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Theorem lsmelval 19667
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
lsmelval ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑋 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑋 = (𝑦 + 𝑧)))
Distinct variable groups:   𝑦,𝑧, +   𝑦,𝑇,𝑧   𝑦,𝑈,𝑧   𝑦,𝐺,𝑧   𝑦,𝑋,𝑧
Allowed substitution hints:   (𝑦,𝑧)

Proof of Theorem lsmelval
StepHypRef Expression
1 subgrcl 19149 . 2 (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2 eqid 2737 . . 3 (Base‘𝐺) = (Base‘𝐺)
32subgss 19145 . 2 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
42subgss 19145 . 2 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
5 lsmelval.a . . 3 + = (+g𝐺)
6 lsmelval.p . . 3 = (LSSum‘𝐺)
72, 5, 6lsmelvalx 19658 . 2 ((𝐺 ∈ Grp ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑋 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑋 = (𝑦 + 𝑧)))
81, 3, 4, 7syl2an3an 1424 1 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑋 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑋 = (𝑦 + 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  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:  lsmelvalm  19669  lsmsubg  19672  lsmcom2  19673  lsmmod  19693  lsmdisj2  19700  pj1eu  19714  lsmcl  21082  lsmspsn  21083  lsmelval2  21084  lsmcv  21143  lindsunlem  33675  lsmsat  39009  lshpsmreu  39110  dvhopellsm  41119  diblsmopel  41173  cdlemn11c  41211  dihord11c  41226  hdmapglem7a  41929
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