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Theorem iscrng 18835
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 6379 . . . 4 (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
2 ringmgp.g . . . 4 𝐺 = (mulGrp‘𝑅)
31, 2syl6eqr 2817 . . 3 (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝐺)
43eleq1d 2829 . 2 (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ CMnd ↔ 𝐺 ∈ CMnd))
5 df-cring 18831 . 2 CRing = {𝑟 ∈ Ring ∣ (mulGrp‘𝑟) ∈ CMnd}
64, 5elrab2 3525 1 (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ CMnd))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  wcel 2155  cfv 6070  CMndccmn 18473  mulGrpcmgp 18770  Ringcrg 18828  CRingccrg 18829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-iota 6033  df-fv 6078  df-cring 18831
This theorem is referenced by:  crngmgp  18836  crngring  18839  iscrng2  18844  crngpropd  18864  iscrngd  18867  prdscrngd  18894  subrgcrng  19067  psrcrng  19701
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