| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > lmod0cl | Structured version Visualization version GIF version | ||
| Description: The ring zero in a left module belongs to the set of scalars. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
| Ref | Expression |
|---|---|
| lmod0cl.f | ⊢ 𝐹 = (Scalar‘𝑊) |
| lmod0cl.k | ⊢ 𝐾 = (Base‘𝐹) |
| lmod0cl.z | ⊢ 0 = (0g‘𝐹) |
| Ref | Expression |
|---|---|
| lmod0cl | ⊢ (𝑊 ∈ LMod → 0 ∈ 𝐾) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lmod0cl.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
| 2 | 1 | lmodring 20866 | . 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
| 3 | lmod0cl.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
| 4 | lmod0cl.z | . . 3 ⊢ 0 = (0g‘𝐹) | |
| 5 | 3, 4 | ring0cl 20264 | . 2 ⊢ (𝐹 ∈ Ring → 0 ∈ 𝐾) |
| 6 | 2, 5 | syl 17 | 1 ⊢ (𝑊 ∈ LMod → 0 ∈ 𝐾) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 Basecbs 17247 Scalarcsca 17300 0gc0g 17484 Ringcrg 20230 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-ring 20232 df-lmod 20860 |
| This theorem is referenced by: lmodfopnelem2 20897 lmodfopne 20898 lss1d 20961 lspsolvlem 21144 iporthcom 21653 lfl0f 39070 lfl1dim 39122 lfl1dim2N 39123 lkrss2N 39170 baerlem5blem1 41711 hdmap14lem2a 41869 hdmap14lem4a 41873 hdmap14lem6 41875 hgmapval0 41894 hgmapeq0 41906 lincval1 48336 lcosn0 48337 lincvalsc0 48338 lcoc0 48339 linc1 48342 lcoss 48353 el0ldep 48383 |
| Copyright terms: Public domain | W3C validator |