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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 ring base set. (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 20373 | . 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 19998 | . 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 6500 Basecbs 17091 Scalarcsca 17144 0gc0g 17329 Ringcrg 19972 LModclmod 20365 |
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 5260 ax-nul 5267 ax-pr 5388 |
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 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-iota 6452 df-fun 6502 df-fv 6508 df-riota 7317 df-ov 7364 df-0g 17331 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-grp 18759 df-ring 19974 df-lmod 20367 |
This theorem is referenced by: lmodfopnelem2 20403 lmodfopne 20404 lss1d 20468 lspsolvlem 20648 iporthcom 21062 lfl0f 37581 lfl1dim 37633 lfl1dim2N 37634 lkrss2N 37681 baerlem5blem1 40222 hdmap14lem2a 40380 hdmap14lem4a 40384 hdmap14lem6 40386 hgmapval0 40405 hgmapeq0 40417 lincval1 46590 lcosn0 46591 lincvalsc0 46592 lcoc0 46593 linc1 46596 lcoss 46607 el0ldep 46637 |
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