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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 30936 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 20843 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
2 | 0vcl.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | 0vcl.z | . . 3 ⊢ 0 = (0g‘𝑊) | |
4 | 2, 3 | grpidcl 18960 | . 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 1534 ∈ wcel 2099 ‘cfv 6554 Basecbs 17213 0gc0g 17454 Grpcgrp 18928 LModclmod 20836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6506 df-fun 6556 df-fv 6562 df-riota 7380 df-ov 7427 df-0g 17456 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-grp 18931 df-lmod 20838 |
This theorem is referenced by: lmodvs0 20872 lmodfopne 20876 lsssn0 20925 lspun0 20988 lsppr0 21070 lspsneq 21103 lspprat 21134 ip0r 21633 ocvlss 21668 nmhmcn 25138 lfl0 38763 lflmul 38766 lkrlss 38793 dochexmid 41167 lcfl8 41201 lcd0vcl 41313 mapdh6bN 41436 mapdh6cN 41437 hdmap1val0 41498 hdmap1l6b 41510 hdmap1l6c 41511 hdmapval0 41532 hdmaprnlem17N 41562 hdmap14lem13 41579 hdmaplkr 41612 lcoel0 47811 |
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