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| 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 | β’ πΉ = (Scalarβπ) |
| lmod0cl.k | β’ πΎ = (BaseβπΉ) |
| lmod0cl.z | β’ 0 = (0gβπΉ) |
| Ref | Expression |
|---|---|
| lmod0cl | β’ (π β LMod β 0 β πΎ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lmod0cl.f | . . 3 β’ πΉ = (Scalarβπ) | |
| 2 | 1 | lmodring 20479 | . 2 β’ (π β LMod β πΉ β Ring) |
| 3 | lmod0cl.k | . . 3 β’ πΎ = (BaseβπΉ) | |
| 4 | lmod0cl.z | . . 3 β’ 0 = (0gβπΉ) | |
| 5 | 3, 4 | ring0cl 20084 | . 2 β’ (πΉ β Ring β 0 β πΎ) |
| 6 | 2, 5 | syl 17 | 1 β’ (π β LMod β 0 β πΎ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βcfv 6544 Basecbs 17144 Scalarcsca 17200 0gc0g 17385 Ringcrg 20056 LModclmod 20471 |
| 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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
| This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-riota 7365 df-ov 7412 df-0g 17387 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-grp 18822 df-ring 20058 df-lmod 20473 |
| This theorem is referenced by: lmodfopnelem2 20509 lmodfopne 20510 lss1d 20574 lspsolvlem 20755 iporthcom 21188 lfl0f 37939 lfl1dim 37991 lfl1dim2N 37992 lkrss2N 38039 baerlem5blem1 40580 hdmap14lem2a 40738 hdmap14lem4a 40742 hdmap14lem6 40744 hgmapval0 40763 hgmapeq0 40775 lincval1 47100 lcosn0 47101 lincvalsc0 47102 lcoc0 47103 linc1 47106 lcoss 47117 el0ldep 47147 |
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