Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > lmodvsubcl | Structured version Visualization version GIF version |
Description: Closure of vector subtraction. (hvsubcl 27258 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lmodvsubcl.v | ⊢ 𝑉 = (Base‘𝑊) |
lmodvsubcl.m | ⊢ − = (-g‘𝑊) |
Ref | Expression |
---|---|
lmodvsubcl | ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodgrp 18693 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
2 | lmodvsubcl.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | lmodvsubcl.m | . . 3 ⊢ − = (-g‘𝑊) | |
4 | 2, 3 | grpsubcl 17318 | . 2 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉) |
5 | 1, 4 | syl3an1 1351 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Grpcgrp 17245 -gcsg 17247 LModclmod 18686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-sbg 17250 df-lmod 18688 |
This theorem is referenced by: lspsnsub 18828 lvecvscan 18932 ip2subdi 19808 ip2eq 19817 ipcau2 22841 nmparlem 22846 minveclem1 23003 minveclem2 23005 minveclem4 23011 minveclem6 23013 pjthlem1 23016 pjthlem2 23017 eqlkr 33404 lkrlsp 33407 mapdpglem1 35979 mapdpglem2 35980 mapdpglem5N 35984 mapdpglem8 35986 mapdpglem9 35987 mapdpglem13 35991 mapdpglem14 35992 mapdpglem27 36006 baerlem3lem2 36017 baerlem5alem2 36018 baerlem5blem2 36019 mapdheq4lem 36038 mapdh6lem1N 36040 mapdh6lem2N 36041 hdmap1l6lem1 36115 hdmap1l6lem2 36116 hdmap11 36158 hdmapinvlem4 36231 |
Copyright terms: Public domain | W3C validator |