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Theorem srgmgp 19735
Description: A semiring is a monoid under multiplication. (Contributed by Thierry Arnoux, 21-Mar-2018.)
Hypothesis
Ref Expression
srgmgp.g 𝐺 = (mulGrp‘𝑅)
Assertion
Ref Expression
srgmgp (𝑅 ∈ SRing → 𝐺 ∈ Mnd)

Proof of Theorem srgmgp
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2738 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 srgmgp.g . . 3 𝐺 = (mulGrp‘𝑅)
3 eqid 2738 . . 3 (+g𝑅) = (+g𝑅)
4 eqid 2738 . . 3 (.r𝑅) = (.r𝑅)
5 eqid 2738 . . 3 (0g𝑅) = (0g𝑅)
61, 2, 3, 4, 5issrg 19732 . 2 (𝑅 ∈ SRing ↔ (𝑅 ∈ CMnd ∧ 𝐺 ∈ 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)))))
76simp2bi 1145 1 (𝑅 ∈ SRing → 𝐺 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  wral 3064  cfv 6428  (class class class)co 7269  Basecbs 16901  +gcplusg 16951  .rcmulr 16952  0gc0g 17139  Mndcmnd 18374  CMndccmn 19375  mulGrpcmgp 19709  SRingcsrg 19730
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3433  df-sbc 3718  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-br 5076  df-iota 6386  df-fv 6436  df-ov 7272  df-srg 19731
This theorem is referenced by:  srgcl  19737  srgass  19738  srgideu  19739  srgidcl  19743  srgidmlem  19745  srg1zr  19754  srgpcomp  19757  srgpcompp  19758  srgpcomppsc  19759  srg1expzeq1  19764  srgbinomlem1  19765  srgbinomlem4  19768  srgbinomlem  19769
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