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Definition df-unit 19388
 Description: Define the set of units in a ring, that is, all elements with a left and right multiplicative inverse. (Contributed by Mario Carneiro, 1-Dec-2014.)
Assertion
Ref Expression
df-unit Unit = (𝑤 ∈ V ↦ (((∥r𝑤) ∩ (∥r‘(oppr𝑤))) “ {(1r𝑤)}))

Detailed syntax breakdown of Definition df-unit
StepHypRef Expression
1 cui 19385 . 2 class Unit
2 vw . . 3 setvar 𝑤
3 cvv 3441 . . 3 class V
42cv 1537 . . . . . . 7 class 𝑤
5 cdsr 19384 . . . . . . 7 class r
64, 5cfv 6324 . . . . . 6 class (∥r𝑤)
7 coppr 19368 . . . . . . . 8 class oppr
84, 7cfv 6324 . . . . . . 7 class (oppr𝑤)
98, 5cfv 6324 . . . . . 6 class (∥r‘(oppr𝑤))
106, 9cin 3880 . . . . 5 class ((∥r𝑤) ∩ (∥r‘(oppr𝑤)))
1110ccnv 5518 . . . 4 class ((∥r𝑤) ∩ (∥r‘(oppr𝑤)))
12 cur 19244 . . . . . 6 class 1r
134, 12cfv 6324 . . . . 5 class (1r𝑤)
1413csn 4525 . . . 4 class {(1r𝑤)}
1511, 14cima 5522 . . 3 class (((∥r𝑤) ∩ (∥r‘(oppr𝑤))) “ {(1r𝑤)})
162, 3, 15cmpt 5110 . 2 class (𝑤 ∈ V ↦ (((∥r𝑤) ∩ (∥r‘(oppr𝑤))) “ {(1r𝑤)}))
171, 16wceq 1538 1 wff Unit = (𝑤 ∈ V ↦ (((∥r𝑤) ∩ (∥r‘(oppr𝑤))) “ {(1r𝑤)}))
 Colors of variables: wff setvar class This definition is referenced by:  isunit  19403
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