Basic proofs / axioms

Reflexive property: A quantity is equal to itself (e.g., $a = a$).

Symmetric property: If a quantity is equal to another, you can swap them (e.g., if $a = b$, then $b = a$).

Transitive property: If one quantity equals a second, and the second equals a third, then the first equals the third (e.g., if $a = b$ and $b = c$, then $a = c$).

Addition property of equality: You may add the same number to both sides of an equation and it will remain true (e.g., if $a = b$, then $a + c = b + c$).

Subtraction property of equality: You may subtract the same number from both sides of an equation and it will remain true (e.g., if $a = b$, then $a - c = b - c$).

Multiplication property of equality: You may multiply the same number on both sides of an equation and it will remain true (e.g., if $a = b$, then $a \times c = b \times c$).

Division property of equality: You may divide the same number on both sides of an equation and it will remain true (e.g., if $a = b$ and $c \neq 0$, then $\frac{a}{c} = \frac{b}{c}$).

Distributive property: Multiplication can be distributed across addition or subtraction (e.g., $a(b + c) = ab + ac$).

Commutative property: You may change the order of numbers without changing the result (e.g., $a + b = b + a$ or $a \times b = b \times a$).

Associative property: You may change the grouping of numbers when adding or multiplying without changing the result (e.g., $(a + b) + c = a + (b + c)$).

Identity property: Adding zero or multiplying by one leaves the number unchanged (e.g., $a + 0 = a$ or $a \times 1 = a$).

Inverse property: A number combined with its opposite is $0$ for addition; for multiplication, it is the number that multiplies it to one (e.g., $a + (-a) = 0$ or $x \times \frac{1}{x} = 1$).