Theorem qusgrp 13683

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 2208	. . 3 |- ( S e. (NrmSGrp ` G ) -> ( Base ` G ) = ( Base ` G ) )
4		eqidd 2208	. . 3 |- ( S e. (NrmSGrp ` G ) -> ( +g ` G ) = ( +g ` G ) )
5		nsgsubg 13656	. . . 4 |- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
6		eqid 2207	. . . . 5 |- ( Base ` G ) = ( Base ` G )
7		eqid 2207	. . . . 5 |- ( G ~QG S ) = ( G ~QG S )
8	6, 7	eqger 13675	. . . 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 13630	. . . 4 |- ( S e. (SubGrp ` G ) -> G e. Grp )
11	5, 10	syl 14	. . 3 |- ( S e. (NrmSGrp ` G ) -> G e. Grp )
12		eqid 2207	. . . 4 |- ( +g ` G ) = ( +g ` G )
13	6, 7, 12	eqgcpbl 13679	. . 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 13455	. . . 4 |- ( ( G e. Grp /\ u e. ( Base ` G ) /\ v e. ( Base ` G ) ) -> ( u ( +g ` G ) v ) e. ( Base ` G ) )
15	11, 14	syl3an1 1283	. . 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 1006	. . . . . . 7 |- ( ( S e. (NrmSGrp ` G ) /\ ( u e. ( Base ` G ) /\ v e. ( Base ` G ) /\ w e. ( Base ` G ) ) ) -> u e. ( Base ` G ) )
19		simpr2 1007	. . . . . . 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 1250	. . . . . 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 1008	. . . . . 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 13455	. . . . . 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 1250	. . . . 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 6663	. . . 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 13456	. . . . 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 4085	. . 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 2207	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
29	6, 28	grpidcl 13476	. . . 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 13478	. . . . 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 6663	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> u ( G ~QG S ) u )
36	32, 35	eqbrtrd 4081	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> ( ( 0g ` G ) ( +g ` G ) u ) ( G ~QG S ) u )
37		eqid 2207	. . . . 5 |- ( invg ` G ) = ( invg ` G )
38	6, 37	grpinvcl 13495	. . . 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 13497	. . . . 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 6663	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ u e. ( Base ` G ) ) -> ( 0g ` G ) ( G ~QG S ) ( 0g ` G ) )
44	41, 43	eqbrtrd 4081	. . 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 13564	. 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 981   = wceq 1373   e. wcel 2178   ` cfv 5290  (class class class)co 5967   Er wer 6640   [cec 6641   Basecbs 12947   +g cplusg 13024   0gc0g 13203   /._s cqus 13247   Grpcgrp 13447   invgcminusg 13448  SubGrpcsubg 13618  NrmSGrpcnsg 13619   ~_QG cqg 13620

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-tp 3651  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-er 6643  df-ec 6645  df-qs 6649  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-3 9131  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-iress 12955  df-plusg 13037  df-mulr 13038  df-0g 13205  df-iimas 13249  df-qus 13250  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-minusg 13451  df-subg 13621  df-nsg 13622  df-eqg 13623

This theorem is referenced by:  qus0  13686  qusinv  13687  qusghm  13733