Composite of Isomorphisms is Isomorphism

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Theorem

Algebraic Structures

Let:

$\struct {S_1, \odot_1, \odot_2, \ldots, \odot_n}$
$\struct {S_2, *_1, *_2, \ldots, *_n}$
$\struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$

be algebraic structures.

Let:

$\phi: \struct {S_1, \odot_1, \odot_2, \ldots, \odot_n} \to \struct {S_2, *_1, *_2, \ldots, *_n}$
$\psi: \struct {S_2, *_1, *_2, \ldots, *_n} \to \struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$

be isomorphisms.

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


R-Algebraic Structures

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.