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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 20282 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
2 | lmodvnegcl.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | lmodvnegcl.n | . . 3 ⊢ 𝑁 = (invg‘𝑊) | |
4 | 2, 3 | grpinvcl 18758 | . 2 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉) → (𝑁‘𝑋) ∈ 𝑉) |
5 | 1, 4 | sylan 580 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘𝑋) ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6493 Basecbs 17043 Grpcgrp 18708 invgcminusg 18709 LModclmod 20275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-riota 7307 df-ov 7354 df-0g 17283 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-grp 18711 df-minusg 18712 df-lmod 20277 |
This theorem is referenced by: lmodvneg1 20318 lspsnneg 20420 lspsntrim 20512 baerlem5amN 40117 baerlem5bmN 40118 baerlem5abmN 40119 hdmapneg 40247 hdmapsub 40248 |
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