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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 | ⊢ 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 20844 | . 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
3 | lmod0cl.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
4 | lmod0cl.z | . . 3 ⊢ 0 = (0g‘𝐹) | |
5 | 3, 4 | ring0cl 20246 | . 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 1534 ∈ wcel 2099 ‘cfv 6554 Basecbs 17213 Scalarcsca 17269 0gc0g 17454 Ringcrg 20216 LModclmod 20836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6506 df-fun 6556 df-fv 6562 df-riota 7380 df-ov 7427 df-0g 17456 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-grp 18931 df-ring 20218 df-lmod 20838 |
This theorem is referenced by: lmodfopnelem2 20875 lmodfopne 20876 lss1d 20940 lspsolvlem 21123 iporthcom 21631 lfl0f 38767 lfl1dim 38819 lfl1dim2N 38820 lkrss2N 38867 baerlem5blem1 41408 hdmap14lem2a 41566 hdmap14lem4a 41570 hdmap14lem6 41572 hgmapval0 41591 hgmapeq0 41603 lincval1 47802 lcosn0 47803 lincvalsc0 47804 lcoc0 47805 linc1 47808 lcoss 47819 el0ldep 47849 |
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