MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eqgid Structured version   Unicode version

Theorem eqgid 16370
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 16365 . . . 4  |-  Rel  .~
3 relelec 7270 . . . 4  |-  ( Rel 
.~  ->  ( x  e. 
[  .0.  ]  .~  <->  .0. 
.~  x ) )
42, 3ax-mp 5 . . 3  |-  ( x  e.  [  .0.  ]  .~ 
<->  .0.  .~  x )
5 subgrcl 16323 . . . . . . . . . 10  |-  ( Y  e.  (SubGrp `  G
)  ->  G  e.  Grp )
65adantr 463 . . . . . . . . 9  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  G  e.  Grp )
7 eqgid.3 . . . . . . . . . 10  |-  .0.  =  ( 0g `  G )
8 eqid 2382 . . . . . . . . . 10  |-  ( invg `  G )  =  ( invg `  G )
97, 8grpinvid 16218 . . . . . . . . 9  |-  ( G  e.  Grp  ->  (
( invg `  G ) `  .0.  )  =  .0.  )
106, 9syl 16 . . . . . . . 8  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
( invg `  G ) `  .0.  )  =  .0.  )
1110oveq1d 6211 . . . . . . 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 2382 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
1412, 13, 7grplid 16197 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  (  .0.  ( +g  `  G ) x )  =  x )
155, 14sylan 469 . . . . . . 7  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (  .0.  ( +g  `  G
) x )  =  x )
1611, 15eqtrd 2423 . . . . . 6  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
( ( invg `  G ) `  .0.  ) ( +g  `  G
) x )  =  x )
1716eleq1d 2451 . . . . 5  |-  ( ( Y  e.  (SubGrp `  G )  /\  x  e.  X )  ->  (
( ( ( invg `  G ) `
 .0.  ) ( +g  `  G ) x )  e.  Y  <->  x  e.  Y ) )
1817pm5.32da 639 . . . 4  |-  ( Y  e.  (SubGrp `  G
)  ->  ( (
x  e.  X  /\  ( ( ( invg `  G ) `
 .0.  ) ( +g  `  G ) x )  e.  Y
)  <->  ( x  e.  X  /\  x  e.  Y ) ) )
1912subgss 16319 . . . . 5  |-  ( Y  e.  (SubGrp `  G
)  ->  Y  C_  X
)
2012, 7grpidcl 16195 . . . . . 6  |-  ( G  e.  Grp  ->  .0.  e.  X )
215, 20syl 16 . . . . 5  |-  ( Y  e.  (SubGrp `  G
)  ->  .0.  e.  X )
2212, 8, 13, 1eqgval 16367 . . . . . . 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 975 . . . . . . 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 1225 . . . 4  |-  ( Y  e.  (SubGrp `  G
)  ->  (  .0.  .~  x  <->  ( x  e.  X  /\  ( ( ( invg `  G ) `  .0.  ) ( +g  `  G
) x )  e.  Y ) ) )
2719sseld 3416 . . . . 5  |-  ( Y  e.  (SubGrp `  G
)  ->  ( x  e.  Y  ->  x  e.  X ) )
2827pm4.71rd 633 . . . 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 2379 1  |-  ( Y  e.  (SubGrp `  G
)  ->  [  .0.  ]  .~  =  Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    C_ wss 3389   class class class wbr 4367   Rel wrel 4918   ` cfv 5496  (class class class)co 6196   [cec 7227   Basecbs 14634   +g cplusg 14702   0gc0g 14847   Grpcgrp 16170   invgcminusg 16171  SubGrpcsubg 16312   ~QG cqg 16314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-ec 7231  df-0g 14849  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-grp 16174  df-minusg 16175  df-subg 16315  df-eqg 16317
This theorem is referenced by:  cldsubg  20694  qustgphaus  20706
  Copyright terms: Public domain W3C validator