Theorem qusabl 19731

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 19713	. . . . 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 19052	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐻 ∈ Grp)
6	3, 5	syl 17	. 2 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Grp)
7		vex 3437	. . . . . . 7 ⊢ 𝑥 ∈ V
8	7	elqs 8683	. . . . . 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 7375	. . . . . . . 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 17436	. . . . . . 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 3437	. . . . . . 7 ⊢ 𝑦 ∈ V
17	16	elqs 8683	. . . . . 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 3201	. . . . 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 19665	. . . . . . . . . . 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 8656	. . . . . . . 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 19054	. . . . . . . . 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 19054	. . . . . . . . 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 7349	. . . . . . . 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 7349	. . . . . . . . 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 3186	. . . . 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 3170	. 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 19656	. 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 3433  ‘cfv 6476  (class class class)co 7340  [cec 8614   / cqs 8615  Basecbs 17107  +_gcplusg 17148   /_s cqus 17396  Grpcgrp 18799  SubGrpcsubg 18986  NrmSGrpcnsg 18987   ~_QG cqg 18988  Abelcabl 19647

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 5214  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5367  ax-un 7662  ax-cnex 11053  ax-resscn 11054  ax-1cn 11055  ax-icn 11056  ax-addcl 11057  ax-addrcl 11058  ax-mulcl 11059  ax-mulrcl 11060  ax-mulcom 11061  ax-addass 11062  ax-mulass 11063  ax-distr 11064  ax-i2m1 11065  ax-1ne0 11066  ax-1rid 11067  ax-rnegex 11068  ax-rrecex 11069  ax-cnre 11070  ax-pre-lttri 11071  ax-pre-lttrn 11072  ax-pre-ltadd 11073  ax-pre-mulgt0 11074

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 3343  df-reu 3344  df-rab 3393  df-v 3435  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4857  df-iun 4940  df-br 5089  df-opab 5151  df-mpt 5170  df-tr 5196  df-id 5508  df-eprel 5513  df-po 5521  df-so 5522  df-fr 5566  df-we 5568  df-xp 5619  df-rel 5620  df-cnv 5621  df-co 5622  df-dm 5623  df-rn 5624  df-res 5625  df-ima 5626  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7297  df-ov 7343  df-oprab 7344  df-mpo 7345  df-om 7791  df-1st 7915  df-2nd 7916  df-frecs 8205  df-wrecs 8236  df-recs 8285  df-rdg 8323  df-1o 8379  df-er 8616  df-ec 8618  df-qs 8622  df-en 8864  df-dom 8865  df-sdom 8866  df-fin 8867  df-sup 9320  df-inf 9321  df-pnf 11139  df-mnf 11140  df-xr 11141  df-ltxr 11142  df-le 11143  df-sub 11337  df-neg 11338  df-nn 12117  df-2 12179  df-3 12180  df-4 12181  df-5 12182  df-6 12183  df-7 12184  df-8 12185  df-9 12186  df-n0 12373  df-z 12460  df-dec 12580  df-uz 12724  df-fz 13399  df-struct 17045  df-sets 17062  df-slot 17080  df-ndx 17092  df-base 17108  df-ress 17129  df-plusg 17161  df-mulr 17162  df-sca 17164  df-vsca 17165  df-ip 17166  df-tset 17167  df-ple 17168  df-ds 17170  df-0g 17332  df-imas 17399  df-qus 17400  df-mgm 18501  df-sgrp 18580  df-mnd 18596  df-grp 18802  df-minusg 18803  df-subg 18989  df-nsg 18990  df-eqg 18991  df-cmn 19648  df-abl 19649

This theorem is referenced by: (None)