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Theorem eqglact 16124
Description: A left coset can be expressed as the image of a left action. (Contributed by Mario Carneiro, 20-Sep-2015.)
Hypotheses
Ref Expression
eqger.x  |-  X  =  ( Base `  G
)
eqger.r  |-  .~  =  ( G ~QG  Y )
eqglact.3  |-  .+  =  ( +g  `  G )
Assertion
Ref Expression
eqglact  |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  A  e.  X )  ->  [ A ]  .~  =  ( ( x  e.  X  |->  ( A  .+  x ) ) " Y ) )
Distinct variable groups:    x,  .+    x, 
.~    x, G    x, X    x, A    x, Y

Proof of Theorem eqglact
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 eqger.x . . . . . . 7  |-  X  =  ( Base `  G
)
2 eqid 2467 . . . . . . 7  |-  ( invg `  G )  =  ( invg `  G )
3 eqglact.3 . . . . . . 7  |-  .+  =  ( +g  `  G )
4 eqger.r . . . . . . 7  |-  .~  =  ( G ~QG  Y )
51, 2, 3, 4eqgval 16122 . . . . . 6  |-  ( ( G  e.  Grp  /\  Y  C_  X )  -> 
( A  .~  x  <->  ( A  e.  X  /\  x  e.  X  /\  ( ( ( invg `  G ) `
 A )  .+  x )  e.  Y
) ) )
6 3anass 977 . . . . . 6  |-  ( ( A  e.  X  /\  x  e.  X  /\  ( ( ( invg `  G ) `
 A )  .+  x )  e.  Y
)  <->  ( A  e.  X  /\  ( x  e.  X  /\  (
( ( invg `  G ) `  A
)  .+  x )  e.  Y ) ) )
75, 6syl6bb 261 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  C_  X )  -> 
( A  .~  x  <->  ( A  e.  X  /\  ( x  e.  X  /\  ( ( ( invg `  G ) `
 A )  .+  x )  e.  Y
) ) ) )
87baibd 907 . . . 4  |-  ( ( ( G  e.  Grp  /\  Y  C_  X )  /\  A  e.  X
)  ->  ( A  .~  x  <->  ( x  e.  X  /\  ( ( ( invg `  G ) `  A
)  .+  x )  e.  Y ) ) )
983impa 1191 . . 3  |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  A  e.  X )  ->  ( A  .~  x  <->  ( x  e.  X  /\  (
( ( invg `  G ) `  A
)  .+  x )  e.  Y ) ) )
109abbidv 2603 . 2  |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  A  e.  X )  ->  { x  |  A  .~  x }  =  { x  |  ( x  e.  X  /\  ( ( ( invg `  G ) `  A
)  .+  x )  e.  Y ) } )
11 dfec2 7326 . . 3  |-  ( A  e.  X  ->  [ A ]  .~  =  { x  |  A  .~  x } )
12113ad2ant3 1019 . 2  |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  A  e.  X )  ->  [ A ]  .~  =  { x  |  A  .~  x } )
13 eqid 2467 . . . . . . . . 9  |-  ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) )  =  ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) )
1413, 1, 3, 2grplactcnv 16010 . . . . . . . 8  |-  ( ( 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
) ) ) )
1514simprd 463 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  `' ( ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) ) `  A )  =  ( ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) ) `  ( ( invg `  G
) `  A )
) )
1613, 1grplactfval 16008 . . . . . . . . 9  |-  ( A  e.  X  ->  (
( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x
) ) ) `  A )  =  ( x  e.  X  |->  ( A  .+  x ) ) )
1716adantl 466 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x ) ) ) `
 A )  =  ( x  e.  X  |->  ( A  .+  x
) ) )
1817cnveqd 5184 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  `' ( ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) ) `  A )  =  `' ( x  e.  X  |->  ( A 
.+  x ) ) )
191, 2grpinvcl 15967 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( invg `  G ) `  A
)  e.  X )
2013, 1grplactfval 16008 . . . . . . . 8  |-  ( ( ( 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 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x ) ) ) `
 ( ( invg `  G ) `
 A ) )  =  ( x  e.  X  |->  ( ( ( invg `  G
) `  A )  .+  x ) ) )
2215, 18, 213eqtr3d 2516 . . . . . 6  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  `' ( x  e.  X  |->  ( A  .+  x ) )  =  ( x  e.  X  |->  ( ( ( invg `  G ) `
 A )  .+  x ) ) )
2322cnveqd 5184 . . . . 5  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  `' `' ( x  e.  X  |->  ( A  .+  x ) )  =  `' ( x  e.  X  |->  ( ( ( invg `  G
) `  A )  .+  x ) ) )
24233adant2 1015 . . . 4  |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  A  e.  X )  ->  `' `' ( x  e.  X  |->  ( A  .+  x ) )  =  `' ( x  e.  X  |->  ( ( ( invg `  G
) `  A )  .+  x ) ) )
2524imaeq1d 5342 . . 3  |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  A  e.  X )  ->  ( `' `' ( x  e.  X  |->  ( A  .+  x ) ) " Y )  =  ( `' ( x  e.  X  |->  ( ( ( invg `  G
) `  A )  .+  x ) ) " Y ) )
26 imacnvcnv 5478 . . 3  |-  ( `' `' ( x  e.  X  |->  ( A  .+  x ) ) " Y )  =  ( ( x  e.  X  |->  ( A  .+  x
) ) " Y
)
27 eqid 2467 . . . . 5  |-  ( x  e.  X  |->  ( ( ( invg `  G ) `  A
)  .+  x )
)  =  ( x  e.  X  |->  ( ( ( invg `  G ) `  A
)  .+  x )
)
2827mptpreima 5506 . . . 4  |-  ( `' ( x  e.  X  |->  ( ( ( invg `  G ) `
 A )  .+  x ) ) " Y )  =  {
x  e.  X  | 
( ( ( invg `  G ) `
 A )  .+  x )  e.  Y }
29 df-rab 2826 . . . 4  |-  { x  e.  X  |  (
( ( invg `  G ) `  A
)  .+  x )  e.  Y }  =  {
x  |  ( x  e.  X  /\  (
( ( invg `  G ) `  A
)  .+  x )  e.  Y ) }
3028, 29eqtri 2496 . . 3  |-  ( `' ( x  e.  X  |->  ( ( ( invg `  G ) `
 A )  .+  x ) ) " Y )  =  {
x  |  ( x  e.  X  /\  (
( ( invg `  G ) `  A
)  .+  x )  e.  Y ) }
3125, 26, 303eqtr3g 2531 . 2  |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( A  .+  x
) ) " Y
)  =  { x  |  ( x  e.  X  /\  ( ( ( invg `  G ) `  A
)  .+  x )  e.  Y ) } )
3210, 12, 313eqtr4d 2518 1  |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  A  e.  X )  ->  [ A ]  .~  =  ( ( x  e.  X  |->  ( A  .+  x ) ) " Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {cab 2452   {crab 2821    C_ wss 3481   class class class wbr 4453    |-> cmpt 4511   `'ccnv 5004   "cima 5008   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6295   [cec 7321   Basecbs 14507   +g cplusg 14572   Grpcgrp 15925   invgcminusg 15926   ~QG cqg 16069
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-ec 7325  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-eqg 16072
This theorem is referenced by:  eqgen  16126  cldsubg  20477  tgpconcompeqg  20478  snclseqg  20482
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