lmod0vcl - Metamath Proof Explorer

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Theorem lmod0vcl 20152

Description: The zero vector is a vector. (ax-hv0cl 29365 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 20130	. 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
2		0vcl.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
3		0vcl.z	. . 3 ⊢ 0 = (0_g‘𝑊)
4	2, 3	grpidcl 18607	. 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 1539 ∈ wcel 2106 ‘cfv 6433 Basecbs 16912 0_gc0g 17150 Grpcgrp 18577 LModclmod 20123

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-iota 6391 df-fun 6435 df-fv 6441 df-riota 7232 df-ov 7278 df-0g 17152 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-grp 18580 df-lmod 20125

This theorem is referenced by: lmodvs0 20157 lmodfopne 20161 lsssn0 20209 lspun0 20273 lsppr0 20354 lspsneq 20384 lspprat 20415 ip0r 20842 ocvlss 20877 nmhmcn 24283 lfl0 37079 lflmul 37082 lkrlss 37109 dochexmid 39482 lcfl8 39516 lcd0vcl 39628 mapdh6bN 39751 mapdh6cN 39752 hdmap1val0 39813 hdmap1l6b 39825 hdmap1l6c 39826 hdmapval0 39847 hdmaprnlem17N 39877 hdmap14lem13 39894 hdmaplkr 39927 lcoel0 45769