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Theorem tgplacthmeo 18056
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 18032 . . 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 18055 . . 3  |-  ( ( G  e. TopMnd  /\  A  e.  X )  ->  F  e.  ( J  Cn  J
) )
71, 6sylan 458 . 2  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  F  e.  ( J  Cn  J
) )
8 tgpgrp 18031 . . . . . 6  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
9 eqid 2389 . . . . . . 7  |-  ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) )  =  ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) )
10 eqid 2389 . . . . . . 7  |-  ( inv g `  G )  =  ( inv g `  G )
119, 3, 4, 10grplactcnv 14816 . . . . . 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 458 . . . . 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 450 . . . 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 14814 . . . . . . 7  |-  ( A  e.  X  ->  (
( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x
) ) ) `  A )  =  ( x  e.  X  |->  ( A  .+  x ) ) )
1514adantl 453 . . . . . 6  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x
) ) ) `  A )  =  ( x  e.  X  |->  ( A  .+  x ) ) )
1615, 2syl6eqr 2439 . . . . 5  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x
) ) ) `  A )  =  F )
1716cnveqd 4990 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  `' ( ( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x ) ) ) `
 A )  =  `' F )
183, 10grpinvcl 14779 . . . . . 6  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( inv g `  G ) `  A
)  e.  X )
198, 18sylan 458 . . . . 5  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( inv g `  G ) `  A
)  e.  X )
209, 3grplactfval 14814 . . . . 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 16 . . . 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 2429 . . 3  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  `' F  =  ( x  e.  X  |->  ( ( ( inv g `  G ) `  A
)  .+  x )
) )
23 eqid 2389 . . . . . 6  |-  ( x  e.  X  |->  ( ( ( inv g `  G ) `  A
)  .+  x )
)  =  ( x  e.  X  |->  ( ( ( inv g `  G ) `  A
)  .+  x )
)
2423, 3, 4, 5tmdlactcn 18055 . . . . 5  |-  ( ( G  e. TopMnd  /\  (
( inv g `  G ) `  A
)  e.  X )  ->  ( x  e.  X  |->  ( ( ( inv g `  G
) `  A )  .+  x ) )  e.  ( J  Cn  J
) )
251, 24sylan 458 . . . 4  |-  ( ( G  e.  TopGrp  /\  (
( inv g `  G ) `  A
)  e.  X )  ->  ( x  e.  X  |->  ( ( ( inv g `  G
) `  A )  .+  x ) )  e.  ( J  Cn  J
) )
2619, 25syldan 457 . . 3  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
x  e.  X  |->  ( ( ( inv g `  G ) `  A
)  .+  x )
)  e.  ( J  Cn  J ) )
2722, 26eqeltrd 2463 . 2  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  `' F  e.  ( J  Cn  J ) )
28 ishmeo 17714 . 2  |-  ( F  e.  ( J  Homeo  J )  <->  ( F  e.  ( J  Cn  J
)  /\  `' F  e.  ( J  Cn  J
) ) )
297, 27, 28sylanbrc 646 1  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  F  e.  ( J  Homeo  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    e. cmpt 4209   `'ccnv 4819   -1-1-onto->wf1o 5395   ` cfv 5396  (class class class)co 6022   Basecbs 13398   +g cplusg 13458   TopOpenctopn 13578   Grpcgrp 14614   inv gcminusg 14615    Cn ccn 17212    Homeo chmeo 17708  TopMndctmd 18023   TopGrpctgp 18024
This theorem is referenced by:  subgntr  18059  opnsubg  18060  cldsubg  18063  tgpconcompeqg  18064  tgpconcomp  18065  snclseqg  18068  divstgpopn  18072  tsmsxplem1  18105
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-map 6958  df-topgen 13596  df-0g 13656  df-mnd 14619  df-plusf 14620  df-grp 14741  df-minusg 14742  df-top 16888  df-bases 16890  df-topon 16891  df-topsp 16892  df-cn 17215  df-cnp 17216  df-tx 17517  df-hmeo 17710  df-tmd 18025  df-tgp 18026
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