Theorem lsmcom 19764

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 2729	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
3	2	subgss 19035	. 2 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
4	2	subgss 19035	. 2 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
5		lsmcom.s	. . 3 ⊢ ⊕ = (LSSum‘𝐺)
6	2, 5	lsmcomx 19762	. 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 1540   ∈ wcel 2109   ⊆ wss 3911  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  SubGrpcsubg 19028  LSSumclsm 19540  Abelcabl 19687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-subg 19031  df-lsm 19542  df-cmn 19688  df-abl 19689

This theorem is referenced by:  lsm4  19766  pgpfac1lem4  19986  pgpfaclem1  19989  lspprabs  20978  ocvpj  21602  idlsrgcmnd  33459  lcvexchlem3  39002  lcvexchlem4  39003  lcvexchlem5  39004  lsatcvatlem  39015  lsatcvat  39016  lsatcvat3  39018  l1cvat  39021  dia2dimlem5  41035  dihjatc3  41280  dihmeetlem9N  41282  dihjat  41390  lclkrlem2b  41475