Definition df-asslaw 41824

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 >. | A. x e. m A. y e. m A. z e. m ( ( x o y ) o z ) = ( x o ( y o z ) ) }

Distinct variable group:   m, o, x, y, z

Detailed syntax breakdown of Definition df-asslaw

Step	Hyp	Ref	Expression
1		casslaw 41820	. 2 class assLaw
2		vx	. . . . . . . . . 10 setvar x
3	2	cv 1482	. . . . . . . . 9 class x
4		vy	. . . . . . . . . 10 setvar y
5	4	cv 1482	. . . . . . . . 9 class y
6		vo	. . . . . . . . . 10 setvar o
7	6	cv 1482	. . . . . . . . 9 class o
8	3, 5, 7	co 6650	. . . . . . . 8 class ( x o y )
9		vz	. . . . . . . . 9 setvar z
10	9	cv 1482	. . . . . . . 8 class z
11	8, 10, 7	co 6650	. . . . . . 7 class ( ( x o y ) o z )
12	5, 10, 7	co 6650	. . . . . . . 8 class ( y o z )
13	3, 12, 7	co 6650	. . . . . . 7 class ( x o ( y o z ) )
14	11, 13	wceq 1483	. . . . . 6 wff ( ( x o y ) o z ) = ( x o ( y o z ) )
15		vm	. . . . . . 7 setvar m
16	15	cv 1482	. . . . . 6 class m
17	14, 9, 16	wral 2912	. . . . 5 wff A. z e. m ( ( x o y ) o z ) = ( x o ( y o z ) )
18	17, 4, 16	wral 2912	. . . 4 wff A. y e. m A. z e. m ( ( x o y ) o z ) = ( x o ( y o z ) )
19	18, 2, 16	wral 2912	. . 3 wff A. x e. m A. y e. m A. z e. m ( ( x o y ) o z ) = ( x o ( y o z ) )
20	19, 6, 15	copab 4712	. 2 class { <. o , m >. | A. x e. m A. y e. m A. z e. m ( ( x o y ) o z ) = ( x o ( y o z ) ) }
21	1, 20	wceq 1483	1 wff assLaw = { <. o , m >. | A. x e. m A. y e. m A. z e. m ( ( x o y ) o z ) = ( x o ( y o z ) ) }

This definition is referenced by:  isasslaw  41828  asslawass  41829