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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 30828 | . 2 ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing) |
3 | srgmnd 19251 | . 2 ⊢ (𝐹 ∈ SRing → 𝐹 ∈ Mnd) | |
4 | slmdsn0.b | . . 3 ⊢ 𝐵 = (Base‘𝐹) | |
5 | 4 | mndbn0 17919 | . 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 1531 ∈ wcel 2108 ≠ wne 3014 ∅c0 4289 ‘cfv 6348 Basecbs 16475 Scalarcsca 16560 Mndcmnd 17903 SRingcsrg 19247 SLModcslmd 30821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-riota 7106 df-ov 7151 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-cmn 18900 df-srg 19248 df-slmd 30822 |
This theorem is referenced by: (None) |
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