Theorem slmdvacl 33300

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	⊢ + = (+g‘𝑊)

Assertion

Ref	Expression
slmdvacl	⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉)

Proof of Theorem slmdvacl

Step	Hyp	Ref	Expression
1		slmdmnd 33294	. 2 ⊢ (𝑊 ∈ SLMod → 𝑊 ∈ Mnd)
2		slmdvacl.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
3		slmdvacl.a	. . 3 ⊢ + = (+g‘𝑊)
4	2, 3	mndcl 18708	. 2 ⊢ ((𝑊 ∈ Mnd ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉)
5	1, 4	syl3an1 1169	1 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉)

Syntax hints:   → wi 4   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  +_gcplusg 17218  Mndcmnd 18700  SLModcslmd 33288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-nul 5235

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-iota 6448  df-fv 6500  df-ov 7366  df-mgm 18606  df-sgrp 18685  df-mnd 18701  df-cmn 19755  df-slmd 33289

This theorem is referenced by: (None)