Definition df-comlaw

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)

	Ref	Expression
Assertion	df-comlaw	|- comLaw = { <. o , m >. | A. x e. m A. y e. m ( x o y ) = ( y o x ) }

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		ccomlaw	|- comLaw
1		vo	|- o
2		vm	|- m
3		vx	|- x
4	2	cv	|- m
5		vy	|- y
6	3	cv	|- x
7	1	cv	|- o
8	5	cv	|- y
9	6 8 7	co	|- ( x o y )
10	8 6 7	co	|- ( y o x )
11	9 10	wceq	|- ( x o y ) = ( y o x )
12	11 5 4	wral	|- A. y e. m ( x o y ) = ( y o x )
13	12 3 4	wral	|- A. x e. m A. y e. m ( x o y ) = ( y o x )
14	13 1 2	copab	|- { <. o , m >. | A. x e. m A. y e. m ( x o y ) = ( y o x ) }
15	0 14	wceq	|- comLaw = { <. o , m >. | A. x e. m A. y e. m ( x o y ) = ( y o x ) }