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Theorem srgcmn 19389
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 2739 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2739 . . 3 (mulGrp‘𝑅) = (mulGrp‘𝑅)
3 eqid 2739 . . 3 (+g𝑅) = (+g𝑅)
4 eqid 2739 . . 3 (.r𝑅) = (.r𝑅)
5 eqid 2739 . . 3 (0g𝑅) = (0g𝑅)
61, 2, 3, 4, 5issrg 19388 . 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 399   = wceq 1542  wcel 2114  wral 3054  cfv 6349  (class class class)co 7182  Basecbs 16598  +gcplusg 16680  .rcmulr 16681  0gc0g 16828  Mndcmnd 18039  CMndccmn 19036  mulGrpcmgp 19370  SRingcsrg 19386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-nul 5184
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-iota 6307  df-fv 6357  df-ov 7185  df-srg 19387
This theorem is referenced by:  srgmnd  19390  srgcom  19406  srgsummulcr  19418  sgsummulcl  19419  srgbinomlem3  19423  srgbinomlem4  19424  srgbinomlem  19425  gsumvsca2  31069
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