lmodvacl - Intuitionistic Logic Explorer

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Theorem lmodvacl 13618

Description: Closure of vector addition for a left module. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

**Hypotheses**
Ref	Expression
lmodvacl.v	⊢ 𝑉 = (Base‘𝑊)
lmodvacl.a	⊢ + = (+_g‘𝑊)

**Assertion**
Ref	Expression
lmodvacl	⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉)

**Proof of Theorem lmodvacl**
Step	Hyp	Ref	Expression
1		lmodgrp 13610	. 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
2		lmodvacl.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
3		lmodvacl.a	. . 3 ⊢ + = (+_g‘𝑊)
4	2, 3	grpcl 12953	. 2 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉)
5	1, 4	syl3an1 1282	1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ w3a 980 = wceq 1364 ∈ wcel 2160 ‘cfv 5235 (class class class)co 5896 Basecbs 12512 +_gcplusg 12589 Grpcgrp 12945 LModclmod 13603

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-cnex 7932 ax-resscn 7933 ax-1re 7935 ax-addrcl 7938

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-iota 5196 df-fun 5237 df-fn 5238 df-fv 5243 df-ov 5899 df-inn 8950 df-2 9008 df-3 9009 df-4 9010 df-5 9011 df-6 9012 df-ndx 12515 df-slot 12516 df-base 12518 df-plusg 12602 df-mulr 12603 df-sca 12605 df-vsca 12606 df-mgm 12832 df-sgrp 12865 df-mnd 12878 df-grp 12948 df-lmod 13605

This theorem is referenced by: lmodcom 13649 lss1 13678

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