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Theorem srgmgp 19189
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 2818 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 srgmgp.g . . 3 𝐺 = (mulGrp‘𝑅)
3 eqid 2818 . . 3 (+g𝑅) = (+g𝑅)
4 eqid 2818 . . 3 (.r𝑅) = (.r𝑅)
5 eqid 2818 . . 3 (0g𝑅) = (0g𝑅)
61, 2, 3, 4, 5issrg 19186 . 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 1138 1 (𝑅 ∈ SRing → 𝐺 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  .rcmulr 16554  0gc0g 16701  Mndcmnd 17899  CMndccmn 18835  mulGrpcmgp 19168  SRingcsrg 19184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-srg 19185
This theorem is referenced by:  srgcl  19191  srgass  19192  srgideu  19193  srgidcl  19197  srgidmlem  19199  srg1zr  19208  srgpcomp  19211  srgpcompp  19212  srgpcomppsc  19213  srg1expzeq1  19218  srgbinomlem1  19219  srgbinomlem4  19222  srgbinomlem  19223
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