Natural number minimum


THEOREM   

Zero is a left zero element of natural number minimum.

$$\forall x \in \mathbb{N} : \min(0, x) = 0$$
Proof.    
THEOREM   

Zero is a right zero element of natural number minimum.

$$\forall x \in \mathbb{N} : \min(x, 0) = 0$$
Proof.    
THEOREM   

Natural number minimum is idempotent.

$$\forall x \in \mathbb{N} : \min(x, x) = x$$
Proof.    
THEOREM   

Natural number multiplication is left-distributive over minimum.

$$\forall x,y,z \in \mathbb{N} : x\min(y, z) = \min(xy, xz)$$
Proof.    
THEOREM   

Natural number multiplication is right-distributive over minimum.

$$\forall x,y,z \in \mathbb{N} : \min(y, z)x = \min(yx, zx)$$
Proof.    

Proper supersets