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Theorem srgmnd 20013
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 20012 . 2 (𝑅 ∈ SRing → 𝑅 ∈ CMnd)
2 cmnmnd 19665 . 2 (𝑅 ∈ CMnd → 𝑅 ∈ Mnd)
31, 2syl 17 1 (𝑅 ∈ SRing → 𝑅 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Mndcmnd 18625  CMndccmn 19648  SRingcsrg 20009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-cmn 19650  df-srg 20010
This theorem is referenced by:  srg0cl  20023  srgacl  20028  srgcom4  20037  srg1zr  20038  srgmulgass  20040  srgpcomppsc  20043  srglmhm  20044  srgrmhm  20045  srgsummulcr  20046  sgsummulcl  20047  srgbinomlem2  20050  srgbinomlem3  20051  srgbinomlem4  20052  srgbinomlem  20053  srgbinom  20054  slmdacl  32354  slmdsn0  32356
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