ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-cring GIF version

Definition df-cring 13180
Description: Define class of all commutative rings. (Contributed by Mario Carneiro, 7-Jan-2015.)
Assertion
Ref Expression
df-cring CRing = {𝑓 ∈ Ring ∣ (mulGrp‘𝑓) ∈ CMnd}

Detailed syntax breakdown of Definition df-cring
StepHypRef Expression
1 ccrg 13178 . 2 class CRing
2 vf . . . . . 6 setvar 𝑓
32cv 1352 . . . . 5 class 𝑓
4 cmgp 13128 . . . . 5 class mulGrp
53, 4cfv 5216 . . . 4 class (mulGrp‘𝑓)
6 ccmn 13086 . . . 4 class CMnd
75, 6wcel 2148 . . 3 wff (mulGrp‘𝑓) ∈ CMnd
8 crg 13177 . . 3 class Ring
97, 2, 8crab 2459 . 2 class {𝑓 ∈ Ring ∣ (mulGrp‘𝑓) ∈ CMnd}
101, 9wceq 1353 1 wff CRing = {𝑓 ∈ Ring ∣ (mulGrp‘𝑓) ∈ CMnd}
Colors of variables: wff set class
This definition is referenced by:  iscrng  13184
  Copyright terms: Public domain W3C validator