Theorem lsmcom 19927

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 23	. 2 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Abel)
2		eqid 2769	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
3	2	subgss 19192	. 2 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
4	2	subgss 19192	. 2 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
5		lsmcom.s	. . 3 ⊢ ⊕ = (LSSum‘𝐺)
6	2, 5	lsmcomx 19925	. 2 ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))
7	1, 3, 4, 6	syl3an 1176	1 ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))

Syntax hints:   → wi 4   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ⊆ wss 3913  ‘cfv 6537  (class class class)co 7411  Basecbs 17268  SubGrpcsubg 19185  LSSumclsm 19703  Abelcabl 19850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7985  df-2nd 7986  df-subg 19188  df-lsm 19705  df-cmn 19851  df-abl 19852

This theorem is referenced by:  lsm4  19929  pgpfac1lem4  20149  pgpfaclem1  20152  lspprabs  21193  ocvpj  21835  idlsrgcmnd  33749  lcvexchlem3  39699  lcvexchlem4  39700  lcvexchlem5  39701  lsatcvatlem  39712  lsatcvat  39713  lsatcvat3  39715  l1cvat  39718  dia2dimlem5  41731  dihjatc3  41976  dihmeetlem9N  41978  dihjat  42086  lclkrlem2b  42171