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Theorem divsgrp 15729
Description: If  Y is a normal subgroup of  G, then  H  =  G  /  Y is a group, called the quotient of  G by  Y. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypothesis
Ref Expression
divsgrp.h  |-  H  =  ( G  /.s  ( G ~QG  S
) )
Assertion
Ref Expression
divsgrp  |-  ( S  e.  (NrmSGrp `  G
)  ->  H  e.  Grp )

Proof of Theorem divsgrp
Dummy variables  a 
b  c  d  u  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 divsgrp.h . . . 4  |-  H  =  ( G  /.s  ( G ~QG  S
) )
21a1i 11 . . 3  |-  ( S  e.  (NrmSGrp `  G
)  ->  H  =  ( G  /.s  ( G ~QG  S ) ) )
3 eqidd 2442 . . 3  |-  ( S  e.  (NrmSGrp `  G
)  ->  ( Base `  G )  =  (
Base `  G )
)
4 eqidd 2442 . . 3  |-  ( S  e.  (NrmSGrp `  G
)  ->  ( +g  `  G )  =  ( +g  `  G ) )
5 nsgsubg 15706 . . . 4  |-  ( S  e.  (NrmSGrp `  G
)  ->  S  e.  (SubGrp `  G ) )
6 eqid 2441 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
7 eqid 2441 . . . . 5  |-  ( G ~QG  S )  =  ( G ~QG  S )
86, 7eqger 15724 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  ( G ~QG  S
)  Er  ( Base `  G ) )
95, 8syl 16 . . 3  |-  ( S  e.  (NrmSGrp `  G
)  ->  ( G ~QG  S
)  Er  ( Base `  G ) )
10 subgrcl 15679 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
115, 10syl 16 . . 3  |-  ( S  e.  (NrmSGrp `  G
)  ->  G  e.  Grp )
12 eqid 2441 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
136, 7, 12eqgcpbl 15728 . . 3  |-  ( S  e.  (NrmSGrp `  G
)  ->  ( (
a ( G ~QG  S ) c  /\  b ( G ~QG  S ) d )  ->  ( a ( +g  `  G ) b ) ( G ~QG  S ) ( c ( +g  `  G ) d ) ) )
146, 12grpcl 15544 . . . 4  |-  ( ( G  e.  Grp  /\  u  e.  ( Base `  G )  /\  v  e.  ( Base `  G
) )  ->  (
u ( +g  `  G
) v )  e.  ( Base `  G
) )
1511, 14syl3an1 1246 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  u  e.  ( Base `  G )  /\  v  e.  ( Base `  G ) )  ->  ( u ( +g  `  G ) v )  e.  (
Base `  G )
)
169adantr 462 . . . . 5  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  ( u  e.  ( Base `  G
)  /\  v  e.  ( Base `  G )  /\  w  e.  ( Base `  G ) ) )  ->  ( G ~QG  S
)  Er  ( Base `  G ) )
1711adantr 462 . . . . . 6  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  ( u  e.  ( Base `  G
)  /\  v  e.  ( Base `  G )  /\  w  e.  ( Base `  G ) ) )  ->  G  e.  Grp )
18 simpr1 989 . . . . . . 7  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  ( u  e.  ( Base `  G
)  /\  v  e.  ( Base `  G )  /\  w  e.  ( Base `  G ) ) )  ->  u  e.  ( Base `  G )
)
19 simpr2 990 . . . . . . 7  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  ( u  e.  ( Base `  G
)  /\  v  e.  ( Base `  G )  /\  w  e.  ( Base `  G ) ) )  ->  v  e.  ( Base `  G )
)
2017, 18, 19, 14syl3anc 1213 . . . . . 6  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  ( u  e.  ( Base `  G
)  /\  v  e.  ( Base `  G )  /\  w  e.  ( Base `  G ) ) )  ->  ( u
( +g  `  G ) v )  e.  (
Base `  G )
)
21 simpr3 991 . . . . . 6  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  ( u  e.  ( Base `  G
)  /\  v  e.  ( Base `  G )  /\  w  e.  ( Base `  G ) ) )  ->  w  e.  ( Base `  G )
)
226, 12grpcl 15544 . . . . . 6  |-  ( ( G  e.  Grp  /\  ( u ( +g  `  G ) v )  e.  ( Base `  G
)  /\  w  e.  ( Base `  G )
)  ->  ( (
u ( +g  `  G
) v ) ( +g  `  G ) w )  e.  (
Base `  G )
)
2317, 20, 21, 22syl3anc 1213 . . . . 5  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  ( u  e.  ( Base `  G
)  /\  v  e.  ( Base `  G )  /\  w  e.  ( Base `  G ) ) )  ->  ( (
u ( +g  `  G
) v ) ( +g  `  G ) w )  e.  (
Base `  G )
)
2416, 23erref 7117 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  ( u  e.  ( Base `  G
)  /\  v  e.  ( Base `  G )  /\  w  e.  ( Base `  G ) ) )  ->  ( (
u ( +g  `  G
) v ) ( +g  `  G ) w ) ( G ~QG  S ) ( ( u ( +g  `  G
) v ) ( +g  `  G ) w ) )
