Theorem lsmcom 19874

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 2756	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
3	2	subgss 19145	. 2 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
4	2	subgss 19145	. 2 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
5		lsmcom.s	. . 3 ⊢ ⊕ = (LSSum‘𝐺)
6	2, 5	lsmcomx 19872	. 2 ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))
7	1, 3, 4, 6	syl3an 1169	1 ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))

Syntax hints:   → wi 4   ∧ w3a 1095   = wceq 1554   ∈ wcel 2136   ⊆ wss 3899  ‘cfv 6510  (class class class)co 7385  Basecbs 17221  SubGrpcsubg 19138  LSSumclsm 19650  Abelcabl 19797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-rep 5221  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-ov 7388  df-oprab 7389  df-mpo 7390  df-1st 7959  df-2nd 7960  df-subg 19141  df-lsm 19652  df-cmn 19798  df-abl 19799

This theorem is referenced by:  lsm4  19876  pgpfac1lem4  20096  pgpfaclem1  20099  lspprabs  21135  ocvpj  21742  idlsrgcmnd  33665  lcvexchlem3  39608  lcvexchlem4  39609  lcvexchlem5  39610  lsatcvatlem  39621  lsatcvat  39622  lsatcvat3  39624  l1cvat  39627  dia2dimlem5  41640  dihjatc3  41885  dihmeetlem9N  41887  dihjat  41995  lclkrlem2b  42080