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Theorem iscrng 19385
 Description: A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
Hypothesis
Ref Expression
ringmgp.g 𝐺 = (mulGrp‘𝑅)
Assertion
Ref Expression
iscrng (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ CMnd))

Proof of Theorem iscrng
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6663 . . . 4 (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
2 ringmgp.g . . . 4 𝐺 = (mulGrp‘𝑅)
31, 2eqtr4di 2811 . . 3 (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝐺)
43eleq1d 2836 . 2 (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ CMnd ↔ 𝐺 ∈ CMnd))
5 df-cring 19381 . 2 CRing = {𝑟 ∈ Ring ∣ (mulGrp‘𝑟) ∈ CMnd}
64, 5elrab2 3607 1 (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ CMnd))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ‘cfv 6340  CMndccmn 18986  mulGrpcmgp 19320  Ringcrg 19378  CRingccrg 19379 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 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-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-rab 3079  df-v 3411  df-un 3865  df-in 3867  df-ss 3877  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-iota 6299  df-fv 6348  df-cring 19381 This theorem is referenced by:  crngmgp  19386  crngring  19390  iscrng2  19397  crngpropd  19417  iscrngd  19420  prdscrngd  19447  subrgcrng  19620  psrcrng  20754  cntrcrng  30860
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