Theorem qusabl 19794

Description: If 𝑌 is a subgroup of the abelian group 𝐺, then 𝐻 = 𝐺 / 𝑌 is an abelian group. (Contributed by Mario Carneiro, 26-Apr-2016.)

Hypothesis

Ref	Expression
qusabl.h	⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆))

Assertion

Ref	Expression
qusabl	⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Abel)

Proof of Theorem qusabl

Dummy variables 𝑎 𝑏 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ablnsg 19776	. . . . 5 ⊢ (𝐺 ∈ Abel → (NrmSGrp‘𝐺) = (SubGrp‘𝐺))
2	1	eleq2d 2822	. . . 4 ⊢ (𝐺 ∈ Abel → (𝑆 ∈ (NrmSGrp‘𝐺) ↔ 𝑆 ∈ (SubGrp‘𝐺)))
3	2	biimpar 477	. . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ∈ (NrmSGrp‘𝐺))
4		qusabl.h	. . . 4 ⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆))
5	4	qusgrp 19115	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐻 ∈ Grp)
6	3, 5	syl 17	. 2 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Grp)
7		vex 3444	. . . . . . 7 ⊢ 𝑥 ∈ V
8	7	elqs 8702	. . . . . 6 ⊢ (𝑥 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑆)) ↔ ∃𝑎 ∈ (Base‘𝐺)𝑥 = [𝑎](𝐺 ~QG 𝑆))
9	4	a1i 11	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆)))
10		eqidd 2737	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (Base‘𝐺) = (Base‘𝐺))
11		ovexd 7393	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐺 ~QG 𝑆) ∈ V)
12		simpl 482	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Abel)
13	9, 10, 11, 12	qusbas 17466	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((Base‘𝐺) / (𝐺 ~QG 𝑆)) = (Base‘𝐻))
14	13	eleq2d 2822	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑆)) ↔ 𝑥 ∈ (Base‘𝐻)))
15	8, 14	bitr3id 285	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (∃𝑎 ∈ (Base‘𝐺)𝑥 = [𝑎](𝐺 ~QG 𝑆) ↔ 𝑥 ∈ (Base‘𝐻)))
16		vex 3444	. . . . . . 7 ⊢ 𝑦 ∈ V
17	16	elqs 8702	. . . . . 6 ⊢ (𝑦 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑆)) ↔ ∃𝑏 ∈ (Base‘𝐺)𝑦 = [𝑏](𝐺 ~QG 𝑆))
18	13	eleq2d 2822	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑦 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑆)) ↔ 𝑦 ∈ (Base‘𝐻)))
19	17, 18	bitr3id 285	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (∃𝑏 ∈ (Base‘𝐺)𝑦 = [𝑏](𝐺 ~QG 𝑆) ↔ 𝑦 ∈ (Base‘𝐻)))
20	15, 19	anbi12d 632	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((∃𝑎 ∈ (Base‘𝐺)𝑥 = [𝑎](𝐺 ~QG 𝑆) ∧ ∃𝑏 ∈ (Base‘𝐺)𝑦 = [𝑏](𝐺 ~QG 𝑆)) ↔ (𝑥 ∈ (Base‘𝐻) ∧ 𝑦 ∈ (Base‘𝐻))))
21		reeanv 3208	. . . . 5 ⊢ (∃𝑎 ∈ (Base‘𝐺)∃𝑏 ∈ (Base‘𝐺)(𝑥 = [𝑎](𝐺 ~QG 𝑆) ∧ 𝑦 = [𝑏](𝐺 ~QG 𝑆)) ↔ (∃𝑎 ∈ (Base‘𝐺)𝑥 = [𝑎](𝐺 ~QG 𝑆) ∧ ∃𝑏 ∈ (Base‘𝐺)𝑦 = [𝑏](𝐺 ~QG 𝑆)))
22		eqid 2736	. . . . . . . . . . . 12 ⊢ (Base‘𝐺) = (Base‘𝐺)
23		eqid 2736	. . . . . . . . . . . 12 ⊢ (+g‘𝐺) = (+g‘𝐺)
24	22, 23	ablcom 19728	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Abel ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g‘𝐺)𝑏) = (𝑏(+g‘𝐺)𝑎))
25	24	3expb 1120	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Abel ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g‘𝐺)𝑏) = (𝑏(+g‘𝐺)𝑎))
26	25	adantlr 715	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g‘𝐺)𝑏) = (𝑏(+g‘𝐺)𝑎))
27	26	eceq1d 8675	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → [(𝑎(+g‘𝐺)𝑏)](𝐺 ~QG 𝑆) = [(𝑏(+g‘𝐺)𝑎)](𝐺 ~QG 𝑆))
28	3	adantr 480	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → 𝑆 ∈ (NrmSGrp‘𝐺))
29		simprl 770	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → 𝑎 ∈ (Base‘𝐺))
30		simprr 772	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → 𝑏 ∈ (Base‘𝐺))
