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Theorem eqgen 14993
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 2436 . 2  |-  ( X /.  .~  )  =  ( X /.  .~  )
2 breq2 4216 . 2  |-  ( [ x ]  .~  =  A  ->  ( Y  ~~  [ x ]  .~  <->  Y  ~~  A ) )
3 simpl 444 . . . 4  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  Y  e.  (SubGrp `  G )
)
4 subgrcl 14949 . . . . . . 7  |-  ( Y  e.  (SubGrp `  G
)  ->  G  e.  Grp )
5 eqger.x . . . . . . . 8  |-  X  =  ( Base `  G
)
65subgss 14945 . . . . . . 7  |-  ( Y  e.  (SubGrp `  G
)  ->  Y  C_  X
)
74, 6jca 519 . . . . . 6  |-  ( Y  e.  (SubGrp `  G
)  ->  ( G  e.  Grp  /\  Y  C_  X ) )
8 eqger.r . . . . . . . 8  |-  .~  =  ( G ~QG  Y )
9 eqid 2436 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
105, 8, 9eqglact 14991 . . . . . . 7  |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  x  e.  X )  ->  [ x ]  .~  =  ( ( z  e.  X  |->  ( x ( +g  `  G
) z ) )
" Y ) )
11103expa 1153 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  Y  C_  X )  /\  x  e.  X
)  ->  [ x ]  .~  =  ( ( z  e.  X  |->  ( x ( +g  `  G
) z ) )
" Y ) )
127, 11sylan 458 . . . . 5  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  [ x ]  .~  =  ( ( z  e.  X  |->  ( x ( +g  `  G
) z ) )
" Y ) )
13 ovex 6106 . . . . . . 7  |-  ( G ~QG  Y )  e.  _V
148, 13eqeltri 2506 . . . . . 6  |-  .~  e.  _V
15 ecexg 6909 . . . . . 6  |-  (  .~  e.  _V  ->  [ x ]  .~  e.  _V )
1614, 15ax-mp 8 . . . . 5  |-  [ x ]  .~  e.  _V
1712, 16syl6eqelr 2525 . . . 4  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
( z  e.  X  |->  ( x ( +g  `  G ) z ) ) " Y )  e.  _V )
18 eqid 2436 . . . . . . . . 9  |-  ( y  e.  X  |->  ( z  e.  X  |->  ( y ( +g  `  G
) z ) ) )  =  ( y  e.  X  |->  ( z  e.  X  |->  ( y ( +g  `  G
) z ) ) )
1918, 5, 9grplactf1o 14888 . . . . . . . 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 14885 . . . . . . . . . 10  |-  ( x  e.  X  ->  (
( y  e.  X  |->  ( z  e.  X  |->  ( y ( +g  `  G ) z ) ) ) `  x
)  =  ( z  e.  X  |->  ( x ( +g  `  G
) z ) ) )
2120adantl 453 . . . . . . . . 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 5665 . . . . . . . . 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 16 . . . . . . . 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 202 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( z  e.  X  |->  ( x ( +g  `  G ) z ) ) : X -1-1-onto-> X )
254, 24sylan 458 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
z  e.  X  |->  ( x ( +g  `  G
) z ) ) : X -1-1-onto-> X )
26 f1of1 5673 . . . . . 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 16 . . . . 5  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
z  e.  X  |->  ( x ( +g  `  G
) z ) ) : X -1-1-> X )
286adantr 452 . . . . 5  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  Y  C_  X )
29 f1ores 5689 . . . . 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 643 . . . 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 7124 . . . 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 1184 . . 3  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  Y  ~~  ( ( z  e.  X  |->  ( x ( +g  `  G ) z ) ) " Y ) )
3332, 12breqtrrd 4238 . 2  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  Y  ~~  [ x ]  .~  )
341, 2, 33ectocld 6971 1  |-  ( ( Y  e.  (SubGrp `  G )  /\  A  e.  ( X /.  .~  ) )  ->  Y  ~~  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956    C_ wss 3320   class class class wbr 4212    e. cmpt 4266    |` cres 4880   "cima 4881   -1-1->wf1 5451   -1-1-onto->wf1o 5453   ` cfv 5454  (class class class)co 6081   [cec 6903   /.cqs 6904    ~~ cen 7106   Basecbs 13469   +g cplusg 13529   Grpcgrp 14685  SubGrpcsubg 14938   ~QG cqg 14940
This theorem is referenced by:  lagsubg2  15001  sylow2blem1  15254
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-ec 6907  df-qs 6911  df-en 7110  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-subg 14941  df-eqg 14943
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