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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 31022 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 20865 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
| 2 | 0vcl.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
| 3 | 0vcl.z | . . 3 ⊢ 0 = (0g‘𝑊) | |
| 4 | 2, 3 | grpidcl 18983 | . 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 1540 ∈ wcel 2108 ‘cfv 6561 Basecbs 17247 0gc0g 17484 Grpcgrp 18951 LModclmod 20858 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-riota 7388 df-ov 7434 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-lmod 20860 |
| This theorem is referenced by: lmodvs0 20894 lmodfopne 20898 lsssn0 20946 lspun0 21009 lsppr0 21091 lspsneq 21124 lspprat 21155 ip0r 21655 ocvlss 21690 nmhmcn 25153 lfl0 39066 lflmul 39069 lkrlss 39096 dochexmid 41470 lcfl8 41504 lcd0vcl 41616 mapdh6bN 41739 mapdh6cN 41740 hdmap1val0 41801 hdmap1l6b 41813 hdmap1l6c 41814 hdmapval0 41835 hdmaprnlem17N 41865 hdmap14lem13 41882 hdmaplkr 41915 lcoel0 48345 |
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