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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 19642 | . 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
3 | lmod0cl.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
4 | lmod0cl.z | . . 3 ⊢ 0 = (0g‘𝐹) | |
5 | 3, 4 | ring0cl 19319 | . 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 1537 ∈ wcel 2114 ‘cfv 6355 Basecbs 16483 Scalarcsca 16568 0gc0g 16713 Ringcrg 19297 LModclmod 19634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-riota 7114 df-ov 7159 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-ring 19299 df-lmod 19636 |
This theorem is referenced by: lmodfopnelem2 19671 lmodfopne 19672 lss1d 19735 lspsolvlem 19914 iporthcom 20779 lfl0f 36220 lfl1dim 36272 lfl1dim2N 36273 lkrss2N 36320 baerlem5blem1 38860 hdmap14lem2a 39018 hdmap14lem4a 39022 hdmap14lem6 39024 hgmapval0 39043 hgmapeq0 39055 lincval1 44494 lcosn0 44495 lincvalsc0 44496 lcoc0 44497 linc1 44500 lcoss 44511 el0ldep 44541 |
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