MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  qusabl Structured version   Visualization version   GIF version

Theorem qusabl 18249
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 18231 . . . . 5 (𝐺 ∈ Abel → (NrmSGrp‘𝐺) = (SubGrp‘𝐺))
21eleq2d 2685 . . . 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 17630 . . 3 (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐻 ∈ Grp)
63, 5syl 17 . 2 ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Grp)
7 vex 3198 . . . . . . 7 𝑥 ∈ V
87elqs 7784 . . . . . 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 2621 . . . . . . . 8 ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (Base‘𝐺) = (Base‘𝐺))
11 ovexd 6665 . . . . . . . 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 16186 . . . . . . 7 ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((Base‘𝐺) / (𝐺 ~QG 𝑆)) = (Base‘𝐻))
1413eleq2d 2685 . . . . . 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 3198 . . . . . . 7 𝑦 ∈ V
1716elqs 7784 . . . . . 6 (𝑦 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑆)) ↔ ∃𝑏 ∈ (Base‘𝐺)𝑦 = [𝑏](𝐺 ~QG 𝑆))
1813eleq2d 2685 . . . . . 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 3102 . . . . 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 2620 . . . . . . . . . . . 12 (Base‘𝐺) = (Base‘𝐺)
23 eqid 2620 . . . . . . . . . . . 12 (+g𝐺) = (+g𝐺)
2422, 23ablcom 18191 . . . . . . . . . . 11 ((𝐺 ∈ Abel ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g𝐺)𝑏) = (𝑏(+g𝐺)𝑎))
25243expb 1264 . . . . . . . . . 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 7768 . . . . . . . 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 2620 . . . . . . . . . 10 (+g𝐻) = (+g𝐻)
324, 22, 23, 31qusadd 17632 . . . . . . . . 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 1324 . . . . . . . 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 17632 . . . . . . . . 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 1324 . . . . . . . 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 2664 . . . . . . 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 6644 . . . . . . . 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 6644 . . . . . . . . 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 2635 . . . . . . 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 3034 . . . . 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 2967 . 2 ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∀𝑥 ∈ (Base‘𝐻)∀𝑦 ∈ (Base‘𝐻)(𝑥(+g𝐻)𝑦) = (𝑦(+g𝐻)𝑥))
46 eqid 2620 . . 3 (Base‘𝐻) = (Base‘𝐻)
4746, 31isabl2 18182 . 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 1481  wcel 1988  wral 2909  wrex 2910  Vcvv 3195  cfv 5876  (class class class)co 6635  [cec 7725   / cqs 7726  Basecbs 15838  +gcplusg 15922   /s cqus 16146  Grpcgrp 17403  SubGrpcsubg 17569  NrmSGrpcnsg 17570   ~QG cqg 17571  Abelcabl 18175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-ec 7729  df-qs 7733  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-sup 8333  df-inf 8334  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-3 11065  df-4 11066  df-5 11067  df-6 11068  df-7 11069  df-8 11070  df-9 11071  df-n0 11278  df-z 11363  df-dec 11479  df-uz 11673  df-fz 12312  df-struct 15840  df-ndx 15841  df-slot 15842  df-base 15844  df-sets 15845  df-ress 15846  df-plusg 15935  df-mulr 15936  df-sca 15938  df-vsca 15939  df-ip 15940  df-tset 15941  df-ple 15942  df-ds 15945  df-0g 16083  df-imas 16149  df-qus 16150  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-grp 17406  df-minusg 17407  df-subg 17572  df-nsg 17573  df-eqg 17574  df-cmn 18176  df-abl 18177
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator