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Theorem divsgrp 15850
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 2453 . . 3  |-  ( S  e.  (NrmSGrp `  G
)  ->  ( Base `  G )  =  (
Base `  G )
)
4 eqidd 2453 . . 3  |-  ( S  e.  (NrmSGrp `  G
)  ->  ( +g  `  G )  =  ( +g  `  G ) )
5 nsgsubg 15827 . . . 4  |-  ( S  e.  (NrmSGrp `  G
)  ->  S  e.  (SubGrp `  G ) )
6 eqid 2452 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
7 eqid 2452 . . . . 5  |-  ( G ~QG  S )  =  ( G ~QG  S )
86, 7eqger 15845 . . . 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 15800 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
115, 10syl 16 . . 3  |-  ( S  e.  (NrmSGrp `  G
)  ->  G  e.  Grp )
12 eqid 2452 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
136, 7, 12eqgcpbl 15849 . . 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 15665 . . . 4  |-  ( ( G  e.  Grp  /\  u  e.  ( Base `  G )  /\  v  e.  ( Base `  G
) )  ->  (
u ( +g  `  G
) v )  e.  ( Base `  G
) )
1511, 14syl3an1 1252 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  u  e.  ( Base `  G )  /\  v  e.  ( Base `  G ) )  ->  ( u ( +g  `  G ) v )  e.  (
Base `  G )
)
169adantr 465 . . . . 5  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  ( u  e.  ( Base `  G
)  /\  v  e.  ( Base `  G )  /\  w  e.  ( Base `  G ) ) )  ->  ( G ~QG  S
)  Er  ( Base `  G ) )
1711adantr 465 . . . . . 6  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  ( u  e.  ( Base `  G
)  /\  v  e.  ( Base `  G )  /\  w  e.  ( Base `  G ) ) )  ->  G  e.  Grp )
18 simpr1 994 . . . . . . 7  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  ( u  e.  ( Base `  G
)  /\  v  e.  ( Base `  G )  /\  w  e.  ( Base `  G ) ) )  ->  u  e.  ( Base `  G )
)
19 simpr2 995 . . . . . . 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 1219 . . . . . 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 996 . . . . . 6  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  ( u  e.  ( Base `  G
)  /\  v  e.  ( Base `  G )  /\  w  e.  ( Base `  G ) ) )  ->  w  e.  ( Base `  G )
)
226, 12grpcl 15665 . . . . . 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 1219 . . . . 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 7226 . . . 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 15666 . . . . 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 471 . . . 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 4419 . . 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 2452 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
296, 28grpidcl 15680 . . . 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 15682 . . . . 5  |-  ( ( G  e.  Grp  /\  u  e.  ( Base `  G ) )  -> 
( ( 0g `  G ) ( +g  `  G ) u )  =  u )
3211, 31sylan 471 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  u  e.  ( Base `  G )
)  ->  ( ( 0g `  G ) ( +g  `  G ) u )  =  u )
339adantr 465 . . . . 5  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  u  e.  ( Base `  G )
)  ->  ( G ~QG  S
)  Er  ( Base `  G ) )
34 simpr 461 . . . . 5  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  u  e.  ( Base `  G )
)  ->  u  e.  ( Base `  G )
)
3533, 34erref 7226 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  u  e.  ( Base `  G )
)  ->  u ( G ~QG  S ) u )
3632, 35eqbrtrd 4415 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  u  e.  ( Base `  G )
)  ->  ( ( 0g `  G ) ( +g  `  G ) u ) ( G ~QG  S ) u )
37 eqid 2452 . . . . 5  |-  ( invg `  G )  =  ( invg `  G )
386, 37grpinvcl 15697 . . . 4  |-  ( ( G  e.  Grp  /\  u  e.  ( Base `  G ) )  -> 
( ( invg `  G ) `  u
)  e.  ( Base `  G ) )
3911, 38sylan 471 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  u  e.  ( Base `  G )
)  ->  ( ( invg `  G ) `
 u )  e.  ( Base `  G
) )
406, 12, 28, 37grplinv 15698 . . . . 5  |-  ( ( G  e.  Grp  /\  u  e.  ( Base `  G ) )  -> 
( ( ( invg `  G ) `
 u ) ( +g  `  G ) u )  =  ( 0g `  G ) )
4111, 40sylan 471 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  u  e.  ( Base `  G )
)  ->  ( (
( invg `  G ) `  u
) ( +g  `  G
) u )  =  ( 0g `  G
) )
4230adantr 465 . . . . 5  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  u  e.  ( Base `  G )
)  ->  ( 0g `  G )  e.  (
Base `  G )
)
4333, 42erref 7226 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  u  e.  ( Base `  G )
)  ->  ( 0g `  G ) ( G ~QG  S ) ( 0g `  G ) )
4441, 43eqbrtrd 4415 . . 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 15787 . 2  |-  ( S  e.  (NrmSGrp `  G
)  ->  ( H  e.  Grp  /\  [ ( 0g `  G ) ] ( G ~QG  S )  =  ( 0g `  H ) ) )
4645simpld 459 1  |-  ( S  e.  (NrmSGrp `  G
)  ->  H  e.  Grp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   ` cfv 5521  (class class class)co 6195    Er wer 7203   [cec 7204   Basecbs 14287   +g cplusg 14352   0gc0g 14492    /.s cqus 14557   Grpcgrp 15524   invgcminusg 15525  SubGrpcsubg 15789  NrmSGrpcnsg 15790   ~QG cqg 15791
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 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-ec 7208  df-qs 7212  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-sup 7797  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-3 10487  df-4 10488  df-5 10489  df-6 10490  df-7 10491  df-8 10492  df-9 10493  df-10 10494  df-n0 10686  df-z 10753  df-dec 10862  df-uz 10968  df-fz 11550  df-struct 14289  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-ress 14294  df-plusg 14365  df-mulr 14366  df-sca 14368  df-vsca 14369  df-ip 14370  df-tset 14371  df-ple 14372  df-ds 14374  df-0g 14494  df-imas 14560  df-divs 14561  df-mnd 15529  df-grp 15659  df-minusg 15660  df-subg 15792  df-nsg 15793  df-eqg 15794
This theorem is referenced by:  divs0  15853  divsinv  15854  divsghm  15897  divsabl  16463  divstgplem  19818
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