Natural Number Multiplication is Associative/Proof 1

Theorem

The operation of multiplication on the set of natural numbers $\N$ is associative:

$\forall x, y, z \in \N: \paren {x \times y} \times z = x \times \paren {y \times z}$

Proof

From Index Laws for Semigroup: Product of Indices we have:

$+^{z \times y} x = \map {+^z} {+^y x}$

By definition of multiplication, this amounts to:

$x \times \paren {z \times y} = \paren {x \times y} \times z$

From Natural Number Multiplication is Commutative, we have:

$x \times \paren {z \times y} = x \times \paren {y \times z}$

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers: Theorem $16.12$