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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > slmdmnd | Structured version Visualization version GIF version |
Description: A semimodule is a monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
Ref | Expression |
---|---|
slmdmnd | ⊢ (𝑊 ∈ SLMod → 𝑊 ∈ Mnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | slmdcmn 29886 | . 2 ⊢ (𝑊 ∈ SLMod → 𝑊 ∈ CMnd) | |
2 | cmnmnd 18254 | . 2 ⊢ (𝑊 ∈ CMnd → 𝑊 ∈ Mnd) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑊 ∈ SLMod → 𝑊 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2030 Mndcmnd 17341 CMndccmn 18239 SLModcslmd 29881 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-nul 4822 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-iota 5889 df-fv 5934 df-ov 6693 df-cmn 18241 df-slmd 29882 |
This theorem is referenced by: slmdbn0 29889 slmdvacl 29893 slmdass 29894 slmd0vcl 29902 slmd0vlid 29903 slmd0vrid 29904 |
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