Theorem lsmcom 19778

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

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Abel)
2		eqid 2733	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
3	2	subgss 19048	. 2 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
4	2	subgss 19048	. 2 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
5		lsmcom.s	. . 3 ⊢ ⊕ = (LSSum‘𝐺)
6	2, 5	lsmcomx 19776	. 2 ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))
7	1, 3, 4, 6	syl3an 1160	1 ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ⊆ wss 3898  ‘cfv 6489  (class class class)co 7355  Basecbs 17127  SubGrpcsubg 19041  LSSumclsm 19554  Abelcabl 19701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-1st 7930  df-2nd 7931  df-subg 19044  df-lsm 19556  df-cmn 19702  df-abl 19703

This theorem is referenced by:  lsm4  19780  pgpfac1lem4  20000  pgpfaclem1  20003  lspprabs  21038  ocvpj  21663  idlsrgcmnd  33524  lcvexchlem3  39208  lcvexchlem4  39209  lcvexchlem5  39210  lsatcvatlem  39221  lsatcvat  39222  lsatcvat3  39224  l1cvat  39227  dia2dimlem5  41240  dihjatc3  41485  dihmeetlem9N  41487  dihjat  41595  lclkrlem2b  41680