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Theorem srgcmn 20186
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 2737 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2737 . . 3 (mulGrp‘𝑅) = (mulGrp‘𝑅)
3 eqid 2737 . . 3 (+g𝑅) = (+g𝑅)
4 eqid 2737 . . 3 (.r𝑅) = (.r𝑅)
5 eqid 2737 . . 3 (0g𝑅) = (0g𝑅)
61, 2, 3, 4, 5issrg 20185 . 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 395   = wceq 1540  wcel 2108  wral 3061  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  .rcmulr 17298  0gc0g 17484  Mndcmnd 18747  CMndccmn 19798  mulGrpcmgp 20137  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-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-srg 20184
This theorem is referenced by:  srgmnd  20187  srgcom  20203  srgsummulcr  20220  sgsummulcl  20221  srgbinomlem3  20225  srgbinomlem4  20226  srgbinomlem  20227  gsumvsca2  33233
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