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Theorem lsmcomx 19765
Description: Subgroup sum commutes (extended domain version). (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
lsmcomx.v 𝐵 = (Base‘𝐺)
lsmcomx.s = (LSSum‘𝐺)
Assertion
Ref Expression
lsmcomx ((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) → (𝑇 𝑈) = (𝑈 𝑇))

Proof of Theorem lsmcomx
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl1 1189 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝐺 ∈ Abel)
2 simpl2 1190 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑇𝐵)
3 simprl 767 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑦𝑇)
42, 3sseldd 3982 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑦𝐵)
5 simpl3 1191 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑈𝐵)
6 simprr 769 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑧𝑈)
75, 6sseldd 3982 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑧𝐵)
8 lsmcomx.v . . . . . . . 8 𝐵 = (Base‘𝐺)
9 eqid 2730 . . . . . . . 8 (+g𝐺) = (+g𝐺)
108, 9ablcom 19708 . . . . . . 7 ((𝐺 ∈ Abel ∧ 𝑦𝐵𝑧𝐵) → (𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦))
111, 4, 7, 10syl3anc 1369 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → (𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦))
1211eqeq2d 2741 . . . . 5 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → (𝑥 = (𝑦(+g𝐺)𝑧) ↔ 𝑥 = (𝑧(+g𝐺)𝑦)))
13122rexbidva 3215 . . . 4 ((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) → (∃𝑦𝑇𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧) ↔ ∃𝑦𝑇𝑧𝑈 𝑥 = (𝑧(+g𝐺)𝑦)))
14 rexcom 3285 . . . 4 (∃𝑦𝑇𝑧𝑈 𝑥 = (𝑧(+g𝐺)𝑦) ↔ ∃𝑧𝑈𝑦𝑇 𝑥 = (𝑧(+g𝐺)𝑦))
1513, 14bitrdi 286 . . 3 ((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) → (∃𝑦𝑇𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧) ↔ ∃𝑧𝑈𝑦𝑇 𝑥 = (𝑧(+g𝐺)𝑦)))
16 lsmcomx.s . . . 4 = (LSSum‘𝐺)
178, 9, 16lsmelvalx 19549 . . 3 ((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) → (𝑥 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
188, 9, 16lsmelvalx 19549 . . . 4 ((𝐺 ∈ Abel ∧ 𝑈𝐵𝑇𝐵) → (𝑥 ∈ (𝑈 𝑇) ↔ ∃𝑧𝑈𝑦𝑇 𝑥 = (𝑧(+g𝐺)𝑦)))
19183com23 1124 . . 3 ((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) → (𝑥 ∈ (𝑈 𝑇) ↔ ∃𝑧𝑈𝑦𝑇 𝑥 = (𝑧(+g𝐺)𝑦)))
2015, 17, 193bitr4d 310 . 2 ((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) → (𝑥 ∈ (𝑇 𝑈) ↔ 𝑥 ∈ (𝑈 𝑇)))
2120eqrdv 2728 1 ((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) → (𝑇 𝑈) = (𝑈 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1085   = wceq 1539  wcel 2104  wrex 3068  wss 3947  cfv 6542  (class class class)co 7411  Basecbs 17148  +gcplusg 17201  LSSumclsm 19543  Abelcabl 19690
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-lsm 19545  df-cmn 19691  df-abl 19692
This theorem is referenced by:  lsmcom  19767
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