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| Mirrors > Home > MPE Home > Th. List > srgmnd | Structured version Visualization version GIF version | ||
| Description: A semiring is a monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.) |
| Ref | Expression |
|---|---|
| srgmnd | ⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | srgcmn 20186 | . 2 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ CMnd) | |
| 2 | cmnmnd 19815 | . 2 ⊢ (𝑅 ∈ CMnd → 𝑅 ∈ Mnd) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 Mndcmnd 18747 CMndccmn 19798 SRingcsrg 20183 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-cmn 19800 df-srg 20184 |
| This theorem is referenced by: srg0cl 20197 srgacl 20202 srgcom4 20211 srg1zr 20212 srgmulgass 20214 srgpcomppsc 20217 srglmhm 20218 srgrmhm 20219 srgsummulcr 20220 sgsummulcl 20221 srgbinomlem2 20224 srgbinomlem3 20225 srgbinomlem4 20226 srgbinomlem 20227 srgbinom 20228 slmdacl 33215 slmdsn0 33217 |
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