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Theorem eqgid 16918
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 16913 . . . 4  |-  Rel  .~
3 relelec 7430 . . . 4  |-  ( Rel 
.~  ->  ( x  e. 
[  .0.  ]  .~  <->  .0. 
.~  x ) )
42, 3ax-mp 5 . . 3  |-  ( x  e.  [  .0.  ]  .~ 
<->  .0.  .~  x )
5 subgrcl 16871 . . . . . . . . . 10  |-  ( Y  e.  (SubGrp `  G
)  ->  G  e.  Grp )
65adantr 471 . . . . . . . . 9  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  G  e.  Grp )
7 eqgid.3 . . . . . . . . . 10  |-  .0.  =  ( 0g `  G )
8 eqid 2462 . . . . . . . . . 10  |-  ( invg `  G )  =  ( invg `  G )
97, 8grpinvid 16766 . . . . . . . . 9  |-  ( G  e.  Grp  ->  (
( invg `  G ) `  .0.  )  =  .0.  )
106, 9syl 17 . . . . . . . 8  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
( invg `  G ) `  .0.  )  =  .0.  )
1110oveq1d 6330 . . . . . . 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 2462 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
1412, 13, 7grplid 16745 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  (  .0.  ( +g  `  G ) x )  =  x )
155, 14sylan 478 . . . . . . 7  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (  .0.  ( +g  `  G
) x )  =  x )
1611, 15eqtrd 2496 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
( ( invg `  G ) `  .0.  ) ( +g  `  G
) x )  =  x )
1716eleq1d 2524 . . . . 5  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
( ( ( invg `  G ) `
 .0.  ) ( +g  `  G ) x )  e.  Y  <->  x  e.  Y ) )
1817pm5.32da 651 . . . 4  |-  ( Y  e.  (SubGrp `  G
)  ->  ( (
x  e.  X  /\  ( ( ( invg `  G ) `
 .0.  ) ( +g  `  G ) x )  e.  Y
)  <->  ( x  e.  X  /\  x  e.  Y ) ) )
1912subgss 16867 . . . . 5  |-  ( Y  e.  (SubGrp `  G
)  ->  Y  C_  X
)
2012, 7grpidcl 16743 . . . . . 6  |-  ( G  e.  Grp  ->  .0.  e.  X )
215, 20syl 17 . . . . 5  |-  ( Y  e.  (SubGrp `  G
)  ->  .0.  e.  X )
2212, 8, 13, 1eqgval 16915 . . . . . . 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 995 . . . . . . 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 269 . . . . . 6  |-  ( ( G  e.  Grp  /\  Y  C_  X )  -> 
(  .0.  .~  x  <->  (  .0.  e.  X  /\  ( x  e.  X  /\  ( ( ( invg `  G ) `
 .0.  ) ( +g  `  G ) x )  e.  Y
) ) ) )
2524baibd 925 . . . . 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 1275 . . . 4  |-  ( Y  e.  (SubGrp `  G
)  ->  (  .0.  .~  x  <->  ( x  e.  X  /\  ( ( ( invg `  G ) `  .0.  ) ( +g  `  G
) x )  e.  Y ) ) )
2719sseld 3443 . . . . 5  |-  ( Y  e.  (SubGrp `  G
)  ->  ( x  e.  Y  ->  x  e.  X ) )
2827pm4.71rd 645 . . . 4  |-  ( Y  e.  (SubGrp `  G
)  ->  ( x  e.  Y  <->  ( x  e.  X  /\  x  e.  Y ) ) )
2918, 26, 283bitr4d 293 . . 3  |-  ( Y  e.  (SubGrp `  G
)  ->  (  .0.  .~  x  <->  x  e.  Y
) )
304, 29syl5bb 265 . 2  |-  ( Y  e.  (SubGrp `  G
)  ->  ( x  e.  [  .0.  ]  .~  <->  x  e.  Y ) )
3130eqrdv 2460 1  |-  ( Y  e.  (SubGrp `  G
)  ->  [  .0.  ]  .~  =  Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898    C_ wss 3416   class class class wbr 4416   Rel wrel 4858   ` cfv 5601  (class class class)co 6315   [cec 7387   Basecbs 15170   +g cplusg 15239   0gc0g 15387   Grpcgrp 16718   invgcminusg 16719  SubGrpcsubg 16860   ~QG cqg 16862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-1st 6820  df-2nd 6821  df-ec 7391  df-0g 15389  df-mgm 16537  df-sgrp 16576  df-mnd 16586  df-grp 16722  df-minusg 16723  df-subg 16863  df-eqg 16865
This theorem is referenced by:  cldsubg  21174  qustgphaus  21186
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