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Theorem lsmelval 18110
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 17646 . . 3 (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
21adantr 480 . 2 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
3 eqid 2651 . . . 4 (Base‘𝐺) = (Base‘𝐺)
43subgss 17642 . . 3 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
54adantr 480 . 2 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑇 ⊆ (Base‘𝐺))
63subgss 17642 . . 3 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
76adantl 481 . 2 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑈 ⊆ (Base‘𝐺))
8 lsmelval.a . . 3 + = (+g𝐺)
9 lsmelval.p . . 3 = (LSSum‘𝐺)
103, 8, 9lsmelvalx 18101 . 2 ((𝐺 ∈ Grp ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑋 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑋 = (𝑦 + 𝑧)))
112, 5, 7, 10syl3anc 1366 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 196  wa 383   = wceq 1523  wcel 2030  wrex 2942  wss 3607  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  Grpcgrp 17469  SubGrpcsubg 17635  LSSumclsm 18095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-subg 17638  df-lsm 18097
This theorem is referenced by:  lsmelvalm  18112  lsmsubg  18115  lsmcom2  18116  lsmmod  18134  lsmdisj2  18141  pj1eu  18155  lsmcl  19131  lsmspsn  19132  lsmelval2  19133  lsmcv  19189  lsmsat  34613  lshpsmreu  34714  dvhopellsm  36723  diblsmopel  36777  cdlemn11c  36815  dihord11c  36830  hdmapglem7a  37536
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