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Theorem lsmcomx 18459
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 1228 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝐺 ∈ Abel)
2 simpl2 1230 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑇𝐵)
3 simprl 811 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑦𝑇)
42, 3sseldd 3745 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑦𝐵)
5 simpl3 1232 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑈𝐵)
6 simprr 813 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑧𝑈)
75, 6sseldd 3745 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → 𝑧𝐵)
8 lsmcomx.v . . . . . . . 8 𝐵 = (Base‘𝐺)
9 eqid 2760 . . . . . . . 8 (+g𝐺) = (+g𝐺)
108, 9ablcom 18410 . . . . . . 7 ((𝐺 ∈ Abel ∧ 𝑦𝐵𝑧𝐵) → (𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦))
111, 4, 7, 10syl3anc 1477 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → (𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦))
1211eqeq2d 2770 . . . . 5 (((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) ∧ (𝑦𝑇𝑧𝑈)) → (𝑥 = (𝑦(+g𝐺)𝑧) ↔ 𝑥 = (𝑧(+g𝐺)𝑦)))
13122rexbidva 3194 . . . 4 ((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) → (∃𝑦𝑇𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧) ↔ ∃𝑦𝑇𝑧𝑈 𝑥 = (𝑧(+g𝐺)𝑦)))
14 rexcom 3237 . . . 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 18255 . . 3 ((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) → (𝑥 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
188, 9, 16lsmelvalx 18255 . . . 4 ((𝐺 ∈ Abel ∧ 𝑈𝐵𝑇𝐵) → (𝑥 ∈ (𝑈 𝑇) ↔ ∃𝑧𝑈𝑦𝑇 𝑥 = (𝑧(+g𝐺)𝑦)))
19183com23 1121 . . 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 2758 1 ((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) → (𝑇 𝑈) = (𝑈 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wrex 3051  wss 3715  cfv 6049  (class class class)co 6813  Basecbs 16059  +gcplusg 16143  LSSumclsm 18249  Abelcabl 18394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-lsm 18251  df-cmn 18395  df-abl 18396
This theorem is referenced by:  lsmcom  18461
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