Canonical Injection is Injection

Theorem
Let $\struct {S_1, \circ_1}$ and $\struct {S_2, \circ_2}$ be algebraic structures with identities $e_1, e_2$ respectively.

The canonical injections:


 * $\inj_1: \struct {S_1, \circ_1} \to \struct {S_1, \circ_1} \times \struct {S_2, \circ_2}: \forall x \in S_1: \map {\inj_1} x = \tuple {x, e_2}$


 * $\inj_2: \struct {S_2, \circ_2} \to \struct {S_1, \circ_1} \times \struct {S_2, \circ_2}: \forall x \in S_2: \map {\inj_2} x = \tuple {e_1, x}$

are injections.

Proof
Let $x, x' \in S_1$.

Suppose that:
 * $\map {\inj_1} x = \map {\inj_1} {x'}$

Then by definition of canonical injection:
 * $\tuple {x, e_2} = \tuple {x', e_2}$

By Equality of Ordered Pairs:
 * $x = x'$

That is, $\inj_1$ is an injection.

Similarly, let $x, x' \in S_2$.

Suppose that:
 * $\map {\inj_2} x = \map {\inj_2} {x'}$

Then by definition of canonical injection:
 * $\tuple {e_1, x} = \tuple {e_1, x'}$

Again by Equality of Ordered Pairs:
 * $x = x'$

That is, $\inj_2$ is an injection.

So $\inj_1$ and $\inj_2$ are injections.