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

Theorem tgplacthmeo 19672
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 19648 . . 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 19671 . . 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 19647 . . . . . 6  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
9 eqid 2441 . . . . . . 7  |-  ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) )  =  ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) )
10 eqid 2441 . . . . . . 7  |-  ( invg `  G )  =  ( invg `  G )
119, 3, 4, 10grplactcnv 15622 . . . . . 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 15620 . . . . . . 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 2491 . . . . 5  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  (
( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x
) ) ) `  A )  =  F )
1716cnveqd 5013 . . . 4  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  `' ( ( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x ) ) ) `
 A )  =  `' F )
183, 10grpinvcl 15581 . . . . . 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 15620 . . . . 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 2481 . . 3  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  `' F  =  ( x  e.  X  |->  ( ( ( invg `  G ) `  A
)  .+  x )
) )
23 eqid 2441 . . . . . 6  |-  ( x  e.  X  |->  ( ( ( invg `  G ) `  A
)  .+  x )
)  =  ( x  e.  X  |->  ( ( ( invg `  G ) `  A
)  .+  x )
)
2423, 3, 4, 5tmdlactcn 19671 . . . . 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 2515 . 2  |-  ( ( G  e.  TopGrp  /\  A  e.  X )  ->  `' F  e.  ( J  Cn  J ) )
28 ishmeo 19330 . 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 1369    e. wcel 1756    e. cmpt 4348   `'ccnv 4837   -1-1-onto->wf1o 5415   ` cfv 5416  (class class class)co 6089   Basecbs 14172   +g cplusg 14236   TopOpenctopn 14358   Grpcgrp 15408   invgcminusg 15409    Cn ccn 18826   Homeochmeo 19324  TopMndctmd 19639   TopGrpctgp 19640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-1st 6575  df-2nd 6576  df-map 7214  df-0g 14378  df-topgen 14380  df-mnd 15413  df-plusf 15414  df-grp 15543  df-minusg 15544  df-top 18501  df-bases 18503  df-topon 18504  df-topsp 18505  df-cn 18829  df-cnp 18830  df-tx 19133  df-hmeo 19326  df-tmd 19641  df-tgp 19642
This theorem is referenced by:  subgntr  19675  opnsubg  19676  cldsubg  19679  tgpconcompeqg  19680  tgpconcomp  19681  snclseqg  19684  divstgpopn  19688  tsmsxplem1  19725
  Copyright terms: Public domain W3C validator