MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tgplacthmeo Structured version   Unicode version

Theorem tgplacthmeo 18125
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 18101 . . 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 18124 . . 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 18100 . . . . . 6  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
9 eqid 2435 . . . . . . 7  |-  ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) )  =  ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) )
10 eqid 2435 . . . . . . 7  |-  ( inv g `  G )  =  ( inv g `  G )
119, 3, 4, 10grplactcnv 14879 . . . . . 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 14877 . . . . . . 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 2485 . . . . 5  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x
) ) ) `  A )  =  F )
1716cnveqd 5040 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  `' ( ( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x ) ) ) `
 A )  =  `' F )
183, 10grpinvcl 14842 . . . . . 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 14877 . . . . 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 2475 . . 3  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  `' F  =  ( x  e.  X  |->  ( ( ( inv g `  G ) `  A
)  .+  x )
) )
23 eqid 2435 . . . . . 6  |-  ( x  e.  X  |->  ( ( ( inv g `  G ) `  A
)  .+  x )
)  =  ( x  e.  X  |->  ( ( ( inv g `  G ) `  A
)  .+  x )
)
2423, 3, 4, 5tmdlactcn 18124 . . . . 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 2509 . 2  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  `' F  e.  ( J  Cn  J ) )
28 ishmeo 17783 . 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 1652    e. wcel 1725    e. cmpt 4258   `'ccnv 4869   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521   TopOpenctopn 13641   Grpcgrp 14677   inv gcminusg 14678    Cn ccn 17280    Homeo chmeo 17777  TopMndctmd 18092   TopGrpctgp 18093
This theorem is referenced by:  subgntr  18128  opnsubg  18129  cldsubg  18132  tgpconcompeqg  18133  tgpconcomp  18134  snclseqg  18137  divstgpopn  18141  tsmsxplem1  18174
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-map 7012  df-topgen 13659  df-0g 13719  df-mnd 14682  df-plusf 14683  df-grp 14804  df-minusg 14805  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cn 17283  df-cnp 17284  df-tx 17586  df-hmeo 17779  df-tmd 18094  df-tgp 18095
  Copyright terms: Public domain W3C validator