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Theorem lsmcomx 19823
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 1188 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝐺 ∈ Abel)
2 simpl2 1189 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑇𝐵)
3 simprl 769 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑦𝑇)
42, 3sseldd 3977 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑦𝐵)
5 simpl3 1190 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑈𝐵)
6 simprr 771 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑧𝑈)
75, 6sseldd 3977 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑧𝐵)
8 lsmcomx.v . . . . . . . 8 𝐵 = (Base‘𝐺)
9 eqid 2725 . . . . . . . 8 (+g𝐺) = (+g𝐺)
108, 9ablcom 19766 . . . . . . 7 ((𝐺 ∈ Abel ∧ 𝑦𝐵𝑧𝐵) → (𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦))
111, 4, 7, 10syl3anc 1368 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → (𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦))
1211eqeq2d 2736 . . . . 5 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → (𝑥 = (𝑦(+g𝐺)𝑧) ↔ 𝑥 = (𝑧(+g𝐺)𝑦)))
13122rexbidva 3207 . . . 4 ((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) → (∃𝑦𝑇𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧) ↔ ∃𝑦𝑇𝑧𝑈 𝑥 = (𝑧(+g𝐺)𝑦)))
14 rexcom 3277 . . . 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 19607 . . 3 ((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) → (𝑥 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
188, 9, 16lsmelvalx 19607 . . . 4 ((𝐺 ∈ Abel ∧ 𝑈𝐵𝑇𝐵) → (𝑥 ∈ (𝑈 𝑇) ↔ ∃𝑧𝑈𝑦𝑇 𝑥 = (𝑧(+g𝐺)𝑦)))
19183com23 1123 . . 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 2723 1 ((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) → (𝑇 𝑈) = (𝑈 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wrex 3059  wss 3944  cfv 6549  (class class class)co 7419  Basecbs 17183  +gcplusg 17236  LSSumclsm 19601  Abelcabl 19748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-lsm 19603  df-cmn 19749  df-abl 19750
This theorem is referenced by:  lsmcom  19825
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