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Mirrors > Home > MPE Home > Th. List > lmodvacl | Structured version Visualization version GIF version |
Description: Closure of vector addition for a left module. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lmodvacl.v | ⊢ 𝑉 = (Base‘𝑊) |
lmodvacl.a | ⊢ + = (+g‘𝑊) |
Ref | Expression |
---|---|
lmodvacl | ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodgrp 20482 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
2 | lmodvacl.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | lmodvacl.a | . . 3 ⊢ + = (+g‘𝑊) | |
4 | 2, 3 | grpcl 18829 | . 2 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉) |
5 | 1, 4 | syl3an1 1163 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ‘cfv 6543 (class class class)co 7411 Basecbs 17146 +gcplusg 17199 Grpcgrp 18821 LModclmod 20475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-nul 5306 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-ov 7414 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-grp 18824 df-lmod 20477 |
This theorem is referenced by: lmodcom 20523 lmodvsghm 20538 lss1 20554 lspprabs 20711 lspabs2 20739 lspabs3 20740 lspfixed 20747 lspexch 20748 lspsolvlem 20761 ipdir 21198 ipdi 21199 ip2di 21200 ocvlss 21231 frlmphl 21342 frlmup1 21359 nmparlem 24763 minveclem2 24950 lsatfixedN 37965 lfl0f 38025 lfladdcl 38027 lflnegcl 38031 lflvscl 38033 lkrlss 38051 lshpkrlem5 38070 lshpkrlem6 38071 dvh3dim2 40405 dvh3dim3N 40406 lcfrlem17 40516 lcfrlem19 40518 lcfrlem20 40519 lcfrlem23 40522 baerlem3lem1 40664 baerlem5alem1 40665 baerlem5blem1 40666 baerlem5alem2 40668 baerlem5blem2 40669 mapdindp0 40676 mapdindp2 40678 mapdindp4 40680 mapdh6lem2N 40691 mapdh6aN 40692 mapdh6dN 40696 mapdh6eN 40697 mapdh6hN 40700 hdmap1l6lem2 40765 hdmap1l6a 40766 hdmap1l6d 40770 hdmap1l6e 40771 hdmap1l6h 40774 hdmap11lem1 40798 hdmap11lem2 40799 hdmapneg 40803 hdmaprnlem3N 40807 hdmaprnlem3uN 40808 hdmaprnlem6N 40811 hdmaprnlem7N 40812 hdmaprnlem9N 40814 hdmaprnlem3eN 40815 hdmap14lem10 40834 hdmapinvlem3 40877 hdmapinvlem4 40878 hdmapglem7b 40885 hlhilphllem 40920 frlmsnic 41192 lincsumcl 47190 |
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