Natural Number Multiplication Distributes over Addition/Proof 1

Theorem

$\forall x, y, z \in \N:$

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

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

$\ds \paren {x + y} \times z$

$=$

$\ds +^z \paren {x + y}$

$\ds $

$=$

$\ds \paren {+^z x} + \paren {+^z y}$

$\ds $

$=$

$\ds x \times z + y \times z$

$\Box$

$\ds z \times \paren {x + y}$

$=$

$\ds +^{x + y} z$

$\ds $

$=$

$\ds \paren {+^x z} + \paren {+^y z}$

$\ds $

$=$

$\ds z \times x + z \times y$

$\blacksquare$

1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers: Theorem $16.9$