![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 20882 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
2 | lmodvacl.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | lmodvacl.a | . . 3 ⊢ + = (+g‘𝑊) | |
4 | 2, 3 | grpcl 18972 | . 2 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉) |
5 | 1, 4 | syl3an1 1162 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 Grpcgrp 18964 LModclmod 20875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-lmod 20877 |
This theorem is referenced by: lmodcom 20923 lmodvsghm 20938 lss1 20954 lspprabs 21112 lspabs2 21140 lspabs3 21141 lspfixed 21148 lspexch 21149 lspsolvlem 21162 ipdir 21675 ipdi 21676 ip2di 21677 ocvlss 21708 frlmphl 21819 frlmup1 21836 nmparlem 25287 minveclem2 25474 lsatfixedN 38991 lfl0f 39051 lfladdcl 39053 lflnegcl 39057 lflvscl 39059 lkrlss 39077 lshpkrlem5 39096 lshpkrlem6 39097 dvh3dim2 41431 dvh3dim3N 41432 lcfrlem17 41542 lcfrlem19 41544 lcfrlem20 41545 lcfrlem23 41548 baerlem3lem1 41690 baerlem5alem1 41691 baerlem5blem1 41692 baerlem5alem2 41694 baerlem5blem2 41695 mapdindp0 41702 mapdindp2 41704 mapdindp4 41706 mapdh6lem2N 41717 mapdh6aN 41718 mapdh6dN 41722 mapdh6eN 41723 mapdh6hN 41726 hdmap1l6lem2 41791 hdmap1l6a 41792 hdmap1l6d 41796 hdmap1l6e 41797 hdmap1l6h 41800 hdmap11lem1 41824 hdmap11lem2 41825 hdmapneg 41829 hdmaprnlem3N 41833 hdmaprnlem3uN 41834 hdmaprnlem6N 41837 hdmaprnlem7N 41838 hdmaprnlem9N 41840 hdmaprnlem3eN 41841 hdmap14lem10 41860 hdmapinvlem3 41903 hdmapinvlem4 41904 hdmapglem7b 41911 hlhilphllem 41946 frlmsnic 42527 lincsumcl 48277 |
Copyright terms: Public domain | W3C validator |