lmodvacl - Intuitionistic Logic Explorer

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Theorem lmodvacl 14231

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 14223	. 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
2		lmodvacl.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
3		lmodvacl.a	. . 3 ⊢ + = (+_g‘𝑊)
4	2, 3	grpcl 13507	. 2 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉)
5	1, 4	syl3an1 1285	1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ w3a 983 = wceq 1375 ∈ wcel 2180 ‘cfv 5294 (class class class)co 5974 Basecbs 12998 +_gcplusg 13076 Grpcgrp 13499 LModclmod 14216

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 713 ax-5 1473 ax-7 1474 ax-gen 1475 ax-ie1 1519 ax-ie2 1520 ax-8 1530 ax-10 1531 ax-11 1532 ax-i12 1533 ax-bndl 1535 ax-4 1536 ax-17 1552 ax-i9 1556 ax-ial 1560 ax-i5r 1561 ax-13 2182 ax-14 2183 ax-ext 2191 ax-sep 4181 ax-pow 4237 ax-pr 4272 ax-un 4501 ax-cnex 8058 ax-resscn 8059 ax-1re 8061 ax-addrcl 8064

This theorem depends on definitions: df-bi 117 df-3an 985 df-tru 1378 df-nf 1487 df-sb 1789 df-eu 2060 df-mo 2061 df-clab 2196 df-cleq 2202 df-clel 2205 df-nfc 2341 df-ral 2493 df-rex 2494 df-rab 2497 df-v 2781 df-sbc 3009 df-un 3181 df-in 3183 df-ss 3190 df-pw 3631 df-sn 3652 df-pr 3653 df-op 3655 df-uni 3868 df-int 3903 df-br 4063 df-opab 4125 df-mpt 4126 df-id 4361 df-xp 4702 df-rel 4703 df-cnv 4704 df-co 4705 df-dm 4706 df-rn 4707 df-res 4708 df-iota 5254 df-fun 5296 df-fn 5297 df-fv 5302 df-ov 5977 df-inn 9079 df-2 9137 df-3 9138 df-4 9139 df-5 9140 df-6 9141 df-ndx 13001 df-slot 13002 df-base 13004 df-plusg 13089 df-mulr 13090 df-sca 13092 df-vsca 13093 df-mgm 13355 df-sgrp 13401 df-mnd 13416 df-grp 13502 df-lmod 14218

This theorem is referenced by: lmodcom 14262 lss1 14291

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