# Left identity element

## Definition

THEOREM

More than one left identity element implies no right identity element.

Proof available

LEMMA

Zero is a left identity element of rational number addition.

$$\forall x \in \mathbb{Q} : 0 + x = x$$

Proof available

THEOREM

True is a left identity element of conjunction.

$$\top \wedge P \dashv\vdash P$$

Proof available

LEMMA

One is a left identity element of natural number multiplication.

$$\forall x \in \mathbb{N} : 1 * x = x$$

Proof available

THEOREM

False is a left identity element of disjunction.

$$\bot \vee P \dashv\vdash P$$

Proof available

LEMMA

Zero is a left identity element of natural number addition.

$$\forall x \in \mathbb{N} : 0 + x = x$$

Proof available

LEMMA

Zero is a left identity element of integer addition.

$$\forall x \in \mathbb{Z} : 0 + x = x$$

Proof available