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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 20207 | . 2 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ CMnd) | |
2 | cmnmnd 19830 | . 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 2106 Mndcmnd 18760 CMndccmn 19813 SRingcsrg 20204 |
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-ext 2706 ax-nul 5312 |
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-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 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-iota 6516 df-fv 6571 df-ov 7434 df-cmn 19815 df-srg 20205 |
This theorem is referenced by: srg0cl 20218 srgacl 20223 srgcom4 20232 srg1zr 20233 srgmulgass 20235 srgpcomppsc 20238 srglmhm 20239 srgrmhm 20240 srgsummulcr 20241 sgsummulcl 20242 srgbinomlem2 20245 srgbinomlem3 20246 srgbinomlem4 20247 srgbinomlem 20248 srgbinom 20249 slmdacl 33198 slmdsn0 33200 |
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