256, 12grpass 15545 . . . . 5  |-  ( ( G  e.  Grp  /\  ( u  e.  ( Base `  G )  /\  v  e.  ( Base `  G )  /\  w  e.  ( Base `  G
) ) )  -> 
( ( u ( +g  `  G ) v ) ( +g  `  G ) w )  =  ( u ( +g  `  G ) ( v ( +g  `  G ) w ) ) )
2611, 25sylan 468 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  ( u  e.  ( Base `  G
)  /\  v  e.  ( Base `  G )  /\  w  e.  ( Base `  G ) ) )  ->  ( (
u ( +g  `  G
) v ) ( +g  `  G ) w )  =  ( u ( +g  `  G
) ( v ( +g  `  G ) w ) ) )
2724, 26breqtrd 4313 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  ( u  e.  ( Base `  G
)  /\  v  e.  ( Base `  G )  /\  w  e.  ( Base `  G ) ) )  ->  ( (
u ( +g  `  G
) v ) ( +g  `  G ) w ) ( G ~QG  S ) ( u ( +g  `  G ) ( v ( +g  `  G ) w ) ) )
28 eqid 2441 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
296, 28grpidcl 15559 . . . 4  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  ( Base `  G
) )
3011, 29syl 16 . . 3  |-  ( S  e.  (NrmSGrp `  G
)  ->  ( 0g `  G )  e.  (
Base `  G )
)
316, 12, 28grplid 15561 . . . . 5  |-  ( ( G  e.  Grp  /\  u  e.  ( Base `  G ) )  -> 
( ( 0g `  G ) ( +g  `  G ) u )  =  u )
3211, 31sylan 468 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  u  e.  ( Base `  G )
)  ->  ( ( 0g `  G ) ( +g  `  G ) u )  =  u )
339adantr 462 . . . . 5  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  u  e.  ( Base `  G )
)  ->  ( G ~QG  S
)  Er  ( Base `  G ) )
34 simpr 458 . . . . 5  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  u  e.  ( Base `  G )
)  ->  u  e.  ( Base `  G )
)
3533, 34erref 7117 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  u  e.  ( Base `  G )
)  ->  u ( G ~QG  S ) u )
3632, 35eqbrtrd 4309 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  u  e.  ( Base `  G )
)  ->  ( ( 0g `  G ) ( +g  `  G ) u ) ( G ~QG  S ) u )
37 eqid 2441 . . . . 5  |-  ( invg `  G )  =  ( invg `  G )
386, 37grpinvcl 15576 . . . 4  |-  ( ( G  e.  Grp  /\  u  e.  ( Base `  G ) )  -> 
( ( invg `  G ) `  u
)  e.  ( Base `  G ) )
3911, 38sylan 468 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  u  e.  ( Base `  G )
)  ->  ( ( invg `  G ) `
 u )  e.  ( Base `  G
) )
406, 12, 28, 37grplinv 15577 . . . . 5  |-  ( ( G  e.  Grp  /\  u  e.  ( Base `  G ) )  -> 
( ( ( invg `  G ) `
 u ) ( +g  `  G ) u )  =  ( 0g `  G ) )
4111, 40sylan 468 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  u  e.  ( Base `  G )
)  ->  ( (
( invg `  G ) `  u
) ( +g  `  G
) u )  =  ( 0g `  G
) )
4230adantr 462 . . . . 5  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  u  e.  ( Base `  G )
)  ->  ( 0g `  G )  e.  (
Base `  G )
)
4333, 42erref 7117 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  u  e.  ( Base `  G )
)  ->  ( 0g `  G ) ( G ~QG  S ) ( 0g `  G ) )
4441, 43eqbrtrd 4309 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  u  e.  ( Base `  G )
)  ->  ( (
( invg `  G ) `  u
) ( +g  `  G
) u ) ( G ~QG  S ) ( 0g
`  G ) )
452, 3, 4, 9, 11, 13, 15, 27, 30, 36, 39, 44divsgrp2 15666 . 2  |-  ( S  e.  (NrmSGrp `  G
)  ->  ( H  e.  Grp  /\  [ ( 0g `  G ) ] ( G ~QG  S )  =  ( 0g `  H ) ) )
4645simpld 456 1  |-  ( S  e.  (NrmSGrp `  G
)  ->  H  e.  Grp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   ` cfv 5415  (class class class)co 6090    Er wer 7094   [cec 7095   Basecbs 14170   +g cplusg 14234   0gc0g 14374    /.s cqus 14439   Grpcgrp 15406   invgcminusg 15407  SubGrpcsubg 15668  NrmSGrpcnsg 15669   ~QG cqg 15670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-ec 7099  df-qs 7103  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-0g 14376  df-imas 14442  df-divs 14443  df-mnd 15411  df-grp 15538  df-minusg 15539  df-subg 15671  df-nsg 15672  df-eqg 15673
This theorem is referenced by:  divs0  15732  divsinv  15733  divsghm  15776  divsabl  16340  divstgplem  19650
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