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Theorem lsmcomx 18972
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 770 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑦𝑇)
42, 3sseldd 3953 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑦𝐵)
5 simpl3 1190 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑈𝐵)
6 simprr 772 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑧𝑈)
75, 6sseldd 3953 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑧𝐵)
8 lsmcomx.v . . . . . . . 8 𝐵 = (Base‘𝐺)
9 eqid 2824 . . . . . . . 8 (+g𝐺) = (+g𝐺)
108, 9ablcom 18920 . . . . . . 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 2835 . . . . 5 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → (𝑥 = (𝑦(+g𝐺)𝑧) ↔ 𝑥 = (𝑧(+g𝐺)𝑦)))
13122rexbidva 3292 . . . 4 ((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) → (∃𝑦𝑇𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧) ↔ ∃𝑦𝑇𝑧𝑈 𝑥 = (𝑧(+g𝐺)𝑦)))
14 rexcom 3347 . . . 4 (∃𝑦𝑇𝑧𝑈 𝑥 = (𝑧(+g𝐺)𝑦) ↔ ∃𝑧𝑈𝑦𝑇 𝑥 = (𝑧(+g𝐺)𝑦))
1513, 14syl6bb 290 . . 3 ((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) → (∃𝑦𝑇𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧) ↔ ∃𝑧𝑈𝑦𝑇 𝑥 = (𝑧(+g𝐺)𝑦)))
16 lsmcomx.s . . . 4 = (LSSum‘𝐺)
178, 9, 16lsmelvalx 18761 . . 3 ((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) → (𝑥 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
188, 9, 16lsmelvalx 18761 . . . 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 314 . 2 ((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) → (𝑥 ∈ (𝑇 𝑈) ↔ 𝑥 ∈ (𝑈 𝑇)))
2120eqrdv 2822 1 ((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) → (𝑇 𝑈) = (𝑈 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wrex 3134  wss 3919  cfv 6343  (class class class)co 7145  Basecbs 16479  +gcplusg 16561  LSSumclsm 18755  Abelcabl 18903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7679  df-2nd 7680  df-lsm 18757  df-cmn 18904  df-abl 18905
This theorem is referenced by:  lsmcom  18974
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