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Mirrors > Home > MPE Home > Th. List > lmodvsubcl | Structured version Visualization version GIF version |
Description: Closure of vector subtraction. (hvsubcl 28788 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 19635 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
2 | lmodvsubcl.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | lmodvsubcl.m | . . 3 ⊢ − = (-g‘𝑊) | |
4 | 2, 3 | grpsubcl 18173 | . 2 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉) |
5 | 1, 4 | syl3an1 1159 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 Grpcgrp 18097 -gcsg 18099 LModclmod 19628 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-sbg 18102 df-lmod 19630 |
This theorem is referenced by: lspsnsub 19773 lvecvscan 19877 ip2subdi 20782 ip2eq 20791 ipcau2 23831 nmparlem 23836 minveclem1 24021 minveclem2 24023 minveclem4 24029 minveclem6 24031 pjthlem1 24034 pjthlem2 24035 eqlkr 36229 lkrlsp 36232 mapdpglem1 38802 mapdpglem2 38803 mapdpglem5N 38807 mapdpglem8 38809 mapdpglem9 38810 mapdpglem13 38814 mapdpglem14 38815 mapdpglem27 38829 baerlem3lem2 38840 baerlem5alem2 38841 baerlem5blem2 38842 mapdheq4lem 38861 mapdh6lem1N 38863 mapdh6lem2N 38864 hdmap1l6lem1 38937 hdmap1l6lem2 38938 hdmap11 38978 hdmapinvlem4 39051 |
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