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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 20865 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
| 2 | lmodvacl.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
| 3 | lmodvacl.a | . . 3 ⊢ + = (+g‘𝑊) | |
| 4 | 2, 3 | grpcl 18959 | . 2 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉) |
| 5 | 1, 4 | syl3an1 1164 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Grpcgrp 18951 LModclmod 20858 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-lmod 20860 |
| This theorem is referenced by: lmodcom 20906 lmodvsghm 20921 lss1 20936 lspprabs 21094 lspabs2 21122 lspabs3 21123 lspfixed 21130 lspexch 21131 lspsolvlem 21144 ipdir 21657 ipdi 21658 ip2di 21659 ocvlss 21690 frlmphl 21801 frlmup1 21818 nmparlem 25273 minveclem2 25460 lsatfixedN 39010 lfl0f 39070 lfladdcl 39072 lflnegcl 39076 lflvscl 39078 lkrlss 39096 lshpkrlem5 39115 lshpkrlem6 39116 dvh3dim2 41450 dvh3dim3N 41451 lcfrlem17 41561 lcfrlem19 41563 lcfrlem20 41564 lcfrlem23 41567 baerlem3lem1 41709 baerlem5alem1 41710 baerlem5blem1 41711 baerlem5alem2 41713 baerlem5blem2 41714 mapdindp0 41721 mapdindp2 41723 mapdindp4 41725 mapdh6lem2N 41736 mapdh6aN 41737 mapdh6dN 41741 mapdh6eN 41742 mapdh6hN 41745 hdmap1l6lem2 41810 hdmap1l6a 41811 hdmap1l6d 41815 hdmap1l6e 41816 hdmap1l6h 41819 hdmap11lem1 41843 hdmap11lem2 41844 hdmapneg 41848 hdmaprnlem3N 41852 hdmaprnlem3uN 41853 hdmaprnlem6N 41856 hdmaprnlem7N 41857 hdmaprnlem9N 41859 hdmaprnlem3eN 41860 hdmap14lem10 41879 hdmapinvlem3 41922 hdmapinvlem4 41923 hdmapglem7b 41930 hlhilphllem 41965 frlmsnic 42550 lincsumcl 48348 |
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