Integer multiplication


Definition

THEOREM   

Integer multiplication is well-defined.

Proof available

THEOREM   

Integer multiplication is closed.

Proof available

THEOREM   

Integer multiplication is commutative.

$$\forall x,y \in \mathbb{Z} : xy = yx$$

Proof available

THEOREM   

Integer multiplication is associative.

$$\forall x,y,z \in \mathbb{Z} : xyz = x(yz)$$

Proof available

THEOREM    \(\forall a,b,c \in \mathbb{Z} : ac = bc \Leftrightarrow c = 0 \vee a = b\)

Proof unavailable

THEOREM    \(\forall a,b,c \in \mathbb{Z} : ca = cb \Leftrightarrow c = 0 \vee a = b\)

Proof unavailable

THEOREM   

Integer multiplication is left-distributive over addition.

$$\forall x,y,z \in \mathbb{Z} : x(y + z) = xy + xz$$

Proof available

THEOREM   

Integer multiplication is right-distributive over addition.

$$\forall x,y,z \in \mathbb{Z} : (y + z)x = yx + zx$$

Proof available

Proper supersets