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Definition df-asslaw 41606
Description: The associative law for binary operations, see definitions of laws A1. and M1. in section 1.1 of [Hall] p. 1, or definition 5 in [BourbakiAlg1] p. 4: the value of a binary operation applied the value of the binary operation applied to two operands and a third operand equals the value of the binary operation applied to the first operand and the value of the binary operation applied to the second and third operand. By this definition, the associative law is expressed as binary relation: a binary operation is related to a set by assLaw if the associative 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 FL, 1-Nov-2009.) (Revised by AV, 13-Jan-2020.)
Assertion
Ref Expression
df-asslaw assLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚𝑧𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
Distinct variable group:   𝑚,𝑜,𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-asslaw
StepHypRef Expression
1 casslaw 41602 . 2 class assLaw
2 vx . . . . . . . . . 10 setvar 𝑥
32cv 1474 . . . . . . . . 9 class 𝑥
4 vy . . . . . . . . . 10 setvar 𝑦
54cv 1474 . . . . . . . . 9 class 𝑦
6 vo . . . . . . . . . 10 setvar 𝑜
76cv 1474 . . . . . . . . 9 class 𝑜
83, 5, 7co 6527 . . . . . . . 8 class (𝑥𝑜𝑦)
9 vz . . . . . . . . 9 setvar 𝑧
109cv 1474 . . . . . . . 8 class 𝑧
118, 10, 7co 6527 . . . . . . 7 class ((𝑥𝑜𝑦)𝑜𝑧)
125, 10, 7co 6527 . . . . . . . 8 class (𝑦𝑜𝑧)
133, 12, 7co 6527 . . . . . . 7 class (𝑥𝑜(𝑦𝑜𝑧))
1411, 13wceq 1475 . . . . . 6 wff ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))
15 vm . . . . . . 7 setvar 𝑚
1615cv 1474 . . . . . 6 class 𝑚
1714, 9, 16wral 2896 . . . . 5 wff 𝑧𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))
1817, 4, 16wral 2896 . . . 4 wff 𝑦𝑚𝑧𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))
1918, 2, 16wral 2896 . . 3 wff 𝑥𝑚𝑦𝑚𝑧𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))
2019, 6, 15copab 4637 . 2 class {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚𝑧𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
211, 20wceq 1475 1 wff assLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚𝑧𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
Colors of variables: wff setvar class
This definition is referenced by:  isasslaw  41610  asslawass  41611
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