Definition df-comlaw 41823

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

Distinct variable group:   m, o, x, y

Detailed syntax breakdown of Definition df-comlaw

Step	Hyp	Ref	Expression
1		ccomlaw 41821	. 2 class comLaw
2		vx	. . . . . . . 8 setvar x
3	2	cv 1482	. . . . . . 7 class x
4		vy	. . . . . . . 8 setvar y
5	4	cv 1482	. . . . . . 7 class y
6		vo	. . . . . . . 8 setvar o
7	6	cv 1482	. . . . . . 7 class o
8	3, 5, 7	co 6650	. . . . . 6 class ( x o y )
9	5, 3, 7	co 6650	. . . . . 6 class ( y o x )
10	8, 9	wceq 1483	. . . . 5 wff ( x o y ) = ( y o x )
11		vm	. . . . . 6 setvar m
12	11	cv 1482	. . . . 5 class m
13	10, 4, 12	wral 2912	. . . 4 wff A. y e. m ( x o y ) = ( y o x )
14	13, 2, 12	wral 2912	. . 3 wff A. x e. m A. y e. m ( x o y ) = ( y o x )
15	14, 6, 11	copab 4712	. 2 class { <. o , m >. | A. x e. m A. y e. m ( x o y ) = ( y o x ) }
16	1, 15	wceq 1483	1 wff comLaw = { <. o , m >. | A. x e. m A. y e. m ( x o y ) = ( y o x ) }

This definition is referenced by:  iscomlaw  41826