) ) `
 ( ( invg `  G ) `
 A ) )  =  ( x  e.  X  |->  ( ( ( invg `  G
) `  A )  .+  x ) ) )
2119, 20syl 16 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x
) ) ) `  ( ( invg `  G ) `  A
) )  =  ( x  e.  X  |->  ( ( ( invg `  G ) `  A
)  .+  x )
) )
2213, 17, 213eqtr3d 2516 . . 3  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  `' F  =  ( x  e.  X  |->  ( ( ( invg `  G ) `  A
)  .+  x )
) )
23 eqid 2467 . . . . . 6  |-  ( x  e.  X  |->  ( ( ( invg `  G ) `  A
)  .+  x )
)  =  ( x  e.  X  |->  ( ( ( invg `  G ) `  A
)  .+  x )
)
2423, 3, 4, 5tmdlactcn 20333 . . . . 5  |-  ( ( G  e. TopMnd  /\  (
( invg `  G ) `  A
)  e.  X )  ->  ( x  e.  X  |->  ( ( ( invg `  G
) `  A )  .+  x ) )  e.  ( J  Cn  J
) )
251, 24sylan 471 . . . 4  |-  ( ( G  e.  TopGrp  /\  (
( invg `  G ) `  A
)  e.  X )  ->  ( x  e.  X  |->  ( ( ( invg `  G
) `  A )  .+  x ) )  e.  ( J  Cn  J
) )
2619, 25syldan 470 . . 3  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
x  e.  X  |->  ( ( ( invg `  G ) `  A
)  .+  x )
)  e.  ( J  Cn  J ) )
2722, 26eqeltrd 2555 . 2  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  `' F  e.  ( J  Cn  J ) )
28 ishmeo 19992 . 2  |-  ( F  e.  ( J Homeo J )  <->  ( F  e.  ( J  Cn  J
)  /\  `' F  e.  ( J  Cn  J
) ) )
297, 27, 28sylanbrc 664 1  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  F  e.  ( J Homeo J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    |-> cmpt 4505   `'ccnv 4998   -1-1-onto->wf1o 5585   ` cfv 5586  (class class class)co 6282   Basecbs 14483   +g cplusg 14548   TopOpenctopn 14670   Grpcgrp 15720   invgcminusg 15721    Cn ccn 19488   Homeochmeo 19986  TopMndctmd 20301   TopGrpctgp 20302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-map 7419  df-0g 14690  df-topgen 14692  df-mnd 15725  df-plusf 15726  df-grp 15855  df-minusg 15856  df-top 19163  df-bases 19165  df-topon 19166  df-topsp 19167  df-cn 19491  df-cnp 19492  df-tx 19795  df-hmeo 19988  df-tmd 20303  df-tgp 20304
This theorem is referenced by:  subgntr  20337  opnsubg  20338  cldsubg  20341  tgpconcompeqg  20342  tgpconcomp  20343  snclseqg  20346  divstgpopn  20350  tsmsxplem1  20387
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