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Mirrors > Home > MPE Home > Th. List > Mathboxes > slmdbn0 | Structured version Visualization version GIF version |
Description: The base set of a semimodule is nonempty. (Contributed by Thierry Arnoux, 1-Apr-2018.) (Proof shortened by AV, 10-Jan-2023.) |
Ref | Expression |
---|---|
slmdbn0.b | ⊢ 𝐵 = (Base‘𝑊) |
Ref | Expression |
---|---|
slmdbn0 | ⊢ (𝑊 ∈ SLMod → 𝐵 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | slmdmnd 30829 | . 2 ⊢ (𝑊 ∈ SLMod → 𝑊 ∈ Mnd) | |
2 | slmdbn0.b | . . 3 ⊢ 𝐵 = (Base‘𝑊) | |
3 | 2 | mndbn0 17921 | . 2 ⊢ (𝑊 ∈ Mnd → 𝐵 ≠ ∅) |
4 | 1, 3 | syl 17 | 1 ⊢ (𝑊 ∈ SLMod → 𝐵 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∅c0 4291 ‘cfv 6350 Basecbs 16477 Mndcmnd 17905 SLModcslmd 30823 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-iota 6309 df-fun 6352 df-fv 6358 df-riota 7108 df-ov 7153 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-cmn 18902 df-slmd 30824 |
This theorem is referenced by: (None) |
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