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Theorem iscrng 18752
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 6350 . . . 4 (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
2 ringmgp.g . . . 4 𝐺 = (mulGrp‘𝑅)
31, 2syl6eqr 2810 . . 3 (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝐺)
43eleq1d 2822 . 2 (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ CMnd ↔ 𝐺 ∈ CMnd))
5 df-cring 18748 . 2 CRing = {𝑟 ∈ Ring ∣ (mulGrp‘𝑟) ∈ CMnd}
64, 5elrab2 3505 1 (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ CMnd))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1630  wcel 2137  cfv 6047  CMndccmn 18391  mulGrpcmgp 18687  Ringcrg 18745  CRingccrg 18746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-rex 3054  df-rab 3057  df-v 3340  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-br 4803  df-iota 6010  df-fv 6055  df-cring 18748
This theorem is referenced by:  crngmgp  18753  crngring  18756  iscrng2  18761  crngpropd  18781  iscrngd  18784  prdscrngd  18811  subrgcrng  18984  psrcrng  19613
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