lmod0vcl - Metamath Proof Explorer

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Theorem lmod0vcl 20877

Description: The zero vector is a vector. (ax-hv0cl 31089 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 20853	. 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
2		0vcl.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
3		0vcl.z	. . 3 ⊢ 0 = (0_g‘𝑊)
4	2, 3	grpidcl 18932	. 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 6492 Basecbs 17170 0_gc0g 17393 Grpcgrp 18900 LModclmod 20846

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 2709 ax-sep 5231 ax-nul 5241 ax-pr 5370

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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-iota 6448 df-fun 6494 df-fv 6500 df-riota 7317 df-ov 7363 df-0g 17395 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18903 df-lmod 20848

This theorem is referenced by: lmodvs0 20882 lmodfopne 20886 lsssn0 20934 lspun0 20997 lsppr0 21079 lspsneq 21112 lspprat 21143 ip0r 21627 ocvlss 21662 nmhmcn 25097 lfl0 39525 lflmul 39528 lkrlss 39555 dochexmid 41928 lcfl8 41962 lcd0vcl 42074 mapdh6bN 42197 mapdh6cN 42198 hdmap1val0 42259 hdmap1l6b 42271 hdmap1l6c 42272 hdmapval0 42293 hdmaprnlem17N 42323 hdmap14lem13 42340 hdmaplkr 42373 lcoel0 48916