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Mirrors > Home > MPE Home > Th. List > lmod0vcl | Structured version Visualization version GIF version |
Description: The zero vector is a vector. (ax-hv0cl 27244 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
0vcl.v | ⊢ 𝑉 = (Base‘𝑊) |
0vcl.z | ⊢ 0 = (0g‘𝑊) |
Ref | Expression |
---|---|
lmod0vcl | ⊢ (𝑊 ∈ LMod → 0 ∈ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodgrp 18693 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
2 | 0vcl.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | 0vcl.z | . . 3 ⊢ 0 = (0g‘𝑊) | |
4 | 2, 3 | grpidcl 17273 | . 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 1475 ∈ wcel 1977 ‘cfv 5804 Basecbs 15695 0gc0g 15923 Grpcgrp 17245 LModclmod 18686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-riota 6511 df-ov 6552 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-lmod 18688 |
This theorem is referenced by: lmodvs0 18720 lmodfopne 18724 lsssn0 18769 lspun0 18832 lsppr0 18913 lspsneq 18943 lspprat 18974 ip0r 19801 ocvlss 19835 nmhmcn 22728 lfl0 33370 lflmul 33373 lkrlss 33400 dochexmid 35775 lcfl8 35809 lcd0vcl 35921 mapdh6bN 36044 mapdh6cN 36045 hdmap1val0 36107 hdmap1l6b 36119 hdmap1l6c 36120 hdmapval0 36143 hdmaprnlem17N 36173 hdmap14lem13 36190 hdmaplkr 36223 lcoel0 42011 |
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