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Theorem lsmelvali 18766
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 18275 . . . 4 (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
21adantr 484 . . 3 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
3 eqid 2822 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
43subgss 18271 . . . 4 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
54adantr 484 . . 3 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑇 ⊆ (Base‘𝐺))
63subgss 18271 . . . 4 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
76adantl 485 . . 3 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑈 ⊆ (Base‘𝐺))
82, 5, 73jca 1125 . 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 18757 . 2 (((𝐺 ∈ Grp ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) ∧ (𝑋𝑇𝑌𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 𝑈))
128, 11sylan 583 1 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑋𝑇𝑌𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2114  wss 3908  cfv 6334  (class class class)co 7140  Basecbs 16474  +gcplusg 16556  Grpcgrp 18094  SubGrpcsubg 18264  LSSumclsm 18750
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
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 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-subg 18267  df-lsm 18752
This theorem is referenced by:  lsmsubg  18770  lsmmod  18792  lsmdisj2  18799  lsmhash  18822  ablfacrp  19179  lsmcl  19846  lsmelval2  19848  lsppreli  19853  lspprabs  19858  lspabs3  19884  pjthlem2  24040  lkrlsp  36356  dia2dimlem5  38322  mapdindp0  38973  hdmaprnlem3eN  39112
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