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Mathbox for Thierry Arnoux |
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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 32854 | . 2 ⊢ (𝑊 ∈ SLMod → 𝑊 ∈ Mnd) | |
2 | slmdbn0.b | . . 3 ⊢ 𝐵 = (Base‘𝑊) | |
3 | 2 | mndbn0 18680 | . 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 2098 ≠ wne 2934 ∅c0 4317 ‘cfv 6536 Basecbs 17150 Mndcmnd 18664 SLModcslmd 32848 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 df-riota 7360 df-ov 7407 df-0g 17393 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-cmn 19699 df-slmd 32849 |
This theorem is referenced by: (None) |
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