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Theorem eqgen 14670
Description: Each coset is equipotent to the subgroup itself (which is also the coset containing the identity). (Contributed by Mario Carneiro, 20-Sep-2015.)
Hypotheses
Ref Expression
eqger.x  |-  X  =  ( Base `  G
)
eqger.r  |-  .~  =  ( G ~QG  Y )
Assertion
Ref Expression
eqgen  |-  ( ( Y  e.  (SubGrp `  G )  /\  A  e.  ( X /.  .~  ) )  ->  Y  ~~  A )

Proof of Theorem eqgen
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . 2  |-  ( X /.  .~  )  =  ( X /.  .~  )
2 breq2 4027 . 2  |-  ( [ x ]  .~  =  A  ->  ( Y  ~~  [ x ]  .~  <->  Y  ~~  A ) )
3 simpl 443 . . . 4  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  Y  e.  (SubGrp `  G )
)
4 subgrcl 14626 . . . . . . 7  |-  ( Y  e.  (SubGrp `  G
)  ->  G  e.  Grp )
5 eqger.x . . . . . . . 8  |-  X  =  ( Base `  G
)
65subgss 14622 . . . . . . 7  |-  ( Y  e.  (SubGrp `  G
)  ->  Y  C_  X
)
74, 6jca 518 . . . . . 6  |-  ( Y  e.  (SubGrp `  G
)  ->  ( G  e.  Grp  /\  Y  C_  X ) )
8 eqger.r . . . . . . . 8  |-  .~  =  ( G ~QG  Y )
9 eqid 2283 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
105, 8, 9eqglact 14668 . . . . . . 7  |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  x  e.  X )  ->  [ x ]  .~  =  ( ( z  e.  X  |->  ( x ( +g  `  G
) z ) )
" Y ) )
11103expa 1151 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  Y  C_  X )  /\  x  e.  X
)  ->  [ x ]  .~  =  ( ( z  e.  X  |->  ( x ( +g  `  G
) z ) )
" Y ) )
127, 11sylan 457 . . . . 5  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  [ x ]  .~  =  ( ( z  e.  X  |->  ( x ( +g  `  G
) z ) )
" Y ) )
13 ovex 5883 . . . . . . 7  |-  ( G ~QG  Y )  e.  _V
148, 13eqeltri 2353 . . . . . 6  |-  .~  e.  _V
15 ecexg 6664 . . . . . 6  |-  (  .~  e.  _V  ->  [ x ]  .~  e.  _V )
1614, 15ax-mp 8 . . . . 5  |-  [ x ]  .~  e.  _V
1712, 16syl6eqelr 2372 . . . 4  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
( z  e.  X  |->  ( x ( +g  `  G ) z ) ) " Y )  e.  _V )
18 eqid 2283 . . . . . . . . 9  |-  ( y  e.  X  |->  ( z  e.  X  |->  ( y ( +g  `  G
) z ) ) )  =  ( y  e.  X  |->  ( z  e.  X  |->  ( y ( +g  `  G
) z ) ) )
1918, 5, 9grplactf1o 14565 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( ( y  e.  X  |->  ( z  e.  X  |->  ( y ( +g  `  G ) z ) ) ) `
 x ) : X -1-1-onto-> X )
2018, 5grplactfval 14562 . . . . . . . . . 10  |-  ( x  e.  X  ->  (
( y  e.  X  |->  ( z  e.  X  |->  ( y ( +g  `  G ) z ) ) ) `  x
)  =  ( z  e.  X  |->  ( x ( +g  `  G
) z ) ) )
2120adantl 452 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( ( y  e.  X  |->  ( z  e.  X  |->  ( y ( +g  `  G ) z ) ) ) `
 x )  =  ( z  e.  X  |->  ( x ( +g  `  G ) z ) ) )
22 f1oeq1 5463 . . . . . . . . 9  |-  ( ( ( y  e.  X  |->  ( z  e.  X  |->  ( y ( +g  `  G ) z ) ) ) `  x
)  =  ( z  e.  X  |->  ( x ( +g  `  G
) z ) )  ->  ( ( ( y  e.  X  |->  ( z  e.  X  |->  ( y ( +g  `  G
) z ) ) ) `  x ) : X -1-1-onto-> X  <->  ( z  e.  X  |->  ( x ( +g  `  G ) z ) ) : X -1-1-onto-> X ) )
2321, 22syl 15 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( ( ( y  e.  X  |->  ( z  e.  X  |->  ( y ( +g  `  G
) z ) ) ) `  x ) : X -1-1-onto-> X  <->  ( z  e.  X  |->  ( x ( +g  `  G ) z ) ) : X -1-1-onto-> X ) )
2419, 23mpbid 201 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( z  e.  X  |->  ( x ( +g  `  G ) z ) ) : X -1-1-onto-> X )
254, 24sylan 457 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
z  e.  X  |->  ( x ( +g  `  G
) z ) ) : X -1-1-onto-> X )
26 f1of1 5471 . . . . . 6  |-  ( ( z  e.  X  |->  ( x ( +g  `  G
) z ) ) : X -1-1-onto-> X  ->  ( z  e.  X  |->  ( x ( +g  `  G
) z ) ) : X -1-1-> X )
2725, 26syl 15 . . . . 5  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
z  e.  X  |->  ( x ( +g  `  G
) z ) ) : X -1-1-> X )
286adantr 451 . . . . 5  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  Y  C_  X )
29 f1ores 5487 . . . . 5  |-  ( ( ( z  e.  X  |->  ( x ( +g  `  G ) z ) ) : X -1-1-> X  /\  Y  C_  X )  ->  ( ( z  e.  X  |->  ( x ( +g  `  G
) z ) )  |`  Y ) : Y -1-1-onto-> (
( z  e.  X  |->  ( x ( +g  `  G ) z ) ) " Y ) )
3027, 28, 29syl2anc 642 . . . 4  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
( z  e.  X  |->  ( x ( +g  `  G ) z ) )  |`  Y ) : Y -1-1-onto-> ( ( z  e.  X  |->  ( x ( +g  `  G ) z ) ) " Y ) )
31 f1oen2g 6878 . . . 4  |-  ( ( Y  e.  (SubGrp `  G )  /\  (
( z  e.  X  |->  ( x ( +g  `  G ) z ) ) " Y )  e.  _V  /\  (
( z  e.  X  |->  ( x ( +g  `  G ) z ) )  |`  Y ) : Y -1-1-onto-> ( ( z  e.  X  |->  ( x ( +g  `  G ) z ) ) " Y ) )  ->  Y  ~~  ( ( z  e.  X  |->  ( x ( +g  `  G
) z ) )
" Y ) )
323, 17, 30, 31syl3anc 1182 . . 3  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  Y  ~~  ( ( z  e.  X  |->  ( x ( +g  `  G ) z ) ) " Y ) )
3332, 12breqtrrd 4049 . 2  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  Y  ~~  [ x ]  .~  )
341, 2, 33ectocld 6726 1  |-  ( ( Y  e.  (SubGrp `  G )  /\  A  e.  ( X /.  .~  ) )  ->  Y  ~~  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   class class class wbr 4023    e. cmpt 4077    |` cres 4691   "cima 4692   -1-1->wf1 5252   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   [cec 6658   /.cqs 6659    ~~ cen 6860   Basecbs 13148   +g cplusg 13208   Grpcgrp 14362  SubGrpcsubg 14615   ~QG cqg 14617
This theorem is referenced by:  lagsubg2  14678  sylow2blem1  14931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-ec 6662  df-qs 6666  df-en 6864  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-subg 14618  df-eqg 14620
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