Image of Canonical Injection is Kernel of Projection

Theorem
Let $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$ be groups with identity elements $e_1$ and $e_2$ respectively.

Let $\struct {G_1 \times G_2, \circ}$ be the group direct product of $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$

Let:
 * $\pr_1: \struct {G_1 \times G_2, \circ} \to \struct {G_1, \circ_1}$ be the first projection from $\struct {G_1 \times G_2, \circ}$ to $\struct {G_1, \circ_1}$
 * $\pr_2: \struct {G_1 \times G_2, \circ} \to \struct {G_2, \circ_2}$ be the second projection from $\struct {G_1 \times G_2, \circ}$ to $\struct {G_2, \circ_2}$.

Let:
 * $\inj_1: \struct {G_1, \circ_1} \to \struct {G_1 \times G_2, \circ}$ be the canonical injection from $\struct {G_1, \circ_1}$ to $\struct {G_1 \times G_2, \circ}$


 * $\inj_2: \struct {G_2, \circ_2} \to \struct {G_1 \times G_2, \circ}$ be the canonical injection from $\struct {G_2, \circ_2}$ to $\struct {G_1 \times G_2, \circ}$.

Then:
 * $(1): \quad \Img {\inj_1} = \map \ker {\pr_2}$
 * $(2): \quad \Img {\inj_2} = \map \ker {\pr_1}$

That is:
 * the image of the (first) canonical injection is the kernel of the second projection
 * the image of the (second) canonical injection is the kernel of the first projection.

Proof
The canonical injection $\inj_1: \struct {G_1, \circ_1} \to \struct {G_1 \times G_2, \circ}$ is defined as:
 * $\forall x \in G_1: \map {\inj_1} x = \tuple {x, e_2}$

Thus:
 * $\Img {\inj_1} = \set {\tuple {x, e_2}: x \in G_1}$

The second projection $\pr_2: \struct {G_1 \times G_2, \circ} \to \struct {G_2, \circ_2}$ is defined as:
 * $\forall \tuple {x, y} \in G_1 \times G_2: \map {\pr_2} {x, y} = y$

Thus by definition of kernel:
 * $\map \ker {\pr_2} = \map {\pr_2^{-1} } {e_2} = \set {\tuple {x, e_2}: x \in G_1}$

As can be seen:
 * $\Img {\inj_1} = \map \ker {\pr_2}$

Similarly:


 * $\Img {\inj_2} = \set {\tuple {e_1, y}: y \in G_2}$
 * $\map \ker {\pr_1} = \map {\pr_1^{-1} } {e_1} = \set {\tuple {e_1, y}: y \in G_2}$

Hence the result.