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Theorem eqglact 15735
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 2443 . . . . . . 7  |-  ( invg `  G )  =  ( invg `  G )
3 eqglact.3 . . . . . . 7  |-  .+  =  ( +g  `  G )
4 eqger.r . . . . . . 7  |-  .~  =  ( G ~QG  Y )
51, 2, 3, 4eqgval 15733 . . . . . 6  |-  ( ( G  e.  Grp  /\  Y  C_  X )  -> 
( A  .~  x  <->  ( A  e.  X  /\  x  e.  X  /\  ( ( ( invg `  G ) `
 A )  .+  x )  e.  Y
) ) )
6 3anass 969 . . . . . 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 900 . . . 4  |-  ( ( ( G  e.  Grp  /\  Y  C_  X )  /\  A  e.  X
)  ->  ( A  .~  x  <->  ( x  e.  X  /\  ( ( ( invg `  G ) `  A
)  .+  x )  e.  Y ) ) )
983impa 1182 . . 3  |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  A  e.  X )  ->  ( A  .~  x  <->  ( x  e.  X  /\  (
( ( invg `  G ) `  A
)  .+  x )  e.  Y ) ) )
109abbidv 2560 . 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 7107 . . 3  |-  ( A  e.  X  ->  [ A ]  .~  =  { x  |  A  .~  x } )
12113ad2ant3 1011 . 2  |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  A  e.  X )  ->  [ A ]  .~  =  { x  |  A  .~  x } )
13 eqid 2443 . . . . . . . . 9  |-  ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) )  =  ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) )
1413, 1, 3, 2grplactcnv 15627 . . . . . . . 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 15625 . . . . . . . . 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 5018 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  `' ( ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) ) `  A )  =  `' ( x  e.  X  |->  ( A 
.+  x ) ) )
191, 2grpinvcl 15586 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( invg `  G ) `  A
)  e.  X )
2013, 1grplactfval 15625 . . . . . . . 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 2483 . . . . . 6  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  `' ( x  e.  X  |->  ( A  .+  x ) )  =  ( x  e.  X  |->  ( ( ( invg `  G ) `
 A )  .+  x ) ) )
2322cnveqd 5018 . . . . 5  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  `' `' ( x  e.  X  |->  ( A  .+  x ) )  =  `' ( x  e.  X  |->  ( ( ( invg `  G
) `  A )  .+  x ) ) )
24233adant2 1007 . . . 4  |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  A  e.  X )  ->  `' `' ( x  e.  X  |->  ( A  .+  x ) )  =  `' ( x  e.  X  |->  ( ( ( invg `  G
) `  A )  .+  x ) ) )
2524imaeq1d 5171 . . 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 5306 . . 3  |-  ( `' `' ( x  e.  X  |->  ( A  .+  x ) ) " Y )  =  ( ( x  e.  X  |->  ( A  .+  x
) ) " Y
)
27 eqid 2443 . . . . 5  |-  ( x  e.  X  |->  ( ( ( invg `  G ) `  A
)  .+  x )
)  =  ( x  e.  X  |->  ( ( ( invg `  G ) `  A
)  .+  x )
)
2827mptpreima 5334 . . . 4  |-  ( `' ( x  e.  X  |->  ( ( ( invg `  G ) `
 A )  .+  x ) ) " Y )  =  {
x  e.  X  | 
( ( ( invg `  G ) `
 A )  .+  x )  e.  Y }
29 df-rab 2727 . . . 4  |-  { x  e.  X  |  (
( ( invg `  G ) `  A
)  .+  x )  e.  Y }  =  {
x  |  ( x  e.  X  /\  (
( ( invg `  G ) `  A
)  .+  x )  e.  Y ) }
3028, 29eqtri 2463 . . 3  |-  ( `' ( x  e.  X  |->  ( ( ( invg `  G ) `
 A )  .+  x ) ) " Y )  =  {
x  |  ( x  e.  X  /\  (
( ( invg `  G ) `  A
)  .+  x )  e.  Y ) }
3125, 26, 303eqtr3g 2498 . 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 2485 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 965    = wceq 1369    e. wcel 1756   {cab 2429   {crab 2722    C_ wss 3331   class class class wbr 4295    e. cmpt 4353   `'ccnv 4842   "cima 4846   -1-1-onto->wf1o 5420   ` cfv 5421  (class class class)co 6094   [cec 7102   Basecbs 14177   +g cplusg 14241   Grpcgrp 15413   invgcminusg 15414   ~QG cqg 15680
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 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-id 4639  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-ec 7106  df-0g 14383  df-mnd 15418  df-grp 15548  df-minusg 15549  df-eqg 15683
This theorem is referenced by:  eqgen  15737  cldsubg  19684  tgpconcompeqg  19685  snclseqg  19689
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