lmod0vcl - Metamath Proof Explorer

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Theorem lmod0vcl 20886

Description: The zero vector is a vector. (ax-hv0cl 31074 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

**Hypotheses**
Ref	Expression
0vcl.v	⊢ 𝑉 = (Base‘𝑊)
0vcl.z	⊢ 0 = (0_g‘𝑊)

**Assertion**
Ref	Expression
lmod0vcl	⊢ (𝑊 ∈ LMod → 0 ∈ 𝑉)

**Proof of Theorem lmod0vcl**
Step	Hyp	Ref	Expression
1		lmodgrp 20862	. 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
2		0vcl.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
3		0vcl.z	. . 3 ⊢ 0 = (0_g‘𝑊)
4	2, 3	grpidcl 18941	. 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 1542 ∈ wcel 2114 ‘cfv 6498 Basecbs 17179 0_gc0g 17402 Grpcgrp 18909 LModclmod 20855

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pr 5375

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-iota 6454 df-fun 6500 df-fv 6506 df-riota 7324 df-ov 7370 df-0g 17404 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-grp 18912 df-lmod 20857

This theorem is referenced by: lmodvs0 20891 lmodfopne 20895 lsssn0 20943 lspun0 21006 lsppr0 21087 lspsneq 21120 lspprat 21151 ip0r 21617 ocvlss 21652 nmhmcn 25087 lfl0 39511 lflmul 39514 lkrlss 39541 dochexmid 41914 lcfl8 41948 lcd0vcl 42060 mapdh6bN 42183 mapdh6cN 42184 hdmap1val0 42245 hdmap1l6b 42257 hdmap1l6c 42258 hdmapval0 42279 hdmaprnlem17N 42309 hdmap14lem13 42326 hdmaplkr 42359 lcoel0 48904