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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 19641 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
2 | lmodbn0.b | . . 3 ⊢ 𝐵 = (Base‘𝑊) | |
3 | 2 | grpbn0 18132 | . 2 ⊢ (𝑊 ∈ Grp → 𝐵 ≠ ∅) |
4 | 1, 3 | syl 17 | 1 ⊢ (𝑊 ∈ LMod → 𝐵 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∅c0 4291 ‘cfv 6355 Basecbs 16483 Grpcgrp 18103 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-lmod 19636 |
This theorem is referenced by: lmodfopnelem1 19670 lss1 19710 lmod0rng 44159 |
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