Theorem qusgrp 13196

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 2190	. . 3 |- ( S e. (NrmSGrp ` G ) -> ( Base ` G ) = ( Base ` G ) )
4		eqidd 2190	. . 3 |- ( S e. (NrmSGrp ` G ) -> ( +g ` G ) = ( +g ` G ) )
5		nsgsubg 13169	. . . 4 |- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
6		eqid 2189	. . . . 5 |- ( Base ` G ) = ( Base ` G )
7		eqid 2189	. . . . 5 |- ( G ~QG S ) = ( G ~QG S )
8	6, 7	eqger 13188	. . . 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 13143	. . . 4 |- ( S e. (SubGrp ` G ) -> G e. Grp )
11	5, 10	syl 14	. . 3 |- ( S e. (NrmSGrp ` G ) -> G e. Grp )
12		eqid 2189	. . . 4 |- ( +g ` G ) = ( +g ` G )
13	6, 7, 12	eqgcpbl 13192	. . 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 12976	. . . 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 12976	. . . . . 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 6583	. . . 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 12977	. . . . 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 4047	. . 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 2189	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
29	6, 28	grpidcl 12996	. . . 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 12998	. . . . 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 6583	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> u ( G ~QG S ) u )
36	32, 35	eqbrtrd 4043	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> ( ( 0g ` G ) ( +g ` G ) u ) ( G ~QG S ) u )
37		eqid 2189	. . . . 5 |- ( invg ` G ) = ( invg ` G )
38	6, 37	grpinvcl 13015	. . . 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 13017	. . . . 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 6583	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> ( 0g ` G ) ( G ~QG S ) ( 0g ` G ) )
44	41, 43	eqbrtrd 4043	. . 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 13078	. 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 2160   ` cfv 5238  (class class class)co 5900   Er wer 6560   [cec 6561   Basecbs 12523   +g cplusg 12600   0gc0g 12772   /._s cqus 12788   Grpcgrp 12968   invgcminusg 12969  SubGrpcsubg 13131  NrmSGrpcnsg 13132   ~_QG cqg 13133

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 2162  ax-14 2163  ax-ext 2171  ax-coll 4136  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-cnex 7937  ax-resscn 7938  ax-1cn 7939  ax-1re 7940  ax-icn 7941  ax-addcl 7942  ax-addrcl 7943  ax-mulcl 7944  ax-addcom 7946  ax-addass 7948  ax-i2m1 7951  ax-0lt1 7952  ax-0id 7954  ax-rnegex 7955  ax-pre-ltirr 7958  ax-pre-lttrn 7960  ax-pre-ltadd 7962

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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3595  df-sn 3616  df-pr 3617  df-tp 3618  df-op 3619  df-uni 3828  df-int 3863  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-riota 5855  df-ov 5903  df-oprab 5904  df-mpo 5905  df-er 6563  df-ec 6565  df-qs 6569  df-pnf 8029  df-mnf 8030  df-ltxr 8032  df-inn 8955  df-2 9013  df-3 9014  df-ndx 12526  df-slot 12527  df-base 12529  df-sets 12530  df-iress 12531  df-plusg 12613  df-mulr 12614  df-0g 12774  df-iimas 12790  df-qus 12791  df-mgm 12843  df-sgrp 12888  df-mnd 12901  df-grp 12971  df-minusg 12972  df-subg 13134  df-nsg 13135  df-eqg 13136

This theorem is referenced by:  qus0  13199  qusinv  13200  qusghm  13246