Theorem qusabl 18091

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 18073	. . . . 5 ⊢ (𝐺 ∈ Abel → (NrmSGrp‘𝐺) = (SubGrp‘𝐺))
2	1	eleq2d 2673	. . . 4 ⊢ (𝐺 ∈ Abel → (𝑆 ∈ (NrmSGrp‘𝐺) ↔ 𝑆 ∈ (SubGrp‘𝐺)))
3	2	biimpar 501	. . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ∈ (NrmSGrp‘𝐺))
4		qusabl.h	. . . 4 ⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆))
5	4	qusgrp 17472	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐻 ∈ Grp)
6	3, 5	syl 17	. 2 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Grp)
7		vex 3176	. . . . . . 7 ⊢ 𝑥 ∈ V
8	7	elqs 7686	. . . . . 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 2611	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (Base‘𝐺) = (Base‘𝐺))
11		ovex 6577	. . . . . . . . 9 ⊢ (𝐺 ~QG 𝑆) ∈ V
12	11	a1i 11	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐺 ~QG 𝑆) ∈ V)
13		simpl 472	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Abel)
14	9, 10, 12, 13	qusbas 16028	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((Base‘𝐺) / (𝐺 ~QG 𝑆)) = (Base‘𝐻))
15	14	eleq2d 2673	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑆)) ↔ 𝑥 ∈ (Base‘𝐻)))
16	8, 15	syl5bbr 273	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (∃𝑎 ∈ (Base‘𝐺)𝑥 = [𝑎](𝐺 ~QG 𝑆) ↔ 𝑥 ∈ (Base‘𝐻)))
17		vex 3176	. . . . . . 7 ⊢ 𝑦 ∈ V
18	17	elqs 7686	. . . . . 6 ⊢ (𝑦 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑆)) ↔ ∃𝑏 ∈ (Base‘𝐺)𝑦 = [𝑏](𝐺 ~QG 𝑆))
19	14	eleq2d 2673	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑦 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑆)) ↔ 𝑦 ∈ (Base‘𝐻)))
20	18, 19	syl5bbr 273	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (∃𝑏 ∈ (Base‘𝐺)𝑦 = [𝑏](𝐺 ~QG 𝑆) ↔ 𝑦 ∈ (Base‘𝐻)))
21	16, 20	anbi12d 743	. . . 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))))
22		reeanv 3086	. . . . 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)))
23		eqid 2610	. . . . . . . . . . . 12 ⊢ (Base‘𝐺) = (Base‘𝐺)
24		eqid 2610	. . . . . . . . . . . 12 ⊢ (+g‘𝐺) = (+g‘𝐺)
25	23, 24	ablcom 18033	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Abel ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g‘𝐺)𝑏) = (𝑏(+g‘𝐺)𝑎))
26	25	3expb 1258	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Abel ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g‘𝐺)𝑏) = (𝑏(+g‘𝐺)𝑎))
27	26	adantlr 747	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g‘𝐺)𝑏) = (𝑏(+g‘𝐺)𝑎))
28	27	eceq1d 7670	. . . . . . . 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))
29	3	adantr 480	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → 𝑆 ∈ (NrmSGrp‘𝐺))
30		simprl 790	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → 𝑎 ∈ (Base‘𝐺))
31		simprr 792	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → 𝑏 ∈ (Base‘𝐺))
32		eqid 2610	. . . . . . . . . 10 ⊢ (+g‘𝐻) = (+g‘𝐻)
33	4, 23, 24, 32	qusadd 17474	. . . . . . . . 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))
34	29, 30, 31, 33	syl3anc 1318	. . . . . . . 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))
35	4, 23, 24, 32	qusadd 17474	. . . . . . . . 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))
36	29, 31, 30, 35	syl3anc 1318	. . . . . . . 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))
37	28, 34, 36	3eqtr4d 2654	. . . . . . 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)))
38		oveq12 6558	. . . . . . . 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)))
39		oveq12 6558	. . . . . . . . 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)))
40	39	ancoms 468	. . . . . . . 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)))
41	38, 40	eqeq12d 2625	. . . . . . 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))))
42	37, 41	syl5ibrcom 236	. . . . . 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)))
43	42	rexlimdvva 3020	. . . . 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)))
44	22, 43	syl5bir 232	. . . 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)))
45	21, 44	sylbird 249	. . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((𝑥 ∈ (Base‘𝐻) ∧ 𝑦 ∈ (Base‘𝐻)) → (𝑥(+g‘𝐻)𝑦) = (𝑦(+g‘𝐻)𝑥)))
46	45	ralrimivv 2953	. 2 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∀𝑥 ∈ (Base‘𝐻)∀𝑦 ∈ (Base‘𝐻)(𝑥(+g‘𝐻)𝑦) = (𝑦(+g‘𝐻)𝑥))
47		eqid 2610	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
48	47, 32	isabl2 18024	. 2 ⊢ (𝐻 ∈ Abel ↔ (𝐻 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝐻)∀𝑦 ∈ (Base‘𝐻)(𝑥(+g‘𝐻)𝑦) = (𝑦(+g‘𝐻)𝑥)))
49	6, 46, 48	sylanbrc 695	1 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Abel)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173  ‘cfv 5804  (class class class)co 6549  [cec 7627   / cqs 7628  Basecbs 15695  +_gcplusg 15768   /_s cqus 15988  Grpcgrp 17245  SubGrpcsubg 17411  NrmSGrpcnsg 17412   ~_QG cqg 17413  Abelcabl 18017

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-ec 7631  df-qs 7635  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-0g 15925  df-imas 15991  df-qus 15992  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-subg 17414  df-nsg 17415  df-eqg 17416  df-cmn 18018  df-abl 18019

This theorem is referenced by: (None)