Theorem qusgrp 18329

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 ∈ (NrmSGrp‘𝐺) → 𝐻 ∈ 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 11	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆)))
3		eqidd 2822	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → (Base‘𝐺) = (Base‘𝐺))
4		eqidd 2822	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → (+g‘𝐺) = (+g‘𝐺))
5		nsgsubg 18304	. . . 4 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
6		eqid 2821	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
7		eqid 2821	. . . . 5 ⊢ (𝐺 ~QG 𝑆) = (𝐺 ~QG 𝑆)
8	6, 7	eqger 18324	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐺 ~QG 𝑆) Er (Base‘𝐺))
9	5, 8	syl 17	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → (𝐺 ~QG 𝑆) Er (Base‘𝐺))
10		subgrcl 18278	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
11	5, 10	syl 17	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐺 ∈ Grp)
12		eqid 2821	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
13	6, 7, 12	eqgcpbl 18328	. . 3 ⊢ (𝑆 ∈ (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 18105	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺)) → (𝑢(+g‘𝐺)𝑣) ∈ (Base‘𝐺))
15	11, 14	syl3an1 1159	. . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺)) → (𝑢(+g‘𝐺)𝑣) ∈ (Base‘𝐺))
16	9	adantr 483	. . . . 5 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺) ∧ 𝑤 ∈ (Base‘𝐺))) → (𝐺 ~QG 𝑆) Er (Base‘𝐺))
17	11	adantr 483	. . . . . 6 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺) ∧ 𝑤 ∈ (Base‘𝐺))) → 𝐺 ∈ Grp)
18		simpr1 1190	. . . . . . 7 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺) ∧ 𝑤 ∈ (Base‘𝐺))) → 𝑢 ∈ (Base‘𝐺))
19		simpr2 1191	. . . . . . 7 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺) ∧ 𝑤 ∈ (Base‘𝐺))) → 𝑣 ∈ (Base‘𝐺))
20	17, 18, 19, 14	syl3anc 1367	. . . . . 6 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺) ∧ 𝑤 ∈ (Base‘𝐺))) → (𝑢(+g‘𝐺)𝑣) ∈ (Base‘𝐺))
21		simpr3 1192	. . . . . 6 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺) ∧ 𝑤 ∈ (Base‘𝐺))) → 𝑤 ∈ (Base‘𝐺))
22	6, 12	grpcl 18105	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑢(+g‘𝐺)𝑣) ∈ (Base‘𝐺) ∧ 𝑤 ∈ (Base‘𝐺)) → ((𝑢(+g‘𝐺)𝑣)(+g‘𝐺)𝑤) ∈ (Base‘𝐺))
23	17, 20, 21, 22	syl3anc 1367	. . . . 5 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺) ∧ 𝑤 ∈ (Base‘𝐺))) → ((𝑢(+g‘𝐺)𝑣)(+g‘𝐺)𝑤) ∈ (Base‘𝐺))
24	16, 23	erref 8303	. . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺) ∧ 𝑤 ∈ (Base‘𝐺))) → ((𝑢(+g‘𝐺)𝑣)(+g‘𝐺)𝑤)(𝐺 ~QG 𝑆)((𝑢(+g‘𝐺)𝑣)(+g‘𝐺)𝑤))
25	6, 12	grpass 18106	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺) ∧ 𝑤 ∈ (Base‘𝐺))) → ((𝑢(+g‘𝐺)𝑣)(+g‘𝐺)𝑤) = (𝑢(+g‘𝐺)(𝑣(+g‘𝐺)𝑤)))
26	11, 25	sylan 582	. . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺) ∧ 𝑤 ∈ (Base‘𝐺))) → ((𝑢(+g‘𝐺)𝑣)(+g‘𝐺)𝑤) = (𝑢(+g‘𝐺)(𝑣(+g‘𝐺)𝑤)))
27	24, 26	breqtrd 5085	. . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺) ∧ 𝑤 ∈ (Base‘𝐺))) → ((𝑢(+g‘𝐺)𝑣)(+g‘𝐺)𝑤)(𝐺 ~QG 𝑆)(𝑢(+g‘𝐺)(𝑣(+g‘𝐺)𝑤)))
28		eqid 2821	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
29	6, 28	grpidcl 18125	. . . 4 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺))
30	11, 29	syl 17	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → (0g‘𝐺) ∈ (Base‘𝐺))
31	6, 12, 28	grplid 18127	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐺)) → ((0g‘𝐺)(+g‘𝐺)𝑢) = 𝑢)
32	11, 31	sylan 582	. . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺)) → ((0g‘𝐺)(+g‘𝐺)𝑢) = 𝑢)
33	9	adantr 483	. . . . 5 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺)) → (𝐺 ~QG 𝑆) Er (Base‘𝐺))
34		simpr 487	. . . . 5 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺)) → 𝑢 ∈ (Base‘𝐺))
35	33, 34	erref 8303	. . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺)) → 𝑢(𝐺 ~QG 𝑆)𝑢)
36	32, 35	eqbrtrd 5081	. . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺)) → ((0g‘𝐺)(+g‘𝐺)𝑢)(𝐺 ~QG 𝑆)𝑢)
37		eqid 2821	. . . . 5 ⊢ (invg‘𝐺) = (invg‘𝐺)
38	6, 37	grpinvcl 18145	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑢) ∈ (Base‘𝐺))
39	11, 38	sylan 582	. . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑢) ∈ (Base‘𝐺))
40	6, 12, 28, 37	grplinv 18146	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐺)) → (((invg‘𝐺)‘𝑢)(+g‘𝐺)𝑢) = (0g‘𝐺))
41	11, 40	sylan 582	. . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺)) → (((invg‘𝐺)‘𝑢)(+g‘𝐺)𝑢) = (0g‘𝐺))
42	30	adantr 483	. . . . 5 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺)) → (0g‘𝐺) ∈ (Base‘𝐺))
43	33, 42	erref 8303	. . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺)) → (0g‘𝐺)(𝐺 ~QG 𝑆)(0g‘𝐺))
44	41, 43	eqbrtrd 5081	. . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑢 ∈ (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 18211	. 2 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → (𝐻 ∈ Grp ∧ [(0g‘𝐺)](𝐺 ~QG 𝑆) = (0g‘𝐻)))
46	45	simpld 497	1 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐻 ∈ Grp)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ‘cfv 6350  (class class class)co 7150   Er wer 8280  [cec 8281  Basecbs 16477  +_gcplusg 16559  0_gc0g 16707   /_s cqus 16772  Grpcgrp 18097  inv_gcminusg 18098  SubGrpcsubg 18267  NrmSGrpcnsg 18268   ~_QG cqg 18269

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8283  df-ec 8285  df-qs 8289  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-sup 8900  df-inf 8901  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-dec 12093  df-uz 12238  df-fz 12887  df-struct 16479  df-ndx 16480  df-slot 16481  df-base 16483  df-sets 16484  df-ress 16485  df-plusg 16572  df-mulr 16573  df-sca 16575  df-vsca 16576  df-ip 16577  df-tset 16578  df-ple 16579  df-ds 16581  df-0g 16709  df-imas 16775  df-qus 16776  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-grp 18100  df-minusg 18101  df-subg 18270  df-nsg 18271  df-eqg 18272

This theorem is referenced by:  qus0  18332  qusinv  18333  qusghm  18389  qusabl  18979  rzgrp  20761  qustgplem  22723