Theorem qusgrp 14035

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 2239	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → (Base‘𝐺) = (Base‘𝐺))
4		eqidd 2239	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → (+g‘𝐺) = (+g‘𝐺))
5		nsgsubg 14008	. . . 4 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
6		eqid 2238	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
7		eqid 2238	. . . . 5 ⊢ (𝐺 ~QG 𝑆) = (𝐺 ~QG 𝑆)
8	6, 7	eqger 14027	. . . 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 13982	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
11	5, 10	syl 14	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐺 ∈ Grp)
12		eqid 2238	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
13	6, 7, 12	eqgcpbl 14031	. . 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 13813	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺)) → (𝑢(+g‘𝐺)𝑣) ∈ (Base‘𝐺))
15	11, 14	syl3an1 1311	. . 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 1034	. . . . . . 7 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺) ∧ 𝑤 ∈ (Base‘𝐺))) → 𝑢 ∈ (Base‘𝐺))
19		simpr2 1035	. . . . . . 7 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺) ∧ 𝑤 ∈ (Base‘𝐺))) → 𝑣 ∈ (Base‘𝐺))
20	17, 18, 19, 14	syl3anc 1278	. . . . . 6 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺) ∧ 𝑤 ∈ (Base‘𝐺))) → (𝑢(+g‘𝐺)𝑣) ∈ (Base‘𝐺))
21		simpr3 1036	. . . . . 6 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺) ∧ 𝑤 ∈ (Base‘𝐺))) → 𝑤 ∈ (Base‘𝐺))
22	6, 12	grpcl 13813	. . . . . 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 1278	. . . . 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 6827	. . . 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 13814	. . . . 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 4156	. . 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 2238	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
29	6, 28	grpidcl 13834	. . . 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 13836	. . . . 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 6827	. . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺)) → 𝑢(𝐺 ~QG 𝑆)𝑢)
36	32, 35	eqbrtrd 4152	. . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺)) → ((0g‘𝐺)(+g‘𝐺)𝑢)(𝐺 ~QG 𝑆)𝑢)
37		eqid 2238	. . . . 5 ⊢ (invg‘𝐺) = (invg‘𝐺)
38	6, 37	grpinvcl 13853	. . . 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 13855	. . . . 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 6827	. . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺)) → (0g‘𝐺)(𝐺 ~QG 𝑆)(0g‘𝐺))
44	41, 43	eqbrtrd 4152	. . 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 13916	. 2 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → (𝐻 ∈ Grp ∧ [(0g‘𝐺)](𝐺 ~QG 𝑆) = (0g‘𝐻)))
46	45	simpld 112	1 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐻 ∈ Grp)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ‘cfv 5377  (class class class)co 6085   Er wer 6804  [cec 6805  Basecbs 13352  +_gcplusg 13431  0_gc0g 13610   /_s cqus 13623  Grpcgrp 13805  inv_gcminusg 13806  SubGrpcsubg 13970  NrmSGrpcnsg 13971   ~_QG cqg 13972

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-er 6807  df-ec 6809  df-qs 6813  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-3 9364  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-iress 13360  df-plusg 13444  df-mulr 13445  df-0g 13612  df-iimas 13624  df-qus 13625  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-grp 13808  df-minusg 13809  df-subg 13973  df-nsg 13974  df-eqg 13975

This theorem is used by:  qus0  14038  qusinv  14039  qusghm  14085