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Definition df-ablo 27527
Description: Define the class of all Abelian group operations. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
Assertion
Ref Expression
df-ablo AbelOp = {𝑔 ∈ GrpOp ∣ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥)}
Distinct variable group:   𝑥,𝑔,𝑦

Detailed syntax breakdown of Definition df-ablo
StepHypRef Expression
1 cablo 27526 . 2 class AbelOp
2 vx . . . . . . . 8 setvar 𝑥
32cv 1522 . . . . . . 7 class 𝑥
4 vy . . . . . . . 8 setvar 𝑦
54cv 1522 . . . . . . 7 class 𝑦
6 vg . . . . . . . 8 setvar 𝑔
76cv 1522 . . . . . . 7 class 𝑔
83, 5, 7co 6690 . . . . . 6 class (𝑥𝑔𝑦)
95, 3, 7co 6690 . . . . . 6 class (𝑦𝑔𝑥)
108, 9wceq 1523 . . . . 5 wff (𝑥𝑔𝑦) = (𝑦𝑔𝑥)
117crn 5144 . . . . 5 class ran 𝑔
1210, 4, 11wral 2941 . . . 4 wff 𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥)
1312, 2, 11wral 2941 . . 3 wff 𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥)
14 cgr 27471 . . 3 class GrpOp
1513, 6, 14crab 2945 . 2 class {𝑔 ∈ GrpOp ∣ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥)}
161, 15wceq 1523 1 wff AbelOp = {𝑔 ∈ GrpOp ∣ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥)}
Colors of variables: wff setvar class
This definition is referenced by:  isablo  27528
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