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Theorem lsmelval 19558
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 19047 . 2 (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2 eqid 2730 . . 3 (Base‘𝐺) = (Base‘𝐺)
32subgss 19043 . 2 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
42subgss 19043 . 2 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
5 lsmelval.a . . 3 + = (+g𝐺)
6 lsmelval.p . . 3 = (LSSum‘𝐺)
72, 5, 6lsmelvalx 19549 . 2 ((𝐺 ∈ Grp ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑋 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑋 = (𝑦 + 𝑧)))
81, 3, 4, 7syl2an3an 1420 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 205  wa 394   = wceq 1539  wcel 2104  wrex 3068  wss 3947  cfv 6542  (class class class)co 7411  Basecbs 17148  +gcplusg 17201  Grpcgrp 18855  SubGrpcsubg 19036  LSSumclsm 19543
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-subg 19039  df-lsm 19545
This theorem is referenced by:  lsmelvalm  19560  lsmsubg  19563  lsmcom2  19564  lsmmod  19584  lsmdisj2  19591  pj1eu  19605  lsmcl  20838  lsmspsn  20839  lsmelval2  20840  lsmcv  20899  lindsunlem  32997  lsmsat  38181  lshpsmreu  38282  dvhopellsm  40291  diblsmopel  40345  cdlemn11c  40383  dihord11c  40398  hdmapglem7a  41101
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