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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 | ⊢ 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 18694 | . 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
3 | lmod0cl.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
4 | lmod0cl.z | . . 3 ⊢ 0 = (0g‘𝐹) | |
5 | 3, 4 | ring0cl 18392 | . 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 1475 ∈ wcel 1977 ‘cfv 5804 Basecbs 15695 Scalarcsca 15771 0gc0g 15923 Ringcrg 18370 LModclmod 18686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-riota 6511 df-ov 6552 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-ring 18372 df-lmod 18688 |
This theorem is referenced by: lmodfopnelem2 18723 lmodfopne 18724 lss1d 18784 lspsolvlem 18963 iporthcom 19799 lfl0f 33374 lfl1dim 33426 lfl1dim2N 33427 lkrss2N 33474 baerlem5blem1 36016 hdmap14lem2a 36177 hdmap14lem4a 36181 hdmap14lem6 36183 hgmapval0 36202 hgmapeq0 36214 lincval1 42002 lcosn0 42003 lincvalsc0 42004 lcoc0 42005 linc1 42008 lcoss 42019 el0ldep 42049 |
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