lmod0vcl - Metamath Proof Explorer

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Theorem lmod0vcl 19102

Description: The zero vector is a vector. (ax-hv0cl 28200 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 19080	. 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
2		0vcl.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
3		0vcl.z	. . 3 ⊢ 0 = (0_g‘𝑊)
4	2, 3	grpidcl 17658	. 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 1631 ∈ wcel 2145 ‘cfv 6030 Basecbs 16064 0_gc0g 16308 Grpcgrp 17630 LModclmod 19073

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-iota 5993 df-fun 6032 df-fv 6038 df-riota 6757 df-ov 6799 df-0g 16310 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-grp 17633 df-lmod 19075

This theorem is referenced by: lmodvs0 19107 lmodfopne 19111 lsssn0 19158 lspun0 19224 lsppr0 19305 lspsneq 19335 lspprat 19368 ip0r 20199 ocvlss 20233 nmhmcn 23139 lfl0 34874 lflmul 34877 lkrlss 34904 dochexmid 37278 lcfl8 37312 lcd0vcl 37424 mapdh6bN 37547 mapdh6cN 37548 hdmap1val0 37609 hdmap1l6b 37621 hdmap1l6c 37622 hdmapval0 37643 hdmaprnlem17N 37673 hdmap14lem13 37690 hdmaplkr 37723 lcoel0 42742