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Theorem lsmelvali 19668
Description: Subgroup sum membership (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
lsmelval.a + = (+g𝐺)
lsmelval.p = (LSSum‘𝐺)
Assertion
Ref Expression
lsmelvali (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑋𝑇𝑌𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 𝑈))

Proof of Theorem lsmelvali
StepHypRef Expression
1 subgrcl 19149 . . . 4 (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
21adantr 480 . . 3 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
3 eqid 2737 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
43subgss 19145 . . . 4 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
54adantr 480 . . 3 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑇 ⊆ (Base‘𝐺))
63subgss 19145 . . . 4 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
76adantl 481 . . 3 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑈 ⊆ (Base‘𝐺))
82, 5, 73jca 1129 . 2 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝐺 ∈ Grp ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)))
9 lsmelval.a . . 3 + = (+g𝐺)
10 lsmelval.p . . 3 = (LSSum‘𝐺)
113, 9, 10lsmelvalix 19659 . 2 (((𝐺 ∈ Grp ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) ∧ (𝑋𝑇𝑌𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 𝑈))
128, 11sylan 580 1 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑋𝑇𝑌𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wss 3951  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Grpcgrp 18951  SubGrpcsubg 19138  LSSumclsm 19652
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-subg 19141  df-lsm 19654
This theorem is referenced by:  lsmsubg  19672  lsmmod  19693  lsmdisj2  19700  lsmhash  19723  ablfacrp  20086  lsmcl  21082  lsmelval2  21084  lsppreli  21089  lspprabs  21094  lspabs3  21123  pjthlem2  25472  lkrlsp  39103  dia2dimlem5  41070  mapdindp0  41721  hdmaprnlem3eN  41860
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