MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  divs0 Unicode version

Theorem divs0 14961
Description: Value of the group identity operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
Hypotheses
Ref Expression
divsgrp.h  |-  H  =  ( G  /.s  ( G ~QG  S
) )
divs0.p  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
divs0  |-  ( S  e.  (NrmSGrp `  G
)  ->  [  .0.  ] ( G ~QG  S )  =  ( 0g `  H ) )

Proof of Theorem divs0
StepHypRef Expression
1 nsgsubg 14935 . . . . . . 7  |-  ( S  e.  (NrmSGrp `  G
)  ->  S  e.  (SubGrp `  G ) )
2 subgrcl 14912 . . . . . . 7  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
31, 2syl 16 . . . . . 6  |-  ( S  e.  (NrmSGrp `  G
)  ->  G  e.  Grp )
4 eqid 2412 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
5 divs0.p . . . . . . 7  |-  .0.  =  ( 0g `  G )
64, 5grpidcl 14796 . . . . . 6  |-  ( G  e.  Grp  ->  .0.  e.  ( Base `  G
) )
73, 6syl 16 . . . . 5  |-  ( S  e.  (NrmSGrp `  G
)  ->  .0.  e.  ( Base `  G )
)
8 divsgrp.h . . . . . 6  |-  H  =  ( G  /.s  ( G ~QG  S
) )
9 eqid 2412 . . . . . 6  |-  ( +g  `  G )  =  ( +g  `  G )
10 eqid 2412 . . . . . 6  |-  ( +g  `  H )  =  ( +g  `  H )
118, 4, 9, 10divsadd 14960 . . . . 5  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  .0.  e.  ( Base `  G )  /\  .0.  e.  ( Base `  G ) )  -> 
( [  .0.  ]
( G ~QG  S ) ( +g  `  H ) [  .0.  ] ( G ~QG  S ) )  =  [ (  .0.  ( +g  `  G )  .0.  ) ] ( G ~QG  S ) )
127, 7, 11mpd3an23 1281 . . . 4  |-  ( S  e.  (NrmSGrp `  G
)  ->  ( [  .0.  ] ( G ~QG  S ) ( +g  `  H
) [  .0.  ]
( G ~QG  S ) )  =  [ (  .0.  ( +g  `  G )  .0.  ) ] ( G ~QG  S ) )
134, 9, 5grplid 14798 . . . . . 6  |-  ( ( G  e.  Grp  /\  .0.  e.  ( Base `  G
) )  ->  (  .0.  ( +g  `  G
)  .0.  )  =  .0.  )
143, 7, 13syl2anc 643 . . . . 5  |-  ( S  e.  (NrmSGrp `  G
)  ->  (  .0.  ( +g  `  G )  .0.  )  =  .0.  )
15 eceq1 6908 . . . . 5  |-  ( (  .0.  ( +g  `  G
)  .0.  )  =  .0.  ->  [ (  .0.  ( +g  `  G
)  .0.  ) ] ( G ~QG  S )  =  [  .0.  ] ( G ~QG  S ) )
1614, 15syl 16 . . . 4  |-  ( S  e.  (NrmSGrp `  G
)  ->  [ (  .0.  ( +g  `  G
)  .0.  ) ] ( G ~QG  S )  =  [  .0.  ] ( G ~QG  S ) )
1712, 16eqtrd 2444 . . 3  |-  ( S  e.  (NrmSGrp `  G
)  ->  ( [  .0.  ] ( G ~QG  S ) ( +g  `  H
) [  .0.  ]
( G ~QG  S ) )  =  [  .0.  ] ( G ~QG  S ) )
188divsgrp 14958 . . . 4  |-  ( S  e.  (NrmSGrp `  G
)  ->  H  e.  Grp )
19 eqid 2412 . . . . . 6  |-  ( Base `  H )  =  (
Base `  H )
208, 4, 19divseccl 14959 . . . . 5  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  .0.  e.  ( Base `  G )
)  ->  [  .0.  ] ( G ~QG  S )  e.  (
Base `  H )
)
217, 20mpdan 650 . . . 4  |-  ( S  e.  (NrmSGrp `  G
)  ->  [  .0.  ] ( G ~QG  S )  e.  (
Base `  H )
)
22 eqid 2412 . . . . 5  |-  ( 0g
`  H )  =  ( 0g `  H
)
2319, 10, 22grpid 14803 . . . 4  |-  ( ( H  e.  Grp  /\  [  .0.  ] ( G ~QG  S )  e.  ( Base `  H ) )  -> 
( ( [  .0.  ] ( G ~QG  S ) ( +g  `  H ) [  .0.  ] ( G ~QG  S ) )  =  [  .0.  ] ( G ~QG  S )  <->  ( 0g `  H )  =  [  .0.  ] ( G ~QG  S ) ) )
2418, 21, 23syl2anc 643 . . 3  |-  ( S  e.  (NrmSGrp `  G
)  ->  ( ( [  .0.  ] ( G ~QG  S ) ( +g  `  H
) [  .0.  ]
( G ~QG  S ) )  =  [  .0.  ] ( G ~QG  S )  <->  ( 0g `  H )  =  [  .0.  ] ( G ~QG  S ) ) )
2517, 24mpbid 202 . 2  |-  ( S  e.  (NrmSGrp `  G
)  ->  ( 0g `  H )  =  [  .0.  ] ( G ~QG  S ) )
2625eqcomd 2417 1  |-  ( S  e.  (NrmSGrp `  G
)  ->  [  .0.  ] ( G ~QG  S )  =  ( 0g `  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1721   ` cfv 5421  (class class class)co 6048   [cec 6870   Basecbs 13432   +g cplusg 13492   0gc0g 13686    /.s cqus 13694   Grpcgrp 14648  SubGrpcsubg 14901  NrmSGrpcnsg 14902   ~QG cqg 14903
This theorem is referenced by:  divsinv  14962  divstgphaus  18113
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-ec 6874  df-qs 6878  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-fz 11008  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-sca 13508  df-vsca 13509  df-tset 13511  df-ple 13512  df-ds 13514  df-0g 13690  df-imas 13697  df-divs 13698  df-mnd 14653  df-grp 14775  df-minusg 14776  df-subg 14904  df-nsg 14905  df-eqg 14906
  Copyright terms: Public domain W3C validator