Theorem qusgrp 13769

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 2230	. . 3 |- ( S e. (NrmSGrp ` G ) -> ( Base ` G ) = ( Base ` G ) )
4		eqidd 2230	. . 3 |- ( S e. (NrmSGrp ` G ) -> ( +g ` G ) = ( +g ` G ) )
5		nsgsubg 13742	. . . 4 |- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
6		eqid 2229	. . . . 5 |- ( Base ` G ) = ( Base ` G )
7		eqid 2229	. . . . 5 |- ( G ~QG S ) = ( G ~QG S )
8	6, 7	eqger 13761	. . . 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 13716	. . . 4 |- ( S e. (SubGrp ` G ) -> G e. Grp )
11	5, 10	syl 14	. . 3 |- ( S e. (NrmSGrp ` G ) -> G e. Grp )
12		eqid 2229	. . . 4 |- ( +g ` G ) = ( +g ` G )
13	6, 7, 12	eqgcpbl 13765	. . 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 13541	. . . 4 |- ( ( G e. Grp /\ u e. ( Base ` G ) /\ v e. ( Base ` G ) ) -> ( u ( +g ` G ) v ) e. ( Base ` G ) )
15	11, 14	syl3an1 1304	. . 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 1027	. . . . . . 7 |- ( ( S e. (NrmSGrp ` G ) /\ ( u e. ( Base ` G ) /\ v e. ( Base ` G ) /\ w e. ( Base ` G ) ) ) -> u e. ( Base ` G ) )
19		simpr2 1028	. . . . . . 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 1271	. . . . . 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 1029	. . . . . 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 13541	. . . . . 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 1271	. . . . 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 6700	. . . 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 13542	. . . . 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 4109	. . 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 2229	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
29	6, 28	grpidcl 13562	. . . 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 13564	. . . . 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 6700	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> u ( G ~QG S ) u )
36	32, 35	eqbrtrd 4105	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> ( ( 0g ` G ) ( +g ` G ) u ) ( G ~QG S ) u )
37		eqid 2229	. . . . 5 |- ( invg ` G ) = ( invg ` G )
38	6, 37	grpinvcl 13581	. . . 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 13583	. . . . 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 6700	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> ( 0g ` G ) ( G ~QG S ) ( 0g ` G ) )
44	41, 43	eqbrtrd 4105	. . 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 13650	. 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 1002   = wceq 1395   e. wcel 2200   ` cfv 5318  (class class class)co 6001   Er wer 6677   [cec 6678   Basecbs 13032   +g cplusg 13110   0gc0g 13289   /._s cqus 13333   Grpcgrp 13533   invgcminusg 13534  SubGrpcsubg 13704  NrmSGrpcnsg 13705   ~_QG cqg 13706

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-pre-ltirr 8111  ax-pre-lttrn 8113  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-tp 3674  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-er 6680  df-ec 6682  df-qs 6686  df-pnf 8183  df-mnf 8184  df-ltxr 8186  df-inn 9111  df-2 9169  df-3 9170  df-ndx 13035  df-slot 13036  df-base 13038  df-sets 13039  df-iress 13040  df-plusg 13123  df-mulr 13124  df-0g 13291  df-iimas 13335  df-qus 13336  df-mgm 13389  df-sgrp 13435  df-mnd 13450  df-grp 13536  df-minusg 13537  df-subg 13707  df-nsg 13708  df-eqg 13709

This theorem is referenced by:  qus0  13772  qusinv  13773  qusghm  13819