Theorem slmdvacl 29765

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 e. SLMod /\ X e. V /\ Y e. V ) -> ( X .+ Y ) e. V )

Proof of Theorem slmdvacl

Step	Hyp	Ref	Expression
1		slmdmnd 29759	. 2 |- ( W e. SLMod -> W e. Mnd )
2		slmdvacl.v	. . 3 |- V = ( Base ` W )
3		slmdvacl.a	. . 3 |- .+ = ( +g ` W )
4	2, 3	mndcl 17301	. 2 |- ( ( W e. Mnd /\ X e. V /\ Y e. V ) -> ( X .+ Y ) e. V )
5	1, 4	syl3an1 1359	1 |- ( ( W e. SLMod /\ X e. V /\ Y e. V ) -> ( X .+ Y ) e. V )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   Mndcmnd 17294  SLModcslmd 29753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-cmn 18195  df-slmd 29754

This theorem is referenced by: (None)