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Theorem eqglact 14684
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 2296 . . . . . . 7  |-  ( inv g `  G )  =  ( inv g `  G )
3 eqglact.3 . . . . . . 7  |-  .+  =  ( +g  `  G )
4 eqger.r . . . . . . 7  |-  .~  =  ( G ~QG  Y )
51, 2, 3, 4eqgval 14682 . . . . . 6  |-  ( ( G  e.  Grp  /\  Y  C_  X )  -> 
( A  .~  x  <->  ( A  e.  X  /\  x  e.  X  /\  ( ( ( inv g `  G ) `
 A )  .+  x )  e.  Y
) ) )
6 3anass 938 . . . . . 6  |-  ( ( A  e.  X  /\  x  e.  X  /\  ( ( ( inv g `  G ) `
 A )  .+  x )  e.  Y
)  <->  ( A  e.  X  /\  ( x  e.  X  /\  (
( ( inv g `  G ) `  A
)  .+  x )  e.  Y ) ) )
75, 6syl6bb 252 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  C_  X )  -> 
( A  .~  x  <->  ( A  e.  X  /\  ( x  e.  X  /\  ( ( ( inv g `  G ) `
 A )  .+  x )  e.  Y
) ) ) )
87baibd 875 . . . 4  |-  ( ( ( G  e.  Grp  /\  Y  C_  X )  /\  A  e.  X
)  ->  ( A  .~  x  <->  ( x  e.  X  /\  ( ( ( inv g `  G ) `  A
)  .+  x )  e.  Y ) ) )
983impa 1146 . . 3  |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  A  e.  X )  ->  ( A  .~  x  <->  ( x  e.  X  /\  (
( ( inv g `  G ) `  A
)  .+  x )  e.  Y ) ) )
109abbidv 2410 . 2  |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  A  e.  X )  ->  { x  |  A  .~  x }  =  { x  |  ( x  e.  X  /\  ( ( ( inv g `  G ) `  A
)  .+  x )  e.  Y ) } )
11 dfec2 6679 . . 3  |-  ( A  e.  X  ->  [ A ]  .~  =  { x  |  A  .~  x } )
12113ad2ant3 978 . 2  |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  A  e.  X )  ->  [ A ]  .~  =  { x  |  A  .~  x } )
13 eqid 2296 . . . . . . . . 9  |-  ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) )  =  ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) )
1413, 1, 3, 2grplactcnv 14580 . . . . . . . 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
) ) ) `  ( ( inv g `  G ) `  A
) ) ) )
1514simprd 449 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  `' ( ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) ) `  A )  =  ( ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) ) `  ( ( inv g `  G
) `  A )
) )
1613, 1grplactfval 14578 . . . . . . . . 9  |-  ( A  e.  X  ->  (
( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x
) ) ) `  A )  =  ( x  e.  X  |->  ( A  .+  x ) ) )
1716adantl 452 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x ) ) ) `
 A )  =  ( x  e.  X  |->  ( A  .+  x
) ) )
1817cnveqd 4873 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  `' ( ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) ) `  A )  =  `' ( x  e.  X  |->  ( A 
.+  x ) ) )
191, 2grpinvcl 14543 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( inv g `  G ) `  A
)  e.  X )
2013, 1grplactfval 14578 . . . . . . . 8  |-  ( ( ( 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 15 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x ) ) ) `
 ( ( inv g `  G ) `
 A ) )  =  ( x  e.  X  |->  ( ( ( inv g `  G
) `  A )  .+  x ) ) )
2215, 18, 213eqtr3d 2336 . . . . . 6  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  `' ( x  e.  X  |->  ( A  .+  x ) )  =  ( x  e.  X  |->  ( ( ( inv g `  G ) `
 A )  .+  x ) ) )
2322cnveqd 4873 . . . . 5  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  `' `' ( x  e.  X  |->  ( A  .+  x ) )  =  `' ( x  e.  X  |->  ( ( ( inv g `  G
) `  A )  .+  x ) ) )
24233adant2 974 . . . 4  |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  A  e.  X )  ->  `' `' ( x  e.  X  |->  ( A  .+  x ) )  =  `' ( x  e.  X  |->  ( ( ( inv g `  G
) `  A )  .+  x ) ) )
2524imaeq1d 5027 . . 3  |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  A  e.  X )  ->  ( `' `' ( x  e.  X  |->  ( A  .+  x ) ) " Y )  =  ( `' ( x  e.  X  |->  ( ( ( inv g `  G
) `  A )  .+  x ) ) " Y ) )
26 imacnvcnv 5153 . . 3  |-  ( `' `' ( x  e.  X  |->  ( A  .+  x ) ) " Y )  =  ( ( x  e.  X  |->  ( A  .+  x
) ) " Y
)
27 eqid 2296 . . . . 5  |-  ( x  e.  X  |->  ( ( ( inv g `  G ) `  A
)  .+  x )
)  =  ( x  e.  X  |->  ( ( ( inv g `  G ) `  A
)  .+  x )
)
2827mptpreima 5182 . . . 4  |-  ( `' ( x  e.  X  |->  ( ( ( inv g `  G ) `
 A )  .+  x ) ) " Y )  =  {
x  e.  X  | 
( ( ( inv g `  G ) `
 A )  .+  x )  e.  Y }
29 df-rab 2565 . . . 4  |-  { x  e.  X  |  (
( ( inv g `  G ) `  A
)  .+  x )  e.  Y }  =  {
x  |  ( x  e.  X  /\  (
( ( inv g `  G ) `  A
)  .+  x )  e.  Y ) }
3028, 29eqtri 2316 . . 3  |-  ( `' ( x  e.  X  |->  ( ( ( inv g `  G ) `
 A )  .+  x ) ) " Y )  =  {
x  |  ( x  e.  X  /\  (
( ( inv g `  G ) `  A
)  .+  x )  e.  Y ) }
3125, 26, 303eqtr3g 2351 . 2  |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( A  .+  x
) ) " Y
)  =  { x  |  ( x  e.  X  /\  ( ( ( inv g `  G ) `  A
)  .+  x )  e.  Y ) } )
3210, 12, 313eqtr4d 2338 1  |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  A  e.  X )  ->  [ A ]  .~  =  ( ( x  e.  X  |->  ( A  .+  x ) ) " Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {cab 2282   {crab 2560    C_ wss 3165   class class class wbr 4039    e. cmpt 4093   `'ccnv 4704   "cima 4708   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   [cec 6674   Basecbs 13164   +g cplusg 13224   Grpcgrp 14378   inv gcminusg 14379   ~QG cqg 14633
This theorem is referenced by:  eqgen  14686  cldsubg  17809  tgpconcompeqg  17810  snclseqg  17814
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-ec 6678  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-eqg 14636
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