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Mirrors > Home > MPE Home > Th. List > lmodbn0 | Structured version Visualization version GIF version |
Description: The base set of a left module is nonempty. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lmodbn0.b | ⊢ 𝐵 = (Base‘𝑊) |
Ref | Expression |
---|---|
lmodbn0 | ⊢ (𝑊 ∈ LMod → 𝐵 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodgrp 20282 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
2 | lmodbn0.b | . . 3 ⊢ 𝐵 = (Base‘𝑊) | |
3 | 2 | grpbn0 18739 | . 2 ⊢ (𝑊 ∈ Grp → 𝐵 ≠ ∅) |
4 | 1, 3 | syl 17 | 1 ⊢ (𝑊 ∈ LMod → 𝐵 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ≠ wne 2942 ∅c0 4281 ‘cfv 6494 Basecbs 17043 Grpcgrp 18708 LModclmod 20275 |
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 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-iota 6446 df-fun 6496 df-fv 6502 df-riota 7308 df-ov 7355 df-0g 17283 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-grp 18711 df-lmod 20277 |
This theorem is referenced by: lmodfopnelem1 20311 lss1 20352 lmod0rng 46061 |
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