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Mirrors > Home > MPE Home > Th. List > lmod0vcl | Structured version Visualization version GIF version |
Description: The zero vector is a vector. (ax-hv0cl 31032 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
0vcl.v | ⊢ 𝑉 = (Base‘𝑊) |
0vcl.z | ⊢ 0 = (0g‘𝑊) |
Ref | Expression |
---|---|
lmod0vcl | ⊢ (𝑊 ∈ LMod → 0 ∈ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodgrp 20882 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
2 | 0vcl.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | 0vcl.z | . . 3 ⊢ 0 = (0g‘𝑊) | |
4 | 2, 3 | grpidcl 18996 | . 2 ⊢ (𝑊 ∈ Grp → 0 ∈ 𝑉) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝑊 ∈ LMod → 0 ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 Basecbs 17245 0gc0g 17486 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-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
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-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 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-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-riota 7388 df-ov 7434 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-lmod 20877 |
This theorem is referenced by: lmodvs0 20911 lmodfopne 20915 lsssn0 20964 lspun0 21027 lsppr0 21109 lspsneq 21142 lspprat 21173 ip0r 21673 ocvlss 21708 nmhmcn 25167 lfl0 39047 lflmul 39050 lkrlss 39077 dochexmid 41451 lcfl8 41485 lcd0vcl 41597 mapdh6bN 41720 mapdh6cN 41721 hdmap1val0 41782 hdmap1l6b 41794 hdmap1l6c 41795 hdmapval0 41816 hdmaprnlem17N 41846 hdmap14lem13 41863 hdmaplkr 41896 lcoel0 48274 |
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