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Theorem divs0 15843
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 15817 . . . . . . 7  |-  ( S  e.  (NrmSGrp `  G
)  ->  S  e.  (SubGrp `  G ) )
2 subgrcl 15790 . . . . . . 7  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
31, 2syl 16 . . . . . 6  |-  ( S  e.  (NrmSGrp `  G
)  ->  G  e.  Grp )
4 eqid 2451 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
5 divs0.p . . . . . . 7  |-  .0.  =  ( 0g `  G )
64, 5grpidcl 15670 . . . . . 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 2451 . . . . . 6  |-  ( +g  `  G )  =  ( +g  `  G )
10 eqid 2451 . . . . . 6  |-  ( +g  `  H )  =  ( +g  `  H )
118, 4, 9, 10divsadd 15842 . . . . 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 1317 . . . 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 15672 . . . . . 6  |-  ( ( G  e.  Grp  /\  .0.  e.  ( Base `  G
) )  ->  (  .0.  ( +g  `  G
)  .0.  )  =  .0.  )
143, 7, 13syl2anc 661 . . . . 5  |-  ( S  e.  (NrmSGrp `  G
)  ->  (  .0.  ( +g  `  G )  .0.  )  =  .0.  )
15 eceq1 7239 . . . . 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 2492 . . 3  |-  ( S  e.  (NrmSGrp `  G
)  ->  ( [  .0.  ] ( G ~QG  S ) ( +g  `  H
) [  .0.  ]
( G ~QG  S ) )  =  [  .0.  ] ( G ~QG  S ) )
188divsgrp 15840 . . . 4  |-  ( S  e.  (NrmSGrp `  G
)  ->  H  e.  Grp )
19 eqid 2451 . . . . . 6  |-  ( Base `  H )  =  (
Base `  H )
208, 4, 19divseccl 15841 . . . . 5  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  .0.  e.  ( Base `  G )
)  ->  [  .0.  ] ( G ~QG  S )  e.  (
Base `  H )
)
217, 20mpdan 668 . . . 4  |-  ( S  e.  (NrmSGrp `  G
)  ->  [  .0.  ] ( G ~QG  S )  e.  (
Base `  H )
)
22 eqid 2451 . . . . 5  |-  ( 0g
`  H )  =  ( 0g `  H
)
2319, 10, 22grpid 15677 . . . 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 661 . . 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 210 . 2  |-  ( S  e.  (NrmSGrp `  G
)  ->  ( 0g `  H )  =  [  .0.  ] ( G ~QG  S ) )
2625eqcomd 2459 1  |-  ( S  e.  (NrmSGrp `  G
)  ->  [  .0.  ] ( G ~QG  S )  =  ( 0g `  H ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370    e. wcel 1758   ` cfv 5518  (class class class)co 6192   [cec 7201   Basecbs 14278   +g cplusg 14342   0gc0g 14482    /.s cqus 14547   Grpcgrp 15514  SubGrpcsubg 15779  NrmSGrpcnsg 15780   ~QG cqg 15781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-ec 7205  df-qs 7209  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-sup 7794  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-5 10486  df-6 10487  df-7 10488  df-8 10489  df-9 10490  df-10 10491  df-n0 10683  df-z 10750  df-dec 10859  df-uz 10965  df-fz 11541  df-struct 14280  df-ndx 14281  df-slot 14282  df-base 14283  df-sets 14284  df-ress 14285  df-plusg 14355  df-mulr 14356  df-sca 14358  df-vsca 14359  df-ip 14360  df-tset 14361  df-ple 14362  df-ds 14364  df-0g 14484  df-imas 14550  df-divs 14551  df-mnd 15519  df-grp 15649  df-minusg 15650  df-subg 15782  df-nsg 15783  df-eqg 15784
This theorem is referenced by:  divsinv  15844  divstgphaus  19811
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