![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > slmdsn0 | Structured version Visualization version GIF version |
Description: The set of scalars in a semimodule is nonempty. (Contributed by Thierry Arnoux, 1-Apr-2018.) (Proof shortened by AV, 10-Jan-2023.) |
Ref | Expression |
---|---|
slmdsn0.f | ⊢ 𝐹 = (Scalar‘𝑊) |
slmdsn0.b | ⊢ 𝐵 = (Base‘𝐹) |
Ref | Expression |
---|---|
slmdsn0 | ⊢ (𝑊 ∈ SLMod → 𝐵 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | slmdsn0.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
2 | 1 | slmdsrg 33196 | . 2 ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing) |
3 | srgmnd 20208 | . 2 ⊢ (𝐹 ∈ SRing → 𝐹 ∈ Mnd) | |
4 | slmdsn0.b | . . 3 ⊢ 𝐵 = (Base‘𝐹) | |
5 | 4 | mndbn0 18776 | . 2 ⊢ (𝐹 ∈ Mnd → 𝐵 ≠ ∅) |
6 | 2, 3, 5 | 3syl 18 | 1 ⊢ (𝑊 ∈ SLMod → 𝐵 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∅c0 4339 ‘cfv 6563 Basecbs 17245 Scalarcsca 17301 Mndcmnd 18760 SRingcsrg 20204 SLModcslmd 33189 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-riota 7388 df-ov 7434 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-cmn 19815 df-srg 20205 df-slmd 33190 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |