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Theorem divs0 14885
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 14859 . . . . . . 7  |-  ( S  e.  (NrmSGrp `  G
)  ->  S  e.  (SubGrp `  G ) )
2 subgrcl 14836 . . . . . . 7  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
31, 2syl 15 . . . . . 6  |-  ( S  e.  (NrmSGrp `  G
)  ->  G  e.  Grp )
4 eqid 2366 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
5 divs0.p . . . . . . 7  |-  .0.  =  ( 0g `  G )
64, 5grpidcl 14720 . . . . . 6  |-  ( G  e.  Grp  ->  .0.  e.  ( Base `  G
) )
73, 6syl 15 . . . . 5  |-  ( S  e.  (NrmSGrp `  G
)  ->  .0.  e.  ( Base `  G )
)
8 divsgrp.h . . . . . 6  |-  H  =  ( G  /.s  ( G ~QG  S
) )
9 eqid 2366 . . . . . 6  |-  ( +g  `  G )  =  ( +g  `  G )
10 eqid 2366 . . . . . 6  |-  ( +g  `  H )  =  ( +g  `  H )
118, 4, 9, 10divsadd 14884 . . . . 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 1280 . . . 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 14722 . . . . . 6  |-  ( ( G  e.  Grp  /\  .0.  e.  ( Base `  G
) )  ->  (  .0.  ( +g  `  G
)  .0.  )  =  .0.  )
143, 7, 13syl2anc 642 . . . . 5  |-  ( S  e.  (NrmSGrp `  G
)  ->  (  .0.  ( +g  `  G )  .0.  )  =  .0.  )
15 eceq1 6838 . . . . 5  |-  ( (  .0.  ( +g  `  G
)  .0.  )  =  .0.  ->  [ (  .0.  ( +g  `  G
)  .0.  ) ] ( G ~QG  S )  =  [  .0.  ] ( G ~QG  S ) )
1614, 15syl 15 . . . 4  |-  ( S  e.  (NrmSGrp `  G
)  ->  [ (  .0.  ( +g  `  G
)  .0.  ) ] ( G ~QG  S )  =  [  .0.  ] ( G ~QG  S ) )
1712, 16eqtrd 2398 . . 3  |-  ( S  e.  (NrmSGrp `  G
)  ->  ( [  .0.  ] ( G ~QG  S ) ( +g  `  H
) [  .0.  ]
( G ~QG  S ) )  =  [  .0.  ] ( G ~QG  S ) )
188divsgrp 14882 . . . 4  |-  ( S  e.  (NrmSGrp `  G
)  ->  H  e.  Grp )
19 eqid 2366 . . . . . 6  |-  ( Base `  H )  =  (
Base `  H )
208, 4, 19divseccl 14883 . . . . 5  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  .0.  e.  ( Base `  G )
)  ->  [  .0.  ] ( G ~QG  S )  e.  (
Base `  H )
)
217, 20mpdan 649 . . . 4  |-  ( S  e.  (NrmSGrp `  G
)  ->  [  .0.  ] ( G ~QG  S )  e.  (
Base `  H )
)
22 eqid 2366 . . . . 5  |-  ( 0g
`  H )  =  ( 0g `  H
)
2319, 10, 22grpid 14727 . . . 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 642 . . 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 201 . 2  |-  ( S  e.  (NrmSGrp `  G
)  ->  ( 0g `  H )  =  [  .0.  ] ( G ~QG  S ) )
2625eqcomd 2371 1  |-  ( S  e.  (NrmSGrp `  G
)  ->  [  .0.  ] ( G ~QG  S )  =  ( 0g `  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1647    e. wcel 1715   ` cfv 5358  (class class class)co 5981   [cec 6800   Basecbs 13356   +g cplusg 13416   0gc0g 13610    /.s cqus 13618   Grpcgrp 14572  SubGrpcsubg 14825  NrmSGrpcnsg 14826   ~QG cqg 14827
This theorem is referenced by:  divsinv  14886  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  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  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-nel 2532  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-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  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-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-er 6802  df-ec 6804  df-qs 6808  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-sup 7341  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-5 9954  df-6 9955  df-7 9956  df-8 9957  df-9 9958  df-10 9959  df-n0 10115  df-z 10176  df-dec 10276  df-uz 10382  df-fz 10936  df-struct 13358  df-ndx 13359  df-slot 13360  df-base 13361  df-sets 13362  df-ress 13363  df-plusg 13429  df-mulr 13430  df-sca 13432  df-vsca 13433  df-tset 13435  df-ple 13436  df-ds 13438  df-0g 13614  df-imas 13621  df-divs 13622  df-mnd 14577  df-grp 14699  df-minusg 14700  df-subg 14828  df-nsg 14829  df-eqg 14830
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