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Theorem qusabl 18200
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.
StepHypRef Expression
1 ablnsg 18182 . . . . 5 (𝐺 ∈ Abel → (NrmSGrp‘𝐺) = (SubGrp‘𝐺))
21eleq2d 2684 . . . 4 (𝐺 ∈ Abel → (𝑆 ∈ (NrmSGrp‘𝐺) ↔ 𝑆 ∈ (SubGrp‘𝐺)))
32biimpar 502 . . 3 ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ∈ (NrmSGrp‘𝐺))
4 qusabl.h . . . 4 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆))
54qusgrp 17581 . . 3 (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐻 ∈ Grp)
63, 5syl 17 . 2 ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Grp)
7 vex 3192 . . . . . . 7 𝑥 ∈ V
87elqs 7751 . . . . . 6 (𝑥 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑆)) ↔ ∃𝑎 ∈ (Base‘𝐺)𝑥 = [𝑎](𝐺 ~QG 𝑆))
94a1i 11 . . . . . . . 8 ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆)))
10 eqidd 2622 . . . . . . . 8 ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (Base‘𝐺) = (Base‘𝐺))
11 ovexd 6640 . . . . . . . 8 ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐺 ~QG 𝑆) ∈ V)
12 simpl 473 . . . . . . . 8 ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Abel)
139, 10, 11, 12qusbas 16137 . . . . . . 7 ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((Base‘𝐺) / (𝐺 ~QG 𝑆)) = (Base‘𝐻))
1413eleq2d 2684 . . . . . 6 ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑆)) ↔ 𝑥 ∈ (Base‘𝐻)))
158, 14syl5bbr 274 . . . . 5 ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (∃𝑎 ∈ (Base‘𝐺)𝑥 = [𝑎](𝐺 ~QG 𝑆) ↔ 𝑥 ∈ (Base‘𝐻)))
16 vex 3192 . . . . . . 7 𝑦 ∈ V
1716elqs 7751 . . . . . 6 (𝑦 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑆)) ↔ ∃𝑏 ∈ (Base‘𝐺)𝑦 = [𝑏](𝐺 ~QG 𝑆))
1813eleq2d 2684 . . . . . 6 ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑦 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑆)) ↔ 𝑦 ∈ (Base‘𝐻)))
1917, 18syl5bbr 274 . . . . 5 ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (∃𝑏 ∈ (Base‘𝐺)𝑦 = [𝑏](𝐺 ~QG 𝑆) ↔ 𝑦 ∈ (Base‘𝐻)))
2015, 19anbi12d 746 . . . 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 3100 . . . . 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 2621 . . . . . . . . . . . 12 (Base‘𝐺) = (Base‘𝐺)
23 eqid 2621 . . . . . . . . . . . 12 (+g𝐺) = (+g𝐺)
2422, 23ablcom 18142 . . . . . . . . . . 11 ((𝐺 ∈ Abel ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g𝐺)𝑏) = (𝑏(+g𝐺)𝑎))
25243expb 1263 . . . . . . . . . 10 ((𝐺 ∈ Abel ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g𝐺)𝑏) = (𝑏(+g𝐺)𝑎))
2625adantlr 750 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g𝐺)𝑏) = (𝑏(+g𝐺)𝑎))
2726eceq1d 7735 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → [(𝑎(+g𝐺)𝑏)](𝐺 ~QG 𝑆) = [(𝑏(+g𝐺)𝑎)](𝐺 ~QG 𝑆))
283adantr 481 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → 𝑆 ∈ (NrmSGrp‘𝐺))
29 simprl 793 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → 𝑎 ∈ (Base‘𝐺))
30 simprr 795 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → 𝑏 ∈ (Base‘𝐺))
31 eqid 2621 . . . . . . . . . 10 (+g𝐻) = (+g𝐻)
324, 22, 23, 31qusadd 17583 . . . . . . . . 9 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → ([𝑎](𝐺 ~QG 𝑆)(+g𝐻)[𝑏](𝐺 ~QG 𝑆)) = [(𝑎(+g𝐺)𝑏)](𝐺 ~QG 𝑆))
