lmod0vcl - Metamath Proof Explorer

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Theorem lmod0vcl 20722

Description: The zero vector is a vector. (ax-hv0cl 30680 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 20698	. 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
2		0vcl.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
3		0vcl.z	. . 3 ⊢ 0 = (0_g‘𝑊)
4	2, 3	grpidcl 18882	. 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 1533 ∈ wcel 2098 ‘cfv 6533 Basecbs 17140 0_gc0g 17381 Grpcgrp 18850 LModclmod 20691

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-iota 6485 df-fun 6535 df-fv 6541 df-riota 7357 df-ov 7404 df-0g 17383 df-mgm 18560 df-sgrp 18639 df-mnd 18655 df-grp 18853 df-lmod 20693

This theorem is referenced by: lmodvs0 20727 lmodfopne 20731 lsssn0 20780 lspun0 20843 lsppr0 20925 lspsneq 20958 lspprat 20989 ip0r 21490 ocvlss 21525 nmhmcn 24957 lfl0 38391 lflmul 38394 lkrlss 38421 dochexmid 40795 lcfl8 40829 lcd0vcl 40941 mapdh6bN 41064 mapdh6cN 41065 hdmap1val0 41126 hdmap1l6b 41138 hdmap1l6c 41139 hdmapval0 41160 hdmaprnlem17N 41190 hdmap14lem13 41207 hdmaplkr 41240 lcoel0 47263