Theorem qusabl 19795

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 19777	. . . . 5 ⊢ (𝐺 ∈ Abel → (NrmSGrp‘𝐺) = (SubGrp‘𝐺))
2	1	eleq2d 2814	. . . 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 19118	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐻 ∈ Grp)
6	3, 5	syl 17	. 2 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Grp)
7		vex 3451	. . . . . . 7 ⊢ 𝑥 ∈ V
8	7	elqs 8738	. . . . . 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 2730	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (Base‘𝐺) = (Base‘𝐺))
11		ovexd 7422	. . . . . . . 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 17508	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((Base‘𝐺) / (𝐺 ~QG 𝑆)) = (Base‘𝐻))
14	13	eleq2d 2814	. . . . . 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 3451	. . . . . . 7 ⊢ 𝑦 ∈ V
17	16	elqs 8738	. . . . . 6 ⊢ (𝑦 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑆)) ↔ ∃𝑏 ∈ (Base‘𝐺)𝑦 = [𝑏](𝐺 ~QG 𝑆))
18	13	eleq2d 2814	. . . . . 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 3209	. . . . 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 2729	. . . . . . . . . . . 12 ⊢ (Base‘𝐺) = (Base‘𝐺)
23		eqid 2729	. . . . . . . . . . . 12 ⊢ (+g‘𝐺) = (+g‘𝐺)
24	22, 23	ablcom 19729	. . . . . . . . . . 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 8711	. . . . . . . 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 2729	. . . . . . . . . 10 ⊢ (+g‘𝐻) = (+g‘𝐻)
32	4, 22, 23, 31	qusadd 19120	. . . . . . . . 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 19120	. . . . . . . . 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 2774	. . . . . . 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 7396	. . . . . . . 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 7396	. . . . . . . . 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 2745	. . . . . . 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 3194	. . . . 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 3178	. 2 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∀𝑥 ∈ (Base‘𝐻)∀𝑦 ∈ (Base‘𝐻)(𝑥(+g‘𝐻)𝑦) = (𝑦(+g‘𝐻)𝑥))
46		eqid 2729	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
47	46, 31	isabl2 19720	. 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 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3447  ‘cfv 6511  (class class class)co 7387  [cec 8669   / cqs 8670  Basecbs 17179  +_gcplusg 17220   /_s cqus 17468  Grpcgrp 18865  SubGrpcsubg 19052  NrmSGrpcnsg 19053   ~_QG cqg 19054  Abelcabl 19711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-ec 8673  df-qs 8677  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-sup 9393  df-inf 9394  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-0g 17404  df-imas 17471  df-qus 17472  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-grp 18868  df-minusg 18869  df-subg 19055  df-nsg 19056  df-eqg 19057  df-cmn 19712  df-abl 19713

This theorem is referenced by: (None)