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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 20483 | . 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 20086 | . 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 1541 β wcel 2106 βcfv 6543 Basecbs 17146 Scalarcsca 17202 0gc0g 17387 Ringcrg 20058 LModclmod 20475 |
| 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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-riota 7367 df-ov 7414 df-0g 17389 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-grp 18824 df-ring 20060 df-lmod 20477 |
| This theorem is referenced by: lmodfopnelem2 20514 lmodfopne 20515 lss1d 20579 lspsolvlem 20761 iporthcom 21194 lfl0f 38025 lfl1dim 38077 lfl1dim2N 38078 lkrss2N 38125 baerlem5blem1 40666 hdmap14lem2a 40824 hdmap14lem4a 40828 hdmap14lem6 40830 hgmapval0 40849 hgmapeq0 40861 lincval1 47178 lcosn0 47179 lincvalsc0 47180 lcoc0 47181 linc1 47184 lcoss 47195 el0ldep 47225 |
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