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Theorem srgmgp 19251
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 2822 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 srgmgp.g . . 3 𝐺 = (mulGrp‘𝑅)
3 eqid 2822 . . 3 (+g𝑅) = (+g𝑅)
4 eqid 2822 . . 3 (.r𝑅) = (.r𝑅)
5 eqid 2822 . . 3 (0g𝑅) = (0g𝑅)
61, 2, 3, 4, 5issrg 19248 . 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 1143 1 (𝑅 ∈ SRing → 𝐺 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  wral 3130  cfv 6334  (class class class)co 7140  Basecbs 16474  +gcplusg 16556  .rcmulr 16557  0gc0g 16704  Mndcmnd 17902  CMndccmn 18897  mulGrpcmgp 19230  SRingcsrg 19246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-nul 5186
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-iota 6293  df-fv 6342  df-ov 7143  df-srg 19247
This theorem is referenced by:  srgcl  19253  srgass  19254  srgideu  19255  srgidcl  19259  srgidmlem  19261  srg1zr  19270  srgpcomp  19273  srgpcompp  19274  srgpcomppsc  19275  srg1expzeq1  19280  srgbinomlem1  19281  srgbinomlem4  19284  srgbinomlem  19285
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