Theorem qusgrp 13302

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
qusgrp.h	|- H = ( G /.s ( G ~QG S ) )

Assertion

Ref	Expression
qusgrp	|- ( S e. (NrmSGrp ` G ) -> H e. Grp )

Proof of Theorem qusgrp

Dummy variables a b c d u v w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		qusgrp.h	. . . 4 |- H = ( G /.s ( G ~QG S ) )
2	1	a1i 9	. . 3 |- ( S e. (NrmSGrp ` G ) -> H = ( G /.s ( G ~QG S ) ) )
3		eqidd 2194	. . 3 |- ( S e. (NrmSGrp ` G ) -> ( Base ` G ) = ( Base ` G ) )
4		eqidd 2194	. . 3 |- ( S e. (NrmSGrp ` G ) -> ( +g ` G ) = ( +g ` G ) )
5		nsgsubg 13275	. . . 4 |- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
6		eqid 2193	. . . . 5 |- ( Base ` G ) = ( Base ` G )
7		eqid 2193	. . . . 5 |- ( G ~QG S ) = ( G ~QG S )
8	6, 7	eqger 13294	. . . 4 |- ( S e. (SubGrp ` G ) -> ( G ~QG S ) Er ( Base ` G ) )
9	5, 8	syl 14	. . 3 |- ( S e. (NrmSGrp ` G ) -> ( G ~QG S ) Er ( Base ` G ) )
10		subgrcl 13249	. . . 4 |- ( S e. (SubGrp ` G ) -> G e. Grp )
11	5, 10	syl 14	. . 3 |- ( S e. (NrmSGrp ` G ) -> G e. Grp )
12		eqid 2193	. . . 4 |- ( +g ` G ) = ( +g ` G )
13	6, 7, 12	eqgcpbl 13298	. . 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 ) ) )
14	6, 12	grpcl 13080	. . . 4 |- ( ( G e. Grp /\ u e. ( Base ` G ) /\ v e. ( Base ` G ) ) -> ( u ( +g ` G ) v ) e. ( Base ` G ) )
15	11, 14	syl3an1 1282	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ u e. ( Base ` G ) /\ v e. ( Base ` G ) ) -> ( u ( +g ` G ) v ) e. ( Base ` G ) )
16	9	adantr 276	. . . . 5 |- ( ( S e. (NrmSGrp ` G ) /\ ( u e. ( Base ` G ) /\ v e. ( Base ` G ) /\ w e. ( Base ` G ) ) ) -> ( G ~QG S ) Er ( Base ` G ) )
17	11	adantr 276	. . . . . 6 |- ( ( S e. (NrmSGrp ` G ) /\ ( u e. ( Base ` G ) /\ v e. ( Base ` G ) /\ w e. ( Base ` G ) ) ) -> G e. Grp )
18		simpr1 1005	. . . . . . 7 |- ( ( S e. (NrmSGrp ` G ) /\ ( u e. ( Base ` G ) /\ v e. ( Base ` G ) /\ w e. ( Base ` G ) ) ) -> u e. ( Base ` G ) )
19		simpr2 1006	. . . . . . 7 |- ( ( S e. (NrmSGrp ` G ) /\ ( u e. ( Base ` G ) /\ v e. ( Base ` G ) /\ w e. ( Base ` G ) ) ) -> v e. ( Base ` G ) )
20	17, 18, 19, 14	syl3anc 1249	. . . . . 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 1007	. . . . . 6 |- ( ( S e. (NrmSGrp ` G ) /\ ( u e. ( Base ` G ) /\ v e. ( Base ` G ) /\ w e. ( Base ` G ) ) ) -> w e. ( Base ` G ) )
22	6, 12	grpcl 13080	. . . . . 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 ) )
23	17, 20, 21, 22	syl3anc 1249	. . . . 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 ) )
24	16, 23	erref 6607	. . . 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 ) )
25	6, 12	grpass 13081	. . . . 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 ) ) )
26	11, 25	sylan 283	. . . 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 ) ) )
27	24, 26	breqtrd 4055	. . 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 2193	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
29	6, 28	grpidcl 13101	. . . 4 |- ( G e. Grp -> ( 0g ` G ) e. ( Base ` G ) )
30	11, 29	syl 14	. . 3 |- ( S e. (NrmSGrp ` G ) -> ( 0g ` G ) e. ( Base ` G ) )
31	6, 12, 28	grplid 13103	. . . . 5 |- ( ( G e. Grp /\ u e. ( Base ` G ) ) -> ( ( 0g ` G ) ( +g ` G ) u ) = u )
32	11, 31	sylan 283	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> ( ( 0g ` G ) ( +g ` G ) u ) = u )
33	9	adantr 276	. . . . 5 |- ( ( S e. (NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> ( G ~QG S ) Er ( Base ` G ) )
34		simpr 110	. . . . 5 |- ( ( S e. (NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> u e. ( Base ` G ) )
35	33, 34	erref 6607	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> u ( G ~QG S ) u )
36	32, 35	eqbrtrd 4051	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> ( ( 0g ` G ) ( +g ` G ) u ) ( G ~QG S ) u )
37		eqid 2193	. . . . 5 |- ( invg ` G ) = ( invg ` G )
38	6, 37	grpinvcl 13120	. . . 4 |- ( ( G e. Grp /\ u e. ( Base ` G ) ) -> ( ( invg ` G ) ` u ) e. ( Base ` G ) )
39	11, 38	sylan 283	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> ( ( invg ` G ) ` u ) e. ( Base ` G ) )
40	6, 12, 28, 37	grplinv 13122	. . . . 5 |- ( ( G e. Grp /\ u e. ( Base ` G ) ) -> ( ( ( invg ` G ) ` u ) ( +g ` G ) u ) = ( 0g ` G ) )
41	11, 40	sylan 283	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> ( ( ( invg ` G ) ` u ) ( +g ` G ) u ) = ( 0g ` G ) )
42	30	adantr 276	. . . . 5 |- ( ( S e. (NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> ( 0g ` G ) e. ( Base ` G ) )
43	33, 42	erref 6607	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> ( 0g ` G ) ( G ~QG S ) ( 0g ` G ) )
44	41, 43	eqbrtrd 4051	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> ( ( ( invg ` G ) ` u ) ( +g ` G ) u ) ( G ~QG S ) ( 0g ` G ) )
45	2, 3, 4, 9, 11, 13, 15, 27, 30, 36, 39, 44	qusgrp2 13183	. 2 |- ( S e. (NrmSGrp ` G ) -> ( H e. Grp /\ [ ( 0g ` G ) ] ( G ~QG S ) = ( 0g ` H ) ) )
46	45	simpld 112	1 |- ( S e. (NrmSGrp ` G ) -> H e. Grp )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2164   ` cfv 5254  (class class class)co 5918   Er wer 6584   [cec 6585   Basecbs 12618   +g cplusg 12695   0gc0g 12867   /._s cqus 12883   Grpcgrp 13072   invgcminusg 13073  SubGrpcsubg 13237  NrmSGrpcnsg 13238   ~_QG cqg 13239

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-tp 3626  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-er 6587  df-ec 6589  df-qs 6593  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-mulr 12709  df-0g 12869  df-iimas 12885  df-qus 12886  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-subg 13240  df-nsg 13241  df-eqg 13242

This theorem is referenced by:  qus0  13305  qusinv  13306  qusghm  13352