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Theorem lsmcom 18967
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 2824 . . 3 (Base‘𝐺) = (Base‘𝐺)
32subgss 18269 . 2 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
42subgss 18269 . 2 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
5 lsmcom.s . . 3 = (LSSum‘𝐺)
62, 5lsmcomx 18965 . 2 ((𝐺 ∈ Abel ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇 𝑈) = (𝑈 𝑇))
71, 3, 4, 6syl3an 1157 1 ((𝐺 ∈ Abel ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 𝑈) = (𝑈 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2115  wss 3918  cfv 6336  (class class class)co 7138  Basecbs 16472  SubGrpcsubg 18262  LSSumclsm 18748  Abelcabl 18896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7141  df-oprab 7142  df-mpo 7143  df-1st 7672  df-2nd 7673  df-subg 18265  df-lsm 18750  df-cmn 18897  df-abl 18898
This theorem is referenced by:  lsm4  18969  pgpfac1lem4  19189  pgpfaclem1  19192  lspprabs  19853  ocvpj  20847  lcvexchlem3  36232  lcvexchlem4  36233  lcvexchlem5  36234  lsatcvatlem  36245  lsatcvat  36246  lsatcvat3  36248  l1cvat  36251  dia2dimlem5  38264  dihjatc3  38509  dihmeetlem9N  38511  dihjat  38619  lclkrlem2b  38704
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