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Theorem eqgid 14947
Description: The left coset containing the identity is the original subgroup. (Contributed by Mario Carneiro, 20-Sep-2015.)
Hypotheses
Ref Expression
eqger.x  |-  X  =  ( Base `  G
)
eqger.r  |-  .~  =  ( G ~QG  Y )
eqgid.3  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
eqgid  |-  ( Y  e.  (SubGrp `  G
)  ->  [  .0.  ]  .~  =  Y )

Proof of Theorem eqgid
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqger.r . . . . 5  |-  .~  =  ( G ~QG  Y )
21releqg 14942 . . . 4  |-  Rel  .~
3 relelec 6904 . . . 4  |-  ( Rel 
.~  ->  ( x  e. 
[  .0.  ]  .~  <->  .0. 
.~  x ) )
42, 3ax-mp 8 . . 3  |-  ( x  e.  [  .0.  ]  .~ 
<->  .0.  .~  x )
5 subgrcl 14904 . . . . . . . . . 10  |-  ( Y  e.  (SubGrp `  G
)  ->  G  e.  Grp )
65adantr 452 . . . . . . . . 9  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  G  e.  Grp )
7 eqgid.3 . . . . . . . . . 10  |-  .0.  =  ( 0g `  G )
8 eqid 2404 . . . . . . . . . 10  |-  ( inv g `  G )  =  ( inv g `  G )
97, 8grpinvid 14811 . . . . . . . . 9  |-  ( G  e.  Grp  ->  (
( inv g `  G ) `  .0.  )  =  .0.  )
106, 9syl 16 . . . . . . . 8  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
( inv g `  G ) `  .0.  )  =  .0.  )
1110oveq1d 6055 . . . . . . 7  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
( ( inv g `  G ) `  .0.  ) ( +g  `  G
) x )  =  (  .0.  ( +g  `  G ) x ) )
12 eqger.x . . . . . . . . 9  |-  X  =  ( Base `  G
)
13 eqid 2404 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
1412, 13, 7grplid 14790 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  (  .0.  ( +g  `  G ) x )  =  x )
155, 14sylan 458 . . . . . . 7  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (  .0.  ( +g  `  G
) x )  =  x )
1611, 15eqtrd 2436 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
( ( inv g `  G ) `  .0.  ) ( +g  `  G
) x )  =  x )
1716eleq1d 2470 . . . . 5  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
( ( ( inv g `  G ) `
 .0.  ) ( +g  `  G ) x )  e.  Y  <->  x  e.  Y ) )
1817pm5.32da 623 . . . 4  |-  ( Y  e.  (SubGrp `  G
)  ->  ( (
x  e.  X  /\  ( ( ( inv g `  G ) `
 .0.  ) ( +g  `  G ) x )  e.  Y
)  <->  ( x  e.  X  /\  x  e.  Y ) ) )
1912subgss 14900 . . . . 5  |-  ( Y  e.  (SubGrp `  G
)  ->  Y  C_  X
)
2012, 7grpidcl 14788 . . . . . 6  |-  ( G  e.  Grp  ->  .0.  e.  X )
215, 20syl 16 . . . . 5  |-  ( Y  e.  (SubGrp `  G
)  ->  .0.  e.  X )
2212, 8, 13, 1eqgval 14944 . . . . . . 7  |-  ( ( G  e.  Grp  /\  Y  C_  X )  -> 
(  .0.  .~  x  <->  (  .0.  e.  X  /\  x  e.  X  /\  ( ( ( inv g `  G ) `
 .0.  ) ( +g  `  G ) x )  e.  Y
) ) )
23 3anass 940 . . . . . . 7  |-  ( (  .0.  e.  X  /\  x  e.  X  /\  ( ( ( inv g `  G ) `
 .0.  ) ( +g  `  G ) x )  e.  Y
)  <->  (  .0.  e.  X  /\  ( x  e.  X  /\  ( ( ( inv g `  G ) `  .0.  ) ( +g  `  G
) x )  e.  Y ) ) )
2422, 23syl6bb 253 . . . . . 6  |-  ( ( G  e.  Grp  /\  Y  C_  X )  -> 
(  .0.  .~  x  <->  (  .0.  e.  X  /\  ( x  e.  X  /\  ( ( ( inv g `  G ) `
 .0.  ) ( +g  `  G ) x )  e.  Y
) ) ) )
2524baibd 876 . . . . 5  |-  ( ( ( G  e.  Grp  /\  Y  C_  X )  /\  .0.  e.  X )  ->  (  .0.  .~  x 
<->  ( x  e.  X  /\  ( ( ( inv g `  G ) `
 .0.  ) ( +g  `  G ) x )  e.  Y
) ) )
265, 19, 21, 25syl21anc 1183 . . . 4  |-  ( Y  e.  (SubGrp `  G
)  ->  (  .0.  .~  x  <->  ( x  e.  X  /\  ( ( ( inv g `  G ) `  .0.  ) ( +g  `  G
) x )  e.  Y ) ) )
2719sseld 3307 . . . . 5  |-  ( Y  e.  (SubGrp `  G
)  ->  ( x  e.  Y  ->  x  e.  X ) )
2827pm4.71rd 617 . . . 4  |-  ( Y  e.  (SubGrp `  G
)  ->  ( x  e.  Y  <->  ( x  e.  X  /\  x  e.  Y ) ) )
2918, 26, 283bitr4d 277 . . 3  |-  ( Y  e.  (SubGrp `  G
)  ->  (  .0.  .~  x  <->  x  e.  Y
) )
304, 29syl5bb 249 . 2  |-  ( Y  e.  (SubGrp `  G
)  ->  ( x  e.  [  .0.  ]  .~  <->  x  e.  Y ) )
3130eqrdv 2402 1  |-  ( Y  e.  (SubGrp `  G
)  ->  [  .0.  ]  .~  =  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    C_ wss 3280   class class class wbr 4172   Rel wrel 4842   ` cfv 5413  (class class class)co 6040   [cec 6862   Basecbs 13424   +g cplusg 13484   0gc0g 13678   Grpcgrp 14640   inv gcminusg 14641  SubGrpcsubg 14893   ~QG cqg 14895
This theorem is referenced by:  cldsubg  18093  divstgphaus  18105
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-ec 6866  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-subg 14896  df-eqg 14898
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