Theorem qusgrp 13941

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 2233	. . 3 |- ( S e. (NrmSGrp ` G ) -> ( Base ` G ) = ( Base ` G ) )
4		eqidd 2233	. . 3 |- ( S e. (NrmSGrp ` G ) -> ( +g ` G ) = ( +g ` G ) )
5		nsgsubg 13914	. . . 4 |- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
6		eqid 2232	. . . . 5 |- ( Base ` G ) = ( Base ` G )
7		eqid 2232	. . . . 5 |- ( G ~QG S ) = ( G ~QG S )
8	6, 7	eqger 13933	. . . 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 13888	. . . 4 |- ( S e. (SubGrp ` G ) -> G e. Grp )
11	5, 10	syl 14	. . 3 |- ( S e. (NrmSGrp ` G ) -> G e. Grp )
12		eqid 2232	. . . 4 |- ( +g ` G ) = ( +g ` G )
13	6, 7, 12	eqgcpbl 13937	. . 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 13713	. . . 4 |- ( ( G e. Grp /\ u e. ( Base ` G ) /\ v e. ( Base ` G ) ) -> ( u ( +g ` G ) v ) e. ( Base ` G ) )
15	11, 14	syl3an1 1307	. . 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 1030	. . . . . . 7 |- ( ( S e. (NrmSGrp ` G ) /\ ( u e. ( Base ` G ) /\ v e. ( Base ` G ) /\ w e. ( Base ` G ) ) ) -> u e. ( Base ` G ) )
19		simpr2 1031	. . . . . . 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 1274	. . . . . 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 1032	. . . . . 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 13713	. . . . . 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 1274	. . . . 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 6786	. . . 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 13714	. . . . 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 4134	. . 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 2232	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
29	6, 28	grpidcl 13734	. . . 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 13736	. . . . 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 6786	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> u ( G ~QG S ) u )
36	32, 35	eqbrtrd 4130	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> ( ( 0g ` G ) ( +g ` G ) u ) ( G ~QG S ) u )
37		eqid 2232	. . . . 5 |- ( invg ` G ) = ( invg ` G )
38	6, 37	grpinvcl 13753	. . . 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 13755	. . . . 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 6786	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> ( 0g ` G ) ( G ~QG S ) ( 0g ` G ) )
44	41, 43	eqbrtrd 4130	. . 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 13822	. 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 1005   = wceq 1398   e. wcel 2203   ` cfv 5351  (class class class)co 6049   Er wer 6763   [cec 6764   Basecbs 13204   +g cplusg 13282   0gc0g 13461   /._s cqus 13505   Grpcgrp 13705   invgcminusg 13706  SubGrpcsubg 13876  NrmSGrpcnsg 13877   ~_QG cqg 13878

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-addass 8228  ax-i2m1 8231  ax-0lt1 8232  ax-0id 8234  ax-rnegex 8235  ax-pre-ltirr 8238  ax-pre-lttrn 8240  ax-pre-ltadd 8242

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-tp 3696  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-er 6766  df-ec 6768  df-qs 6772  df-pnf 8309  df-mnf 8310  df-ltxr 8312  df-inn 9237  df-2 9295  df-3 9296  df-ndx 13207  df-slot 13208  df-base 13210  df-sets 13211  df-iress 13212  df-plusg 13295  df-mulr 13296  df-0g 13463  df-iimas 13507  df-qus 13508  df-mgm 13561  df-sgrp 13607  df-mnd 13622  df-grp 13708  df-minusg 13709  df-subg 13879  df-nsg 13880  df-eqg 13881

This theorem is referenced by:  qus0  13944  qusinv  13945  qusghm  13991