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Theorem eqglact 15725
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 2441 . . . . . . 7  |-  ( invg `  G )  =  ( invg `  G )
3 eqglact.3 . . . . . . 7  |-  .+  =  ( +g  `  G )
4 eqger.r . . . . . . 7  |-  .~  =  ( G ~QG  Y )
51, 2, 3, 4eqgval 15723 . . . . . 6  |-  ( ( G  e.  Grp  /\  Y  C_  X )  -> 
( A  .~  x  <->  ( A  e.  X  /\  x  e.  X  /\  ( ( ( invg `  G ) `
 A )  .+  x )  e.  Y
) ) )
6 3anass 964 . . . . . 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 895 . . . 4  |-  ( ( ( G  e.  Grp  /\  Y  C_  X )  /\  A  e.  X
)  ->  ( A  .~  x  <->  ( x  e.  X  /\  ( ( ( invg `  G ) `  A
)  .+  x )  e.  Y ) ) )
983impa 1177 . . 3  |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  A  e.  X )  ->  ( A  .~  x  <->  ( x  e.  X  /\  (
( ( invg `  G ) `  A
)  .+  x )  e.  Y ) ) )
109abbidv 2555 . 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 7100 . . 3  |-  ( A  e.  X  ->  [ A ]  .~  =  { x  |  A  .~  x } )
12113ad2ant3 1006 . 2  |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  A  e.  X )  ->  [ A ]  .~  =  { x  |  A  .~  x } )
13 eqid 2441 . . . . . . . . 9  |-  ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) )  =  ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) )
1413, 1, 3, 2grplactcnv 15617 . . . . . . . 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 460 . . . . . . 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 15615 . . . . . . . . 9  |-  ( A  e.  X  ->  (
( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x
) ) ) `  A )  =  ( x  e.  X  |->  ( A  .+  x ) ) )
1716adantl 463 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x ) ) ) `
 A )  =  ( x  e.  X  |->  ( A  .+  x
) ) )
1817cnveqd 5011 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  `' ( ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) ) `  A )  =  `' ( x  e.  X  |->  ( A 
.+  x ) ) )
191, 2grpinvcl 15576 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( invg `  G ) `  A
)  e.  X )
2013, 1grplactfval 15615 . . . . . . . 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 2481 . . . . . 6  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  `' ( x  e.  X  |->  ( A  .+  x ) )  =  ( x  e.  X  |->  ( ( ( invg `  G ) `
 A )  .+  x ) ) )
2322cnveqd 5011 . . . . 5  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  `' `' ( x  e.  X  |->  ( A  .+  x ) )  =  `' ( x  e.  X  |->  ( ( ( invg `  G
) `  A )  .+  x ) ) )
24233adant2 1002 . . . 4  |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  A  e.  X )  ->  `' `' ( x  e.  X  |->  ( A  .+  x ) )  =  `' ( x  e.  X  |->  ( ( ( invg `  G
) `  A )  .+  x ) ) )
2524imaeq1d 5165 . . 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 5300 . . 3  |-  ( `' `' ( x  e.  X  |->  ( A  .+  x ) ) " Y )  =  ( ( x  e.  X  |->  ( A  .+  x
) ) " Y
)
27 eqid 2441 . . . . 5  |-  ( x  e.  X  |->  ( ( ( invg `  G ) `  A
)  .+  x )
)  =  ( x  e.  X  |->  ( ( ( invg `  G ) `  A
)  .+  x )
)
2827mptpreima 5328 . . . 4  |-  ( `' ( x  e.  X  |->  ( ( ( invg `  G ) `
 A )  .+  x ) ) " Y )  =  {
x  e.  X  | 
( ( ( invg `  G ) `
 A )  .+  x )  e.  Y }
29 df-rab 2722 . . . 4  |-  { x  e.  X  |  (
( ( invg `  G ) `  A
)  .+  x )  e.  Y }  =  {
x  |  ( x  e.  X  /\  (
( ( invg `  G ) `  A
)  .+  x )  e.  Y ) }
3028, 29eqtri 2461 . . 3  |-  ( `' ( x  e.  X  |->  ( ( ( invg `  G ) `
 A )  .+  x ) ) " Y )  =  {
x  |  ( x  e.  X  /\  (
( ( invg `  G ) `  A
)  .+  x )  e.  Y ) }
3125, 26, 303eqtr3g 2496 . 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 2483 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 960    = wceq 1364    e. wcel 1761   {cab 2427   {crab 2717    C_ wss 3325   class class class wbr 4289    e. cmpt 4347   `'ccnv 4835   "cima 4839   -1-1-onto->wf1o 5414   ` cfv 5415  (class class class)co 6090   [cec 7095   Basecbs 14170   +g cplusg 14234   Grpcgrp 15406   invgcminusg 15407   ~QG cqg 15670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  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 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-ec 7099  df-0g 14376  df-mnd 15411  df-grp 15538  df-minusg 15539  df-eqg 15673
This theorem is referenced by:  eqgen  15727  cldsubg  19640  tgpconcompeqg  19641  snclseqg  19645
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