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Definition df-drng 18515
Description: Define class of all division rings. A division ring is a ring in which the set of units is exactly the nonzero elements of the ring. (Contributed by NM, 18-Oct-2012.)
Assertion
Ref Expression
df-drng DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g𝑟)})}

Detailed syntax breakdown of Definition df-drng
StepHypRef Expression
1 cdr 18513 . 2 class DivRing
2 vr . . . . . 6 setvar 𝑟
32cv 1473 . . . . 5 class 𝑟
4 cui 18405 . . . . 5 class Unit
53, 4cfv 5787 . . . 4 class (Unit‘𝑟)
6 cbs 15638 . . . . . 6 class Base
73, 6cfv 5787 . . . . 5 class (Base‘𝑟)
8 c0g 15866 . . . . . . 7 class 0g
93, 8cfv 5787 . . . . . 6 class (0g𝑟)
109csn 4121 . . . . 5 class {(0g𝑟)}
117, 10cdif 3533 . . . 4 class ((Base‘𝑟) ∖ {(0g𝑟)})
125, 11wceq 1474 . . 3 wff (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g𝑟)})
13 crg 18313 . . 3 class Ring
1412, 2, 13crab 2896 . 2 class {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g𝑟)})}
151, 14wceq 1474 1 wff DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g𝑟)})}
Colors of variables: wff setvar class
This definition is referenced by:  isdrng  18517
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