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Theorem srgmnd 20208
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 20207 . 2 (𝑅 ∈ SRing → 𝑅 ∈ CMnd)
2 cmnmnd 19830 . 2 (𝑅 ∈ CMnd → 𝑅 ∈ Mnd)
31, 2syl 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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