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Theorem srgmnd 20187
Description: A semiring is a monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.)
Assertion
Ref Expression
srgmnd (𝑅 ∈ SRing → 𝑅 ∈ Mnd)

Proof of Theorem srgmnd
StepHypRef Expression
1 srgcmn 20186 . 2 (𝑅 ∈ SRing → 𝑅 ∈ CMnd)
2 cmnmnd 19815 . 2 (𝑅 ∈ CMnd → 𝑅 ∈ Mnd)
31, 2syl 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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