Theorem srgmgp 20172

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.

Step	Hyp	Ref	Expression
1		eqid 2736	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		srgmgp.g	. . 3 ⊢ 𝐺 = (mulGrp‘𝑅)
3		eqid 2736	. . 3 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		eqid 2736	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
5		eqid 2736	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
6	1, 2, 3, 4, 5	issrg 20169	. 2 ⊢ (𝑅 ∈ SRing ↔ (𝑅 ∈ CMnd ∧ 𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝑅)(∀𝑦 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)(𝑦(+g‘𝑅)𝑧)) = ((𝑥(.r‘𝑅)𝑦)(+g‘𝑅)(𝑥(.r‘𝑅)𝑧)) ∧ ((𝑥(+g‘𝑅)𝑦)(.r‘𝑅)𝑧) = ((𝑥(.r‘𝑅)𝑧)(+g‘𝑅)(𝑦(.r‘𝑅)𝑧))) ∧ (((0g‘𝑅)(.r‘𝑅)𝑥) = (0g‘𝑅) ∧ (𝑥(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅)))))
7	6	simp2bi 1147	1 ⊢ (𝑅 ∈ SRing → 𝐺 ∈ Mnd)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  +_gcplusg 17220  ._rcmulr 17221  0_gc0g 17402  Mndcmnd 18702  CMndccmn 19755  mulGrpcmgp 20121  SRingcsrg 20167

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-ov 7370  df-srg 20168

This theorem is referenced by:  srgcl  20174  srgass  20175  srgideu  20176  srgidcl  20180  srgidmlem  20182  srg1zr  20196  srgpcomp  20199  srgpcompp  20200  srgpcomppsc  20201  srg1expzeq1  20206  srgbinomlem1  20207  srgbinomlem4  20210  srgbinomlem  20211