Composite of Isomorphisms is Isomorphism/R-Algebraic Structure

Theorem

Let:

$\struct {S_1, \ast_1}_R$

$\struct {S_2, \ast_2}_R$

$\struct {S_3, \ast_3}_R$

be $R$-algebraic structures with the same number of operations.

Let:

$\phi: \struct {S_1, \ast_1}_R \to \struct {S_2, \ast_2}_R$

$\psi: \struct {S_2, \ast_2}_R \to \struct {S_3, \ast_3}_R$

be isomorphisms.

Then the composite of $\phi$ and $\psi$ is also an isomorphism.

Proof

If $\phi$ and $\psi$ are both isomorphisms, then they are by definition:

homomorphisms

bijections.

So:

From Composite of Homomorphisms for R-Algebraic Structures is Homomorphism we have that $\phi \circ \psi$ and $\psi \circ \phi$ are both homomorphisms

From Composite of Bijections is Bijection we have that $\phi \circ \psi$ and $\psi \circ \phi$ are both bijections;

and hence by definition also isomorphisms.

$\blacksquare$