Axioms.

Law 1 is "a+d=d+a". The name of Law 1 is "commutative axiom of addition".
Law 2 is "a*b=b*a". This is the commutative axiom of multiplication.
Law 3 is "a+0=a". This is the additive identity axiom.
Law 4 is "a*1=a". This is the multiplicative identity axiom.
Law 5 is "a+(-a)=0". This is the additive inverse axiom.
Law 6 is "a*(1)over(a)=1". This is the multiplicative inverse axiom.
Law 7 is "(a+b)+c=a+(b+c)". This is the associative axiom for addition.
Law 8 is "(a*b)*c=a*(b*c)". This is the associative axiom for multiplication.
Law 9 is "(a+b)*c=a*c+b*c". This is the distributive axiom of multiplication over addition.
Law 10 is "a=a".

Definitions

Law 11 is "a+b+c=(a+b)+c". This the order of operations.
Law 12 is "a*b*c=(a*b)*c". This the order of operations.
Law 13 is "a-b=a+(-b)". This is the definition of subtraction.

Properties

Law 14 is "0+a=a".
Law 15 is "-(-a)=a".
Law 16 is "a-(-b)=a+b".

Axioms for Equations

Law 17 is "if a=b then b=a".
Law 18 is "if a=b then a+c=b+c".
Law 19 is "if a=b then a-c=b-c".
Law 20 is "if a=b then a*c=b*c".
Law 21 is "if a=b then a/c=b/c".
Law 22 is "if a=b then (a)=(b)".
Law 23 is "(a+b)+c=a+b+c". This is the associative axiom for addition.
Law 24 is "(a*b)*c=a*b*c". This is the associative axiom for multiplication.