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Theorem slmdvacl 30875
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 30869 . 2 (𝑊 ∈ SLMod → 𝑊 ∈ Mnd)
2 slmdvacl.v . . 3 𝑉 = (Base‘𝑊)
3 slmdvacl.a . . 3 + = (+g𝑊)
42, 3mndcl 17922 . 2 ((𝑊 ∈ Mnd ∧ 𝑋𝑉𝑌𝑉) → (𝑋 + 𝑌) ∈ 𝑉)
51, 4syl3an1 1160 1 ((𝑊 ∈ SLMod ∧ 𝑋𝑉𝑌𝑉) → (𝑋 + 𝑌) ∈ 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2115  cfv 6344  (class class class)co 7150  Basecbs 16486  +gcplusg 16568  Mndcmnd 17914  SLModcslmd 30863
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-nul 5197
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3483  df-sbc 3760  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-br 5054  df-iota 6303  df-fv 6352  df-ov 7153  df-mgm 17855  df-sgrp 17904  df-mnd 17915  df-cmn 18911  df-slmd 30864
This theorem is referenced by: (None)
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