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Theorem lsmcom 18469
Description: Subgroup sum commutes. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
Hypothesis
Ref Expression
lsmcom.s = (LSSum‘𝐺)
Assertion
Ref Expression
lsmcom ((𝐺 ∈ Abel ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 𝑈) = (𝑈 𝑇))

Proof of Theorem lsmcom
StepHypRef Expression
1 id 22 . 2 (𝐺 ∈ Abel → 𝐺 ∈ Abel)
2 eqid 2771 . . 3 (Base‘𝐺) = (Base‘𝐺)
32subgss 17804 . 2 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
42subgss 17804 . 2 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
5 lsmcom.s . . 3 = (LSSum‘𝐺)
62, 5lsmcomx 18467 . 2 ((𝐺 ∈ Abel ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇 𝑈) = (𝑈 𝑇))
71, 3, 4, 6syl3an 1163 1 ((𝐺 ∈ Abel ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 𝑈) = (𝑈 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071   = wceq 1631  wcel 2145  wss 3724  cfv 6032  (class class class)co 6794  Basecbs 16065  SubGrpcsubg 17797  LSSumclsm 18257  Abelcabl 18402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-1st 7316  df-2nd 7317  df-subg 17800  df-lsm 18259  df-cmn 18403  df-abl 18404
This theorem is referenced by:  lsm4  18471  pgpfac1lem4  18686  pgpfaclem1  18689  lspprabs  19309  ocvpj  20279  lcvexchlem3  34846  lcvexchlem4  34847  lcvexchlem5  34848  lsatcvatlem  34859  lsatcvat  34860  lsatcvat3  34862  l1cvat  34865  dia2dimlem5  36879  dihjatc3  37124  dihmeetlem9N  37126  dihjat  37234  lclkrlem2b  37319
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