Theorem qusgrp 13985

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 9	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆)))
3		eqidd 2235	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → (Base‘𝐺) = (Base‘𝐺))
4		eqidd 2235	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → (+g‘𝐺) = (+g‘𝐺))
5		nsgsubg 13958	. . . 4 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
6		eqid 2234	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
7		eqid 2234	. . . . 5 ⊢ (𝐺 ~QG 𝑆) = (𝐺 ~QG 𝑆)
8	6, 7	eqger 13977	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐺 ~QG 𝑆) Er (Base‘𝐺))
9	5, 8	syl 14	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → (𝐺 ~QG 𝑆) Er (Base‘𝐺))
10		subgrcl 13932	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
11	5, 10	syl 14	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐺 ∈ Grp)
12		eqid 2234	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
13	6, 7, 12	eqgcpbl 13981	. . 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 13763	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺)) → (𝑢(+g‘𝐺)𝑣) ∈ (Base‘𝐺))
15	11, 14	syl3an1 1307	. . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺)) → (𝑢(+g‘𝐺)𝑣) ∈ (Base‘𝐺))
16	9	adantr 276	. . . . 5 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺) ∧ 𝑤 ∈ (Base‘𝐺))) → (𝐺 ~QG 𝑆) Er (Base‘𝐺))
17	11	adantr 276	. . . . . 6 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺) ∧ 𝑤 ∈ (Base‘𝐺))) → 𝐺 ∈ Grp)
18		simpr1 1030	. . . . . . 7 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺) ∧ 𝑤 ∈ (Base‘𝐺))) → 𝑢 ∈ (Base‘𝐺))
19		simpr2 1031	. . . . . . 7 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺) ∧ 𝑤 ∈ (Base‘𝐺))) → 𝑣 ∈ (Base‘𝐺))
20	17, 18, 19, 14	syl3anc 1274	. . . . . 6 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺) ∧ 𝑤 ∈ (Base‘𝐺))) → (𝑢(+g‘𝐺)𝑣) ∈ (Base‘𝐺))
21		simpr3 1032	. . . . . 6 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺) ∧ 𝑤 ∈ (Base‘𝐺))) → 𝑤 ∈ (Base‘𝐺))
22	6, 12	grpcl 13763	. . . . . 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 1274	. . . . 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 6800	. . . 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 13764	. . . . 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 283	. . . 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 4140	. . 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 2234	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
29	6, 28	grpidcl 13784	. . . 4 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺))
30	11, 29	syl 14	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → (0g‘𝐺) ∈ (Base‘𝐺))
31	6, 12, 28	grplid 13786	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐺)) → ((0g‘𝐺)(+g‘𝐺)𝑢) = 𝑢)
32	11, 31	sylan 283	. . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺)) → ((0g‘𝐺)(+g‘𝐺)𝑢) = 𝑢)
33	9	adantr 276	. . . . 5 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺)) → (𝐺 ~QG 𝑆) Er (Base‘𝐺))
34		simpr 110	. . . . 5 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺)) → 𝑢 ∈ (Base‘𝐺))
35	33, 34	erref 6800	. . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺)) → 𝑢(𝐺 ~QG 𝑆)𝑢)
36	32, 35	eqbrtrd 4136	. . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺)) → ((0g‘𝐺)(+g‘𝐺)𝑢)(𝐺 ~QG 𝑆)𝑢)
37		eqid 2234	. . . . 5 ⊢ (invg‘𝐺) = (invg‘𝐺)
38	6, 37	grpinvcl 13803	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑢) ∈ (Base‘𝐺))
39	11, 38	sylan 283	. . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑢) ∈ (Base‘𝐺))
40	6, 12, 28, 37	grplinv 13805	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐺)) → (((invg‘𝐺)‘𝑢)(+g‘𝐺)𝑢) = (0g‘𝐺))
41	11, 40	sylan 283	. . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺)) → (((invg‘𝐺)‘𝑢)(+g‘𝐺)𝑢) = (0g‘𝐺))
42	30	adantr 276	. . . . 5 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺)) → (0g‘𝐺) ∈ (Base‘𝐺))
43	33, 42	erref 6800	. . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺)) → (0g‘𝐺)(𝐺 ~QG 𝑆)(0g‘𝐺))
44	41, 43	eqbrtrd 4136	. . 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 13866	. 2 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → (𝐻 ∈ Grp ∧ [(0g‘𝐺)](𝐺 ~QG 𝑆) = (0g‘𝐻)))
46	45	simpld 112	1 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐻 ∈ Grp)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205  ‘cfv 5357  (class class class)co 6058   Er wer 6777  [cec 6778  Basecbs 13296  +_gcplusg 13374  0_gc0g 13553   /_s cqus 13566  Grpcgrp 13755  inv_gcminusg 13756  SubGrpcsubg 13920  NrmSGrpcnsg 13921   ~_QG cqg 13922

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-tp 3702  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-er 6780  df-ec 6782  df-qs 6786  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-mulr 13388  df-0g 13555  df-iimas 13567  df-qus 13568  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-minusg 13759  df-subg 13923  df-nsg 13924  df-eqg 13925

This theorem is referenced by:  qus0  13988  qusinv  13989  qusghm  14035