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Theorem eqgid 16055
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 16050 . . . 4  |-  Rel  .~
3 relelec 7352 . . . 4  |-  ( Rel 
.~  ->  ( x  e. 
[  .0.  ]  .~  <->  .0. 
.~  x ) )
42, 3ax-mp 5 . . 3  |-  ( x  e.  [  .0.  ]  .~ 
<->  .0.  .~  x )
5 subgrcl 16008 . . . . . . . . . 10  |-  ( Y  e.  (SubGrp `  G
)  ->  G  e.  Grp )
65adantr 465 . . . . . . . . 9  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  G  e.  Grp )
7 eqgid.3 . . . . . . . . . 10  |-  .0.  =  ( 0g `  G )
8 eqid 2467 . . . . . . . . . 10  |-  ( invg `  G )  =  ( invg `  G )
97, 8grpinvid 15908 . . . . . . . . 9  |-  ( G  e.  Grp  ->  (
( invg `  G ) `  .0.  )  =  .0.  )
106, 9syl 16 . . . . . . . 8  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
( invg `  G ) `  .0.  )  =  .0.  )
1110oveq1d 6298 . . . . . . 7  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
( ( invg `  G ) `  .0.  ) ( +g  `  G
) x )  =  (  .0.  ( +g  `  G ) x ) )
12 eqger.x . . . . . . . . 9  |-  X  =  ( Base `  G
)
13 eqid 2467 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
1412, 13, 7grplid 15887 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  (  .0.  ( +g  `  G ) x )  =  x )
155, 14sylan 471 . . . . . . 7  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (  .0.  ( +g  `  G
) x )  =  x )
1611, 15eqtrd 2508 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
( ( invg `  G ) `  .0.  ) ( +g  `  G
) x )  =  x )
1716eleq1d 2536 . . . . 5  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
( ( ( invg `  G ) `
 .0.  ) ( +g  `  G ) x )  e.  Y  <->  x  e.  Y ) )
1817pm5.32da 641 . . . 4  |-  ( Y  e.  (SubGrp `  G
)  ->  ( (
x  e.  X  /\  ( ( ( invg `  G ) `
 .0.  ) ( +g  `  G ) x )  e.  Y
)  <->  ( x  e.  X  /\  x  e.  Y ) ) )
1912subgss 16004 . . . . 5  |-  ( Y  e.  (SubGrp `  G
)  ->  Y  C_  X
)
2012, 7grpidcl 15885 . . . . . 6  |-  ( G  e.  Grp  ->  .0.  e.  X )
215, 20syl 16 . . . . 5  |-  ( Y  e.  (SubGrp `  G
)  ->  .0.  e.  X )
2212, 8, 13, 1eqgval 16052 . . . . . . 7  |-  ( ( G  e.  Grp  /\  Y  C_  X )  -> 
(  .0.  .~  x  <->  (  .0.  e.  X  /\  x  e.  X  /\  ( ( ( invg `  G ) `
 .0.  ) ( +g  `  G ) x )  e.  Y
) ) )
23 3anass 977 . . . . . . 7  |-  ( (  .0.  e.  X  /\  x  e.  X  /\  ( ( ( invg `  G ) `
 .0.  ) ( +g  `  G ) x )  e.  Y
)  <->  (  .0.  e.  X  /\  ( x  e.  X  /\  ( ( ( invg `  G ) `  .0.  ) ( +g  `  G
) x )  e.  Y ) ) )
2422, 23syl6bb 261 . . . . . 6  |-  ( ( G  e.  Grp  /\  Y  C_  X )  -> 
(  .0.  .~  x  <->  (  .0.  e.  X  /\  ( x  e.  X  /\  ( ( ( invg `  G ) `
 .0.  ) ( +g  `  G ) x )  e.  Y
) ) ) )
2524baibd 907 . . . . 5  |-  ( ( ( G  e.  Grp  /\  Y  C_  X )  /\  .0.  e.  X )  ->  (  .0.  .~  x 
<->  ( x  e.  X  /\  ( ( ( invg `  G ) `
 .0.  ) ( +g  `  G ) x )  e.  Y
) ) )
265, 19, 21, 25syl21anc 1227 . . . 4  |-  ( Y  e.  (SubGrp `  G
)  ->  (  .0.  .~  x  <->  ( x  e.  X  /\  ( ( ( invg `  G ) `  .0.  ) ( +g  `  G
) x )  e.  Y ) ) )
2719sseld 3503 . . . . 5  |-  ( Y  e.  (SubGrp `  G
)  ->  ( x  e.  Y  ->  x  e.  X ) )
2827pm4.71rd 635 . . . 4  |-  ( Y  e.  (SubGrp `  G
)  ->  ( x  e.  Y  <->  ( x  e.  X  /\  x  e.  Y ) ) )
2918, 26, 283bitr4d 285 . . 3  |-  ( Y  e.  (SubGrp `  G
)  ->  (  .0.  .~  x  <->  x  e.  Y
) )
304, 29syl5bb 257 . 2  |-  ( Y  e.  (SubGrp `  G
)  ->  ( x  e.  [  .0.  ]  .~  <->  x  e.  Y ) )
3130eqrdv 2464 1  |-  ( Y  e.  (SubGrp `  G
)  ->  [  .0.  ]  .~  =  Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    C_ wss 3476   class class class wbr 4447   Rel wrel 5004   ` cfv 5587  (class class class)co 6283   [cec 7309   Basecbs 14489   +g cplusg 14554   0gc0g 14694   Grpcgrp 15726   invgcminusg 15727  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-1st 6784  df-2nd 6785  df-ec 7313  df-0g 14696  df-mnd 15731  df-grp 15864  df-minusg 15865  df-subg 16000  df-eqg 16002
This theorem is referenced by:  cldsubg  20360  divstgphaus  20372
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