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Theorem srgcmn 19994
Description: A semiring is a commutative monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.)
Assertion
Ref Expression
srgcmn (𝑅 ∈ SRing → 𝑅 ∈ CMnd)

Proof of Theorem srgcmn
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2733 . . 3 (mulGrp‘𝑅) = (mulGrp‘𝑅)
3 eqid 2733 . . 3 (+g𝑅) = (+g𝑅)
4 eqid 2733 . . 3 (.r𝑅) = (.r𝑅)
5 eqid 2733 . . 3 (0g𝑅) = (0g𝑅)
61, 2, 3, 4, 5issrg 19993 . 2 (𝑅 ∈ SRing ↔ (𝑅 ∈ CMnd ∧ (mulGrp‘𝑅) ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝑅)(∀𝑦 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)((𝑥(.r𝑅)(𝑦(+g𝑅)𝑧)) = ((𝑥(.r𝑅)𝑦)(+g𝑅)(𝑥(.r𝑅)𝑧)) ∧ ((𝑥(+g𝑅)𝑦)(.r𝑅)𝑧) = ((𝑥(.r𝑅)𝑧)(+g𝑅)(𝑦(.r𝑅)𝑧))) ∧ (((0g𝑅)(.r𝑅)𝑥) = (0g𝑅) ∧ (𝑥(.r𝑅)(0g𝑅)) = (0g𝑅)))))
76simp1bi 1146 1 (𝑅 ∈ SRing → 𝑅 ∈ CMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3062  cfv 6535  (class class class)co 7396  Basecbs 17131  +gcplusg 17184  .rcmulr 17185  0gc0g 17372  Mndcmnd 18612  CMndccmn 19632  mulGrpcmgp 19970  SRingcsrg 19991
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 5302
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-rab 3434  df-v 3477  df-sbc 3776  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-iota 6487  df-fv 6543  df-ov 7399  df-srg 19992
This theorem is referenced by:  srgmnd  19995  srgcom  20011  srgsummulcr  20028  sgsummulcl  20029  srgbinomlem3  20033  srgbinomlem4  20034  srgbinomlem  20035  gsumvsca2  32343
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