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Theorem eqgen 14719
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 2316 . 2  |-  ( X /.  .~  )  =  ( X /.  .~  )
2 breq2 4064 . 2  |-  ( [ x ]  .~  =  A  ->  ( Y  ~~  [ x ]  .~  <->  Y  ~~  A ) )
3 simpl 443 . . . 4  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  Y  e.  (SubGrp `  G )
)
4 subgrcl 14675 . . . . . . 7  |-  ( Y  e.  (SubGrp `  G
)  ->  G  e.  Grp )
5 eqger.x . . . . . . . 8  |-  X  =  ( Base `  G
)
65subgss 14671 . . . . . . 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 2316 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
105, 8, 9eqglact 14717 . . . . . . 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 5925 . . . . . . 7  |-  ( G ~QG  Y )  e.  _V
148, 13eqeltri 2386 . . . . . 6  |-  .~  e.  _V
15 ecexg 6706 . . . . . 6  |-  (  .~  e.  _V  ->  [ x ]  .~  e.  _V )
1614, 15ax-mp 8 . . . . 5  |-  [ x ]  .~  e.  _V
1712, 16syl6eqelr 2405 . . . 4  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
( z  e.  X  |->  ( x ( +g  `  G ) z ) ) " Y )  e.  _V )
18 eqid 2316 . . . . . . . . 9  |-  ( y  e.  X  |->  ( z  e.  X  |->  ( y ( +g  `  G
) z ) ) )  =  ( y  e.  X  |->  ( z  e.  X  |->  ( y ( +g  `  G
) z ) ) )
1918, 5, 9grplactf1o 14614 . . . . . . . 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 14611 . . . . . . . . . 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 5501 . . . . . . . . 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 5509 . . . . . 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 5525 . . . . 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 6921 . . . 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 4086 . 2  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  Y  ~~  [ x ]  .~  )
341, 2, 33ectocld 6768 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 1633    e. wcel 1701   _Vcvv 2822    C_ wss 3186   class class class wbr 4060    e. cmpt 4114    |` cres 4728   "cima 4729   -1-1->wf1 5289   -1-1-onto->wf1o 5291   ` cfv 5292  (class class class)co 5900   [cec 6700   /.cqs 6701    ~~ cen 6903   Basecbs 13195   +g cplusg 13255   Grpcgrp 14411  SubGrpcsubg 14664   ~QG cqg 14666
This theorem is referenced by:  lagsubg2  14727  sylow2blem1  14980
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-riota 6346  df-ec 6704  df-qs 6708  df-en 6907  df-0g 13453  df-mnd 14416  df-grp 14538  df-minusg 14539  df-subg 14667  df-eqg 14669
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