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Theorem eqglact 13015
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 2177 . . . . . . 7  |-  ( invg `  G )  =  ( invg `  G )
3 eqglact.3 . . . . . . 7  |-  .+  =  ( +g  `  G )
4 eqger.r . . . . . . 7  |-  .~  =  ( G ~QG  Y )
51, 2, 3, 4eqgval 13013 . . . . . 6  |-  ( ( G  e.  Grp  /\  Y  C_  X )  -> 
( A  .~  x  <->  ( A  e.  X  /\  x  e.  X  /\  ( ( ( invg `  G ) `
 A )  .+  x )  e.  Y
) ) )
6 3anass 982 . . . . . 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, 6bitrdi 196 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  C_  X )  -> 
( A  .~  x  <->  ( A  e.  X  /\  ( x  e.  X  /\  ( ( ( invg `  G ) `
 A )  .+  x )  e.  Y
) ) ) )
87baibd 923 . . . 4  |-  ( ( ( G  e.  Grp  /\  Y  C_  X )  /\  A  e.  X
)  ->  ( A  .~  x  <->  ( x  e.  X  /\  ( ( ( invg `  G ) `  A
)  .+  x )  e.  Y ) ) )
983impa 1194 . . 3  |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  A  e.  X )  ->  ( A  .~  x  <->  ( x  e.  X  /\  (
( ( invg `  G ) `  A
)  .+  x )  e.  Y ) ) )
109abbidv 2295 . 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 6535 . . 3  |-  ( A  e.  X  ->  [ A ]  .~  =  { x  |  A  .~  x } )
12113ad2ant3 1020 . 2  |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  A  e.  X )  ->  [ A ]  .~  =  { x  |  A  .~  x } )
13 eqid 2177 . . . . . . . . 9  |-  ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) )  =  ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) )
1413, 1, 3, 2grplactcnv 12904 . . . . . . . 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 114 . . . . . . 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 12903 . . . . . . . . 9  |-  ( A  e.  X  ->  (
( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x
) ) ) `  A )  =  ( x  e.  X  |->  ( A  .+  x ) ) )
1716adantl 277 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( g  e.  X  |->  ( x  e.  X  |->  ( g  .+  x ) ) ) `
 A )  =  ( x  e.  X  |->  ( A  .+  x
) ) )
1817cnveqd 4802 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  `' ( ( g  e.  X  |->  ( x  e.  X  |->  ( g 
.+  x ) ) ) `  A )  =  `' ( x  e.  X  |->  ( A 
.+  x ) ) )
191, 2grpinvcl 12853 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( invg `  G ) `  A
)  e.  X )
2013, 1grplactfval 12903 . . . . . . . 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 14 . . . . . . 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 2218 . . . . . 6  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  `' ( x  e.  X  |->  ( A  .+  x ) )  =  ( x  e.  X  |->  ( ( ( invg `  G ) `
 A )  .+  x ) ) )
2322cnveqd 4802 . . . . 5  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  `' `' ( x  e.  X  |->  ( A  .+  x ) )  =  `' ( x  e.  X  |->  ( ( ( invg `  G
) `  A )  .+  x ) ) )
24233adant2 1016 . . . 4  |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  A  e.  X )  ->  `' `' ( x  e.  X  |->  ( A  .+  x ) )  =  `' ( x  e.  X  |->  ( ( ( invg `  G
) `  A )  .+  x ) ) )
2524imaeq1d 4968 . . 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 5092 . . 3  |-  ( `' `' ( x  e.  X  |->  ( A  .+  x ) ) " Y )  =  ( ( x  e.  X  |->  ( A  .+  x
) ) " Y
)
27 eqid 2177 . . . . 5  |-  ( x  e.  X  |->  ( ( ( invg `  G ) `  A
)  .+  x )
)  =  ( x  e.  X  |->  ( ( ( invg `  G ) `  A
)  .+  x )
)
2827mptpreima 5121 . . . 4  |-  ( `' ( x  e.  X  |->  ( ( ( invg `  G ) `
 A )  .+  x ) ) " Y )  =  {
x  e.  X  | 
( ( ( invg `  G ) `
 A )  .+  x )  e.  Y }
29 df-rab 2464 . . . 4  |-  { x  e.  X  |  (
( ( invg `  G ) `  A
)  .+  x )  e.  Y }  =  {
x  |  ( x  e.  X  /\  (
( ( invg `  G ) `  A
)  .+  x )  e.  Y ) }
3028, 29eqtri 2198 . . 3  |-  ( `' ( x  e.  X  |->  ( ( ( invg `  G ) `
 A )  .+  x ) ) " Y )  =  {
x  |  ( x  e.  X  /\  (
( ( invg `  G ) `  A
)  .+  x )  e.  Y ) }
3125, 26, 303eqtr3g 2233 . 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 2220 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    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   {cab 2163   {crab 2459    C_ wss 3129   class class class wbr 4002    |-> cmpt 4063   `'ccnv 4624   "cima 4628   -1-1-onto->wf1o 5214   ` cfv 5215  (class class class)co 5872   [cec 6530   Basecbs 12454   +g cplusg 12528   Grpcgrp 12809   invgcminusg 12810   ~QG cqg 12960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1re 7902  ax-addrcl 7905
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-ec 6534  df-inn 8916  df-2 8974  df-ndx 12457  df-slot 12458  df-base 12460  df-plusg 12541  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812  df-minusg 12813  df-eqg 12963
This theorem is referenced by:  eqgen  13017
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