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Theorem eqgid 14879
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 14874 . . . 4  |-  Rel  .~
3 relelec 6842 . . . 4  |-  ( Rel 
.~  ->  ( x  e. 
[  .0.  ]  .~  <->  .0. 
.~  x ) )
42, 3ax-mp 8 . . 3  |-  ( x  e.  [  .0.  ]  .~ 
<->  .0.  .~  x )
5 subgrcl 14836 . . . . . . . . . 10  |-  ( Y  e.  (SubGrp `  G
)  ->  G  e.  Grp )
65adantr 451 . . . . . . . . 9  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  G  e.  Grp )
7 eqgid.3 . . . . . . . . . 10  |-  .0.  =  ( 0g `  G )
8 eqid 2366 . . . . . . . . . 10  |-  ( inv g `  G )  =  ( inv g `  G )
97, 8grpinvid 14743 . . . . . . . . 9  |-  ( G  e.  Grp  ->  (
( inv g `  G ) `  .0.  )  =  .0.  )
106, 9syl 15 . . . . . . . 8  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
( inv g `  G ) `  .0.  )  =  .0.  )
1110oveq1d 5996 . . . . . . 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 2366 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
1412, 13, 7grplid 14722 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  (  .0.  ( +g  `  G ) x )  =  x )
155, 14sylan 457 . . . . . . 7  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (  .0.  ( +g  `  G
) x )  =  x )
1611, 15eqtrd 2398 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
( ( inv g `  G ) `  .0.  ) ( +g  `  G
) x )  =  x )
1716eleq1d 2432 . . . . 5  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
( ( ( inv g `  G ) `
 .0.  ) ( +g  `  G ) x )  e.  Y  <->  x  e.  Y ) )
1817pm5.32da 622 . . . 4  |-  ( Y  e.  (SubGrp `  G
)  ->  ( (
x  e.  X  /\  ( ( ( inv g `  G ) `
 .0.  ) ( +g  `  G ) x )  e.  Y
)  <->  ( x  e.  X  /\  x  e.  Y ) ) )
1912subgss 14832 . . . . 5  |-  ( Y  e.  (SubGrp `  G
)  ->  Y  C_  X
)
2012, 7grpidcl 14720 . . . . . 6  |-  ( G  e.  Grp  ->  .0.  e.  X )
215, 20syl 15 . . . . 5  |-  ( Y  e.  (SubGrp `  G
)  ->  .0.  e.  X )
2212, 8, 13, 1eqgval 14876 . . . . . . 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 939 . . . . . . 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 252 . . . . . 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 875 . . . . 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 1182 . . . 4  |-  ( Y  e.  (SubGrp `  G
)  ->  (  .0.  .~  x  <->  ( x  e.  X  /\  ( ( ( inv g `  G ) `  .0.  ) ( +g  `  G
) x )  e.  Y ) ) )
2719sseld 3265 . . . . 5  |-  ( Y  e.  (SubGrp `  G
)  ->  ( x  e.  Y  ->  x  e.  X ) )
2827pm4.71rd 616 . . . 4  |-  ( Y  e.  (SubGrp `  G
)  ->  ( x  e.  Y  <->  ( x  e.  X  /\  x  e.  Y ) ) )
2918, 26, 283bitr4d 276 . . 3  |-  ( Y  e.  (SubGrp `  G
)  ->  (  .0.  .~  x  <->  x  e.  Y
) )
304, 29syl5bb 248 . 2  |-  ( Y  e.  (SubGrp `  G
)  ->  ( x  e.  [  .0.  ]  .~  <->  x  e.  Y ) )
3130eqrdv 2364 1  |-  ( Y  e.  (SubGrp `  G
)  ->  [  .0.  ]  .~  =  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    C_ wss 3238   class class class wbr 4125   Rel wrel 4797   ` cfv 5358  (class class class)co 5981   [cec 6800   Basecbs 13356   +g cplusg 13416   0gc0g 13610   Grpcgrp 14572   inv gcminusg 14573  SubGrpcsubg 14825   ~QG cqg 14827
This theorem is referenced by:  cldsubg  18006  divstgphaus  18018
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-ec 6804  df-0g 13614  df-mnd 14577  df-grp 14699  df-minusg 14700  df-subg 14828  df-eqg 14830
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