31		eqid 2736	. . . . . . . . . 10 ⊢ (+g‘𝐻) = (+g‘𝐻)
32	4, 22, 23, 31	qusadd 19117	. . . . . . . . 9 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → ([𝑎](𝐺 ~QG 𝑆)(+g‘𝐻)[𝑏](𝐺 ~QG 𝑆)) = [(𝑎(+g‘𝐺)𝑏)](𝐺 ~QG 𝑆))
33	28, 29, 30, 32	syl3anc 1373	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → ([𝑎](𝐺 ~QG 𝑆)(+g‘𝐻)[𝑏](𝐺 ~QG 𝑆)) = [(𝑎(+g‘𝐺)𝑏)](𝐺 ~QG 𝑆))
34	4, 22, 23, 31	qusadd 19117	. . . . . . . . 9 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺) ∧ 𝑎 ∈ (Base‘𝐺)) → ([𝑏](𝐺 ~QG 𝑆)(+g‘𝐻)[𝑎](𝐺 ~QG 𝑆)) = [(𝑏(+g‘𝐺)𝑎)](𝐺 ~QG 𝑆))
35	28, 30, 29, 34	syl3anc 1373	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → ([𝑏](𝐺 ~QG 𝑆)(+g‘𝐻)[𝑎](𝐺 ~QG 𝑆)) = [(𝑏(+g‘𝐺)𝑎)](𝐺 ~QG 𝑆))
36	27, 33, 35	3eqtr4d 2781	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → ([𝑎](𝐺 ~QG 𝑆)(+g‘𝐻)[𝑏](𝐺 ~QG 𝑆)) = ([𝑏](𝐺 ~QG 𝑆)(+g‘𝐻)[𝑎](𝐺 ~QG 𝑆)))
37		oveq12 7367	. . . . . . . 8 ⊢ ((𝑥 = [𝑎](𝐺 ~QG 𝑆) ∧ 𝑦 = [𝑏](𝐺 ~QG 𝑆)) → (𝑥(+g‘𝐻)𝑦) = ([𝑎](𝐺 ~QG 𝑆)(+g‘𝐻)[𝑏](𝐺 ~QG 𝑆)))
38		oveq12 7367	. . . . . . . . 9 ⊢ ((𝑦 = [𝑏](𝐺 ~QG 𝑆) ∧ 𝑥 = [𝑎](𝐺 ~QG 𝑆)) → (𝑦(+g‘𝐻)𝑥) = ([𝑏](𝐺 ~QG 𝑆)(+g‘𝐻)[𝑎](𝐺 ~QG 𝑆)))
39	38	ancoms 458	. . . . . . . 8 ⊢ ((𝑥 = [𝑎](𝐺 ~QG 𝑆) ∧ 𝑦 = [𝑏](𝐺 ~QG 𝑆)) → (𝑦(+g‘𝐻)𝑥) = ([𝑏](𝐺 ~QG 𝑆)(+g‘𝐻)[𝑎](𝐺 ~QG 𝑆)))
40	37, 39	eqeq12d 2752	. . . . . . 7 ⊢ ((𝑥 = [𝑎](𝐺 ~QG 𝑆) ∧ 𝑦 = [𝑏](𝐺 ~QG 𝑆)) → ((𝑥(+g‘𝐻)𝑦) = (𝑦(+g‘𝐻)𝑥) ↔ ([𝑎](𝐺 ~QG 𝑆)(+g‘𝐻)[𝑏](𝐺 ~QG 𝑆)) = ([𝑏](𝐺 ~QG 𝑆)(+g‘𝐻)[𝑎](𝐺 ~QG 𝑆))))
41	36, 40	syl5ibrcom 247	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → ((𝑥 = [𝑎](𝐺 ~QG 𝑆) ∧ 𝑦 = [𝑏](𝐺 ~QG 𝑆)) → (𝑥(+g‘𝐻)𝑦) = (𝑦(+g‘𝐻)𝑥)))
42	41	rexlimdvva 3193	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (∃𝑎 ∈ (Base‘𝐺)∃𝑏 ∈ (Base‘𝐺)(𝑥 = [𝑎](𝐺 ~QG 𝑆) ∧ 𝑦 = [𝑏](𝐺 ~QG 𝑆)) → (𝑥(+g‘𝐻)𝑦) = (𝑦(+g‘𝐻)𝑥)))
43	21, 42	biimtrrid 243	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((∃𝑎 ∈ (Base‘𝐺)𝑥 = [𝑎](𝐺 ~QG 𝑆) ∧ ∃𝑏 ∈ (Base‘𝐺)𝑦 = [𝑏](𝐺 ~QG 𝑆)) → (𝑥(+g‘𝐻)𝑦) = (𝑦(+g‘𝐻)𝑥)))
44	20, 43	sylbird 260	. . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((𝑥 ∈ (Base‘𝐻) ∧ 𝑦 ∈ (Base‘𝐻)) → (𝑥(+g‘𝐻)𝑦) = (𝑦(+g‘𝐻)𝑥)))
45	44	ralrimivv 3177	. 2 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∀𝑥 ∈ (Base‘𝐻)∀𝑦 ∈ (Base‘𝐻)(𝑥(+g‘𝐻)𝑦) = (𝑦(+g‘𝐻)𝑥))
46		eqid 2736	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
47	46, 31	isabl2 19719	. 2 ⊢ (𝐻 ∈ Abel ↔ (𝐻 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝐻)∀𝑦 ∈ (Base‘𝐻)(𝑥(+g‘𝐻)𝑦) = (𝑦(+g‘𝐻)𝑥)))
48	6, 45, 47	sylanbrc 583	1 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Abel)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060  Vcvv 3440  ‘cfv 6492  (class class class)co 7358  [cec 8633   / cqs 8634  Basecbs 17136  +_gcplusg 17177   /_s cqus 17426  Grpcgrp 18863  SubGrpcsubg 19050  NrmSGrpcnsg 19051   ~_QG cqg 19052  Abelcabl 19710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-ec 8637  df-qs 8641  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9345  df-inf 9346  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-z 12489  df-dec 12608  df-uz 12752  df-fz 13424  df-struct 17074  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-mulr 17191  df-sca 17193  df-vsca 17194  df-ip 17195  df-tset 17196  df-ple 17197  df-ds 17199  df-0g 17361  df-imas 17429  df-qus 17430  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18866  df-minusg 18867  df-subg 19053  df-nsg 19054  df-eqg 19055  df-cmn 19711  df-abl 19712

This theorem is referenced by: (None)