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Theorem eqgid 13016
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 subgrcl 12970 . . . . 5  |-  ( Y  e.  (SubGrp `  G
)  ->  G  e.  Grp )
2 eqger.r . . . . . 6  |-  .~  =  ( G ~QG  Y )
32releqgg 13011 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  e.  (SubGrp `  G
) )  ->  Rel  .~  )
41, 3mpancom 422 . . . 4  |-  ( Y  e.  (SubGrp `  G
)  ->  Rel  .~  )
5 relelec 6572 . . . 4  |-  ( Rel 
.~  ->  ( x  e. 
[  .0.  ]  .~  <->  .0. 
.~  x ) )
64, 5syl 14 . . 3  |-  ( Y  e.  (SubGrp `  G
)  ->  ( x  e.  [  .0.  ]  .~  <->  .0. 
.~  x ) )
71adantr 276 . . . . . . . . 9  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  G  e.  Grp )
8 eqgid.3 . . . . . . . . . 10  |-  .0.  =  ( 0g `  G )
9 eqid 2177 . . . . . . . . . 10  |-  ( invg `  G )  =  ( invg `  G )
108, 9grpinvid 12862 . . . . . . . . 9  |-  ( G  e.  Grp  ->  (
( invg `  G ) `  .0.  )  =  .0.  )
117, 10syl 14 . . . . . . . 8  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
( invg `  G ) `  .0.  )  =  .0.  )
1211oveq1d 5887 . . . . . . 7  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
( ( invg `  G ) `  .0.  ) ( +g  `  G
) x )  =  (  .0.  ( +g  `  G ) x ) )
13 eqger.x . . . . . . . . 9  |-  X  =  ( Base `  G
)
14 eqid 2177 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
1513, 14, 8grplid 12838 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  (  .0.  ( +g  `  G ) x )  =  x )
161, 15sylan 283 . . . . . . 7  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (  .0.  ( +g  `  G
) x )  =  x )
1712, 16eqtrd 2210 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
( ( invg `  G ) `  .0.  ) ( +g  `  G
) x )  =  x )
1817eleq1d 2246 . . . . 5  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
( ( ( invg `  G ) `
 .0.  ) ( +g  `  G ) x )  e.  Y  <->  x  e.  Y ) )
1918pm5.32da 452 . . . 4  |-  ( Y  e.  (SubGrp `  G
)  ->  ( (
x  e.  X  /\  ( ( ( invg `  G ) `
 .0.  ) ( +g  `  G ) x )  e.  Y
)  <->  ( x  e.  X  /\  x  e.  Y ) ) )
2013subgss 12965 . . . . 5  |-  ( Y  e.  (SubGrp `  G
)  ->  Y  C_  X
)
2113, 8grpidcl 12836 . . . . . 6  |-  ( G  e.  Grp  ->  .0.  e.  X )
221, 21syl 14 . . . . 5  |-  ( Y  e.  (SubGrp `  G
)  ->  .0.  e.  X )
2313, 9, 14, 2eqgval 13013 . . . . . . 7  |-  ( ( G  e.  Grp  /\  Y  C_  X )  -> 
(  .0.  .~  x  <->  (  .0.  e.  X  /\  x  e.  X  /\  ( ( ( invg `  G ) `
 .0.  ) ( +g  `  G ) x )  e.  Y
) ) )
24 3anass 982 . . . . . . 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 ) ) )
2523, 24bitrdi 196 . . . . . 6  |-  ( ( G  e.  Grp  /\  Y  C_  X )  -> 
(  .0.  .~  x  <->  (  .0.  e.  X  /\  ( x  e.  X  /\  ( ( ( invg `  G ) `
 .0.  ) ( +g  `  G ) x )  e.  Y
) ) ) )
2625baibd 923 . . . . 5  |-  ( ( ( G  e.  Grp  /\  Y  C_  X )  /\  .0.  e.  X )  ->  (  .0.  .~  x 
<->  ( x  e.  X  /\  ( ( ( invg `  G ) `
 .0.  ) ( +g  `  G ) x )  e.  Y
) ) )
271, 20, 22, 26syl21anc 1237 . . . 4  |-  ( Y  e.  (SubGrp `  G
)  ->  (  .0.  .~  x  <->  ( x  e.  X  /\  ( ( ( invg `  G ) `  .0.  ) ( +g  `  G
) x )  e.  Y ) ) )
2820sseld 3154 . . . . 5  |-  ( Y  e.  (SubGrp `  G
)  ->  ( x  e.  Y  ->  x  e.  X ) )
2928pm4.71rd 394 . . . 4  |-  ( Y  e.  (SubGrp `  G
)  ->  ( x  e.  Y  <->  ( x  e.  X  /\  x  e.  Y ) ) )
3019, 27, 293bitr4d 220 . . 3  |-  ( Y  e.  (SubGrp `  G
)  ->  (  .0.  .~  x  <->  x  e.  Y
) )
316, 30bitrd 188 . 2  |-  ( Y  e.  (SubGrp `  G
)  ->  ( x  e.  [  .0.  ]  .~  <->  x  e.  Y ) )
3231eqrdv 2175 1  |-  ( Y  e.  (SubGrp `  G
)  ->  [  .0.  ]  .~  =  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148    C_ wss 3129   class class class wbr 4002   Rel wrel 4630   ` cfv 5215  (class class class)co 5872   [cec 6530   Basecbs 12454   +g cplusg 12528   0gc0g 12693   Grpcgrp 12809   invgcminusg 12810  SubGrpcsubg 12958   ~QG cqg 12960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1re 7902  ax-addrcl 7905
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-ec 6534  df-inn 8916  df-2 8974  df-ndx 12457  df-slot 12458  df-base 12460  df-plusg 12541  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812  df-minusg 12813  df-subg 12961  df-eqg 12963
This theorem is referenced by: (None)
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