3328, 29, 30, 32syl3anc 1323 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → ([𝑎](𝐺 ~QG 𝑆)(+g𝐻)[𝑏](𝐺 ~QG 𝑆)) = [(𝑎(+g𝐺)𝑏)](𝐺 ~QG 𝑆))
344, 22, 23, 31qusadd 17583 . . . . . . . . 9 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺) ∧ 𝑎 ∈ (Base‘𝐺)) → ([𝑏](𝐺 ~QG 𝑆)(+g𝐻)[𝑎](𝐺 ~QG 𝑆)) = [(𝑏(+g𝐺)𝑎)](𝐺 ~QG 𝑆))
3528, 30, 29, 34syl3anc 1323 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → ([𝑏](𝐺 ~QG 𝑆)(+g𝐻)[𝑎](𝐺 ~QG 𝑆)) = [(𝑏(+g𝐺)𝑎)](𝐺 ~QG 𝑆))
3627, 33, 353eqtr4d 2665 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → ([𝑎](𝐺 ~QG 𝑆)(+g𝐻)[𝑏](𝐺 ~QG 𝑆)) = ([𝑏](𝐺 ~QG 𝑆)(+g𝐻)[𝑎](𝐺 ~QG 𝑆)))
37 oveq12 6619 . . . . . . . 8 ((𝑥 = [𝑎](𝐺 ~QG 𝑆) ∧ 𝑦 = [𝑏](𝐺 ~QG 𝑆)) → (𝑥(+g𝐻)𝑦) = ([𝑎](𝐺 ~QG 𝑆)(+g𝐻)[𝑏](𝐺 ~QG 𝑆)))
38 oveq12 6619 . . . . . . . . 9 ((𝑦 = [𝑏](𝐺 ~QG 𝑆) ∧ 𝑥 = [𝑎](𝐺 ~QG 𝑆)) → (𝑦(+g𝐻)𝑥) = ([𝑏](𝐺 ~QG 𝑆)(+g𝐻)[𝑎](𝐺 ~QG 𝑆)))
3938ancoms 469 . . . . . . . 8 ((𝑥 = [𝑎](𝐺 ~QG 𝑆) ∧ 𝑦 = [𝑏](𝐺 ~QG 𝑆)) → (𝑦(+g𝐻)𝑥) = ([𝑏](𝐺 ~QG 𝑆)(+g𝐻)[𝑎](𝐺 ~QG 𝑆)))
4037, 39eqeq12d 2636 . . . . . . 7 ((𝑥 = [𝑎](𝐺 ~QG 𝑆) ∧ 𝑦 = [𝑏](𝐺 ~QG 𝑆)) → ((𝑥(+g𝐻)𝑦) = (𝑦(+g𝐻)𝑥) ↔ ([𝑎](𝐺 ~QG 𝑆)(+g𝐻)[𝑏](𝐺 ~QG 𝑆)) = ([𝑏](𝐺 ~QG 𝑆)(+g𝐻)[𝑎](𝐺 ~QG 𝑆))))
4136, 40syl5ibrcom 237 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → ((𝑥 = [𝑎](𝐺 ~QG 𝑆) ∧ 𝑦 = [𝑏](𝐺 ~QG 𝑆)) → (𝑥(+g𝐻)𝑦) = (𝑦(+g𝐻)𝑥)))
4241rexlimdvva 3032 . . . . 5 ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (∃𝑎 ∈ (Base‘𝐺)∃𝑏 ∈ (Base‘𝐺)(𝑥 = [𝑎](𝐺 ~QG 𝑆) ∧ 𝑦 = [𝑏](𝐺 ~QG 𝑆)) → (𝑥(+g𝐻)𝑦) = (𝑦(+g𝐻)𝑥)))
4321, 42syl5bir 233 . . . 4 ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((∃𝑎 ∈ (Base‘𝐺)𝑥 = [𝑎](𝐺 ~QG 𝑆) ∧ ∃𝑏 ∈ (Base‘𝐺)𝑦 = [𝑏](𝐺 ~QG 𝑆)) → (𝑥(+g𝐻)𝑦) = (𝑦(+g𝐻)𝑥)))
4420, 43sylbird 250 . . 3 ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((𝑥 ∈ (Base‘𝐻) ∧ 𝑦 ∈ (Base‘𝐻)) → (𝑥(+g𝐻)𝑦) = (𝑦(+g𝐻)𝑥)))
4544ralrimivv 2965 . 2 ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∀𝑥 ∈ (Base‘𝐻)∀𝑦 ∈ (Base‘𝐻)(𝑥(+g𝐻)𝑦) = (𝑦(+g𝐻)𝑥))
46 eqid 2621 . . 3 (Base‘𝐻) = (Base‘𝐻)
4746, 31isabl2 18133 . 2 (𝐻 ∈ Abel ↔ (𝐻 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝐻)∀𝑦 ∈ (Base‘𝐻)(𝑥(+g𝐻)𝑦) = (𝑦(+g𝐻)𝑥)))
486, 45, 47sylanbrc 697 1 ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Abel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  Vcvv 3189  cfv 5852  (class class class)co 6610  [cec 7692   / cqs 7693  Basecbs 15792  +gcplusg 15873   /s cqus 16097  Grpcgrp 17354  SubGrpcsubg 17520  NrmSGrpcnsg 17521   ~QG cqg 17522  Abelcabl 18126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-ec 7696  df-qs 7700  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-sup 8300  df-inf 8301  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-3 11032  df-4 11033  df-5 11034  df-6 11035  df-7 11036  df-8 11037  df-9 11038  df-n0 11245  df-z 11330  df-dec 11446  df-uz 11640  df-fz 12277  df-struct 15794  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-plusg 15886  df-mulr 15887  df-sca 15889  df-vsca 15890  df-ip 15891  df-tset 15892  df-ple 15893  df-ds 15896  df-0g 16034  df-imas 16100  df-qus 16101  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-grp 17357  df-minusg 17358  df-subg 17523  df-nsg 17524  df-eqg 17525  df-cmn 18127  df-abl 18128
This theorem is referenced by: (None)
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