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Theorem lsmcomx 18180
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 1062 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝐺 ∈ Abel)
2 simpl2 1063 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑇𝐵)
3 simprl 793 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑦𝑇)
42, 3sseldd 3584 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑦𝐵)
5 simpl3 1064 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑈𝐵)
6 simprr 795 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑧𝑈)
75, 6sseldd 3584 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑧𝐵)
8 lsmcomx.v . . . . . . . 8 𝐵 = (Base‘𝐺)
9 eqid 2621 . . . . . . . 8 (+g𝐺) = (+g𝐺)
108, 9ablcom 18131 . . . . . . 7 ((𝐺 ∈ Abel ∧ 𝑦𝐵𝑧𝐵) → (𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦))
111, 4, 7, 10syl3anc 1323 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → (𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦))
1211eqeq2d 2631 . . . . 5 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → (𝑥 = (𝑦(+g𝐺)𝑧) ↔ 𝑥 = (𝑧(+g𝐺)𝑦)))
13122rexbidva 3049 . . . 4 ((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) → (∃𝑦𝑇𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧) ↔ ∃𝑦𝑇𝑧𝑈 𝑥 = (𝑧(+g𝐺)𝑦)))
14 rexcom 3091 . . . 4 (∃𝑦𝑇𝑧𝑈 𝑥 = (𝑧(+g𝐺)𝑦) ↔ ∃𝑧𝑈𝑦𝑇 𝑥 = (𝑧(+g𝐺)𝑦))
1513, 14syl6bb 276 . . 3 ((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) → (∃𝑦𝑇𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧) ↔ ∃𝑧𝑈𝑦𝑇 𝑥 = (𝑧(+g𝐺)𝑦)))
16 lsmcomx.s . . . 4 = (LSSum‘𝐺)
178, 9, 16lsmelvalx 17976 . . 3 ((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) → (𝑥 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
188, 9, 16lsmelvalx 17976 . . . 4 ((𝐺 ∈ Abel ∧ 𝑈𝐵𝑇𝐵) → (𝑥 ∈ (𝑈 𝑇) ↔ ∃𝑧𝑈𝑦𝑇 𝑥 = (𝑧(+g𝐺)𝑦)))
19183com23 1268 . . 3 ((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) → (𝑥 ∈ (𝑈 𝑇) ↔ ∃𝑧𝑈𝑦𝑇 𝑥 = (𝑧(+g𝐺)𝑦)))
2015, 17, 193bitr4d 300 . 2 ((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) → (𝑥 ∈ (𝑇 𝑈) ↔ 𝑥 ∈ (𝑈 𝑇)))
2120eqrdv 2619 1 ((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) → (𝑇 𝑈) = (𝑈 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wrex 2908  wss 3555  cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  LSSumclsm 17970  Abelcabl 18115
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 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-lsm 17972  df-cmn 18116  df-abl 18117
This theorem is referenced by:  lsmcom  18182
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