Theorem nadddid

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

	Ref	Expression
Hypotheses	nadddid.1	$$⊢\phi \to A\in \mathrm{On}$$
	nadddid.2	$$⊢\phi \to B\in \mathrm{On}$$
	nadddid.3	$$⊢\phi \to C\in \mathrm{On}$$
Assertion	nadddid	Could not format assertion : No typesetting found for |- ( ph -> ( A .no ( B +no C ) ) = ( ( A .no B ) +no ( A .no C ) ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		nadddid.1	$$⊢\phi \to A\in \mathrm{On}$$
2		nadddid.2	$$⊢\phi \to B\in \mathrm{On}$$
3		nadddid.3	$$⊢\phi \to C\in \mathrm{On}$$
4		nadddi	Could not format ( ( A e. On /\ B e. On /\ C e. On ) -> ( A