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Theorem slmdvacl 30105
Description: Closure of vector addition for a semiring left module. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
slmdvacl.v 𝑉 = (Base‘𝑊)
slmdvacl.a + = (+g𝑊)
Assertion
Ref Expression
slmdvacl ((𝑊 ∈ SLMod ∧ 𝑋𝑉𝑌𝑉) → (𝑋 + 𝑌) ∈ 𝑉)

Proof of Theorem slmdvacl
StepHypRef Expression
1 slmdmnd 30099 . 2 (𝑊 ∈ SLMod → 𝑊 ∈ Mnd)
2 slmdvacl.v . . 3 𝑉 = (Base‘𝑊)
3 slmdvacl.a . . 3 + = (+g𝑊)
42, 3mndcl 17509 . 2 ((𝑊 ∈ Mnd ∧ 𝑋𝑉𝑌𝑉) → (𝑋 + 𝑌) ∈ 𝑉)
51, 4syl3an1 1166 1 ((𝑊 ∈ SLMod ∧ 𝑋𝑉𝑌𝑉) → (𝑋 + 𝑌) ∈ 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071   = wceq 1631  wcel 2145  cfv 6030  (class class class)co 6796  Basecbs 16064  +gcplusg 16149  Mndcmnd 17502  SLModcslmd 30093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4924
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-iota 5993  df-fv 6038  df-ov 6799  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-cmn 18402  df-slmd 30094
This theorem is referenced by: (None)
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