lmodvacl - Intuitionistic Logic Explorer

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Theorem lmodvacl 14274

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 14266	. 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
2		lmodvacl.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
3		lmodvacl.a	. . 3 ⊢ + = (+_g‘𝑊)
4	2, 3	grpcl 13549	. 2 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉)
5	1, 4	syl3an1 1304	1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ w3a 1002 = wceq 1395 ∈ wcel 2200 ‘cfv 5318 (class class class)co 6007 Basecbs 13040 +_gcplusg 13118 Grpcgrp 13541 LModclmod 14259

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 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-cnex 8098 ax-resscn 8099 ax-1re 8101 ax-addrcl 8104

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ral 2513 df-rex 2514 df-rab 2517 df-v 2801 df-sbc 3029 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-br 4084 df-opab 4146 df-mpt 4147 df-id 4384 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-iota 5278 df-fun 5320 df-fn 5321 df-fv 5326 df-ov 6010 df-inn 9119 df-2 9177 df-3 9178 df-4 9179 df-5 9180 df-6 9181 df-ndx 13043 df-slot 13044 df-base 13046 df-plusg 13131 df-mulr 13132 df-sca 13134 df-vsca 13135 df-mgm 13397 df-sgrp 13443 df-mnd 13458 df-grp 13544 df-lmod 14261

This theorem is referenced by: lmodcom 14305 lss1 14334

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