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Mirrors > Home > MPE Home > Th. List > lmodvnegcl | Structured version Visualization version GIF version |
Description: Closure of vector negative. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lmodvnegcl.v | ⊢ 𝑉 = (Base‘𝑊) |
lmodvnegcl.n | ⊢ 𝑁 = (invg‘𝑊) |
Ref | Expression |
---|---|
lmodvnegcl | ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘𝑋) ∈ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodgrp 19635 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
2 | lmodvnegcl.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | lmodvnegcl.n | . . 3 ⊢ 𝑁 = (invg‘𝑊) | |
4 | 2, 3 | grpinvcl 18145 | . 2 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉) → (𝑁‘𝑋) ∈ 𝑉) |
5 | 1, 4 | sylan 582 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘𝑋) ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ‘cfv 6349 Basecbs 16477 Grpcgrp 18097 invgcminusg 18098 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-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-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-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-lmod 19630 |
This theorem is referenced by: lmodvneg1 19671 lspsnneg 19772 lspsntrim 19864 baerlem5amN 38846 baerlem5bmN 38847 baerlem5abmN 38848 hdmapneg 38976 hdmapsub 38977 |
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