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	$$⊢\mathrm{comLaw}=\left\{⟨o,m⟩|\forall x\in m\forall y\in mxoy=yox\right\}$$

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		ccomlaw	$$class\mathrm{comLaw}$$
1		vo	$$setvaro$$
2		vm	$$setvarm$$
3		vx	$$setvarx$$
4	2	cv	$$setvarm$$
5		vy	$$setvary$$
6	3	cv	$$setvarx$$
7	1	cv	$$setvaro$$
8	5	cv	$$setvary$$
9	6 8 7	co	$$class\left(xoy\right)$$
10	8 6 7	co	$$class\left(yox\right)$$
11	9 10	wceq	$$wffxoy=yox$$
12	11 5 4	wral	$$wff\forall y\in mxoy=yox$$
13	12 3 4	wral	$$wff\forall x\in m\forall y\in mxoy=yox$$
14	13 1 2	copab	$$class\left\{⟨o,m⟩|\forall x\in m\forall y\in mxoy=yox\right\}$$
15	0 14	wceq	$$wff\mathrm{comLaw}=\left\{⟨o,m⟩|\forall x\in m\forall y\in mxoy=yox\right\}$$