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Theorem lsmcom 18621
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 2825 . . 3 (Base‘𝐺) = (Base‘𝐺)
32subgss 17953 . 2 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
42subgss 17953 . 2 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
5 lsmcom.s . . 3 = (LSSum‘𝐺)
62, 5lsmcomx 18619 . 2 ((𝐺 ∈ Abel ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇 𝑈) = (𝑈 𝑇))
71, 3, 4, 6syl3an 1203 1 ((𝐺 ∈ Abel ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 𝑈) = (𝑈 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1111   = wceq 1656  wcel 2164  wss 3798  cfv 6127  (class class class)co 6910  Basecbs 16229  SubGrpcsubg 17946  LSSumclsm 18407  Abelcabl 18554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-subg 17949  df-lsm 18409  df-cmn 18555  df-abl 18556
This theorem is referenced by:  lsm4  18623  pgpfac1lem4  18838  pgpfaclem1  18841  lspprabs  19461  ocvpj  20431  lcvexchlem3  35106  lcvexchlem4  35107  lcvexchlem5  35108  lsatcvatlem  35119  lsatcvat  35120  lsatcvat3  35122  l1cvat  35125  dia2dimlem5  37138  dihjatc3  37383  dihmeetlem9N  37385  dihjat  37493  lclkrlem2b  37578
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