Isomorphism Preserves Commutativity/Proof 1

Proof
Let $\struct {S, \circ}$ be an algebraic structure in which $\circ$ is commutative.

Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be an isomorphism.

As an isomorphism is surjective, it follows that:


 * $\forall u, v \in T: \exists x, y \in S: \map \phi x = u, \map \phi y = v$

So:

As $\phi$ is an isomorphism, it follows from Inverse of Algebraic Structure Isomorphism is Isomorphism that $\phi^{-1}$ is also a isomorphism.

Thus the result for $\phi$ can be applied to $\phi^{-1}$.