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Theorem eqgen 16056
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 2467 . 2  |-  ( X /.  .~  )  =  ( X /.  .~  )
2 breq2 4451 . 2  |-  ( [ x ]  .~  =  A  ->  ( Y  ~~  [ x ]  .~  <->  Y  ~~  A ) )
3 simpl 457 . . . 4  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  Y  e.  (SubGrp `  G )
)
4 subgrcl 16008 . . . . . . 7  |-  ( Y  e.  (SubGrp `  G
)  ->  G  e.  Grp )
5 eqger.x . . . . . . . 8  |-  X  =  ( Base `  G
)
65subgss 16004 . . . . . . 7  |-  ( Y  e.  (SubGrp `  G
)  ->  Y  C_  X
)
74, 6jca 532 . . . . . 6  |-  ( Y  e.  (SubGrp `  G
)  ->  ( G  e.  Grp  /\  Y  C_  X ) )
8 eqger.r . . . . . . . 8  |-  .~  =  ( G ~QG  Y )
9 eqid 2467 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
105, 8, 9eqglact 16054 . . . . . . 7  |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  x  e.  X )  ->  [ x ]  .~  =  ( ( z  e.  X  |->  ( x ( +g  `  G
) z ) )
" Y ) )
11103expa 1196 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  Y  C_  X )  /\  x  e.  X
)  ->  [ x ]  .~  =  ( ( z  e.  X  |->  ( x ( +g  `  G
) z ) )
" Y ) )
127, 11sylan 471 . . . . 5  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  [ x ]  .~  =  ( ( z  e.  X  |->  ( x ( +g  `  G
) z ) )
" Y ) )
13 ovex 6308 . . . . . . 7  |-  ( G ~QG  Y )  e.  _V
148, 13eqeltri 2551 . . . . . 6  |-  .~  e.  _V
15 ecexg 7315 . . . . . 6  |-  (  .~  e.  _V  ->  [ x ]  .~  e.  _V )
1614, 15ax-mp 5 . . . . 5  |-  [ x ]  .~  e.  _V
1712, 16syl6eqelr 2564 . . . 4  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
( z  e.  X  |->  ( x ( +g  `  G ) z ) ) " Y )  e.  _V )
18 eqid 2467 . . . . . . . . 9  |-  ( y  e.  X  |->  ( z  e.  X  |->  ( y ( +g  `  G
) z ) ) )  =  ( y  e.  X  |->  ( z  e.  X  |->  ( y ( +g  `  G
) z ) ) )
1918, 5, 9grplactf1o 15946 . . . . . . . 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 15943 . . . . . . . . . 10  |-  ( x  e.  X  ->  (
( y  e.  X  |->  ( z  e.  X  |->  ( y ( +g  `  G ) z ) ) ) `  x
)  =  ( z  e.  X  |->  ( x ( +g  `  G
) z ) ) )
2120adantl 466 . . . . . . . . 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 5806 . . . . . . . . 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 210 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( z  e.  X  |->  ( x ( +g  `  G ) z ) ) : X -1-1-onto-> X )
254, 24sylan 471 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
z  e.  X  |->  ( x ( +g  `  G
) z ) ) : X -1-1-onto-> X )
26 f1of1 5814 . . . . . 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 465 . . . . 5  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  Y  C_  X )
29 f1ores 5829 . . . . 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 661 . . . 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 7532 . . . 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 1228 . . 3  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  Y  ~~  ( ( z  e.  X  |->  ( x ( +g  `  G ) z ) ) " Y ) )
3332, 12breqtrrd 4473 . 2  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  Y  ~~  [ x ]  .~  )
341, 2, 33ectocld 7378 1  |-  ( ( Y  e.  (SubGrp `  G )  /\  A  e.  ( X /.  .~  ) )  ->  Y  ~~  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    C_ wss 3476   class class class wbr 4447    |-> cmpt 4505    |` cres 5001   "cima 5002   -1-1->wf1 5584   -1-1-onto->wf1o 5586   ` cfv 5587  (class class class)co 6283   [cec 7309   /.cqs 7310    ~~ cen 7513   Basecbs 14489   +g cplusg 14554   Grpcgrp 15726  SubGrpcsubg 15997   ~QG cqg 15999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-ec 7313  df-qs 7317  df-en 7517  df-0g 14696  df-mnd 15731  df-grp 15864  df-minusg 15865  df-subg 16000  df-eqg 16002
This theorem is referenced by:  lagsubg2  16064  sylow2blem1  16443
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