Theorem nadddid

Description: Natural multiplication distributes over natural addition. Deduction form. (Contributed by Scott Fenton, 3-Aug-2026)

	Ref	Expression
Hypotheses	nadddid.1	|- ( ph -> A e. On )
	nadddid.2	|- ( ph -> B e. On )
	nadddid.3	|- ( ph -> C e. On )
Assertion	nadddid	|- ( ph -> ( A .no ( B +no C ) ) = ( ( A .no B ) +no ( A .no C ) ) )

Proof

Step	Hyp	Ref	Expression
1		nadddid.1	|- ( ph -> A e. On )
2		nadddid.2	|- ( ph -> B e. On )
3		nadddid.3	|- ( ph -> C e. On )
4		nadddi	|- ( ( A e. On /\ B e. On /\ C e. On ) -> ( A .no ( B +no C ) ) = ( ( A .no B ) +no ( A .no C ) ) )
5	1 2 3 4	syl3anc	|- ( ph -> ( A .no ( B +no C ) ) = ( ( A .no B ) +no ( A .no C ) ) )