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Theorem lsmcom 19806
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 2728 . . 3 (Base‘𝐺) = (Base‘𝐺)
32subgss 19075 . 2 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
42subgss 19075 . 2 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
5 lsmcom.s . . 3 = (LSSum‘𝐺)
62, 5lsmcomx 19804 . 2 ((𝐺 ∈ Abel ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇 𝑈) = (𝑈 𝑇))
71, 3, 4, 6syl3an 1158 1 ((𝐺 ∈ Abel ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 𝑈) = (𝑈 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1085   = wceq 1534  wcel 2099  wss 3945  cfv 6542  (class class class)co 7414  Basecbs 17173  SubGrpcsubg 19068  LSSumclsm 19582  Abelcabl 19729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-subg 19071  df-lsm 19584  df-cmn 19730  df-abl 19731
This theorem is referenced by:  lsm4  19808  pgpfac1lem4  20028  pgpfaclem1  20031  lspprabs  20973  ocvpj  21644  idlsrgcmnd  33220  lcvexchlem3  38502  lcvexchlem4  38503  lcvexchlem5  38504  lsatcvatlem  38515  lsatcvat  38516  lsatcvat3  38518  l1cvat  38521  dia2dimlem5  40535  dihjatc3  40780  dihmeetlem9N  40782  dihjat  40890  lclkrlem2b  40975
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