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Theorem srgmgp 18504
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 2621 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 srgmgp.g . . 3 𝐺 = (mulGrp‘𝑅)
3 eqid 2621 . . 3 (+g𝑅) = (+g𝑅)
4 eqid 2621 . . 3 (.r𝑅) = (.r𝑅)
5 eqid 2621 . . 3 (0g𝑅) = (0g𝑅)
61, 2, 3, 4, 5issrg 18501 . 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 1076 1 (𝑅 ∈ SRing → 𝐺 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wcel 1989  wral 2911  cfv 5886  (class class class)co 6647  Basecbs 15851  +gcplusg 15935  .rcmulr 15936  0gc0g 16094  Mndcmnd 17288  CMndccmn 18187  mulGrpcmgp 18483  SRingcsrg 18499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-nul 4787
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-iota 5849  df-fv 5894  df-ov 6650  df-srg 18500
This theorem is referenced by:  srgcl  18506  srgass  18507  srgideu  18508  srgidcl  18512  srgidmlem  18514  srg1zr  18523  srgpcomp  18526  srgpcompp  18527  srgpcomppsc  18528  srg1expzeq1  18533  srgbinomlem1  18534  srgbinomlem4  18537  srgbinomlem  18538
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