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Definition df-comlaw 44842
 Description: The commutative law for binary operations, see definitions of laws A2. and M2. in section 1.1 of [Hall] p. 1, or definition 8 in [BourbakiAlg1] p. 7: the value of a binary operation applied to two operands equals the value of a binary operation applied to the two operands in reversed order. By this definition, the commutative law is expressed as binary relation: a binary operation is related to a set by comLaw if the commutative law holds for this binary operation regarding this set. Note that the binary operation needs neither to be closed nor to be a function. (Contributed by AV, 7-Jan-2020.)
Assertion
Ref Expression
df-comlaw comLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥)}
Distinct variable group:   𝑚,𝑜,𝑥,𝑦

Detailed syntax breakdown of Definition df-comlaw
StepHypRef Expression
1 ccomlaw 44840 . 2 class comLaw
2 vx . . . . . . . 8 setvar 𝑥
32cv 1537 . . . . . . 7 class 𝑥
4 vy . . . . . . . 8 setvar 𝑦
54cv 1537 . . . . . . 7 class 𝑦
6 vo . . . . . . . 8 setvar 𝑜
76cv 1537 . . . . . . 7 class 𝑜
83, 5, 7co 7155 . . . . . 6 class (𝑥𝑜𝑦)
95, 3, 7co 7155 . . . . . 6 class (𝑦𝑜𝑥)
108, 9wceq 1538 . . . . 5 wff (𝑥𝑜𝑦) = (𝑦𝑜𝑥)
11 vm . . . . . 6 setvar 𝑚
1211cv 1537 . . . . 5 class 𝑚
1310, 4, 12wral 3070 . . . 4 wff 𝑦𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥)
1413, 2, 12wral 3070 . . 3 wff 𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥)
1514, 6, 11copab 5097 . 2 class {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥)}
161, 15wceq 1538 1 wff comLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥)}
 Colors of variables: wff setvar class This definition is referenced by:  iscomlaw  44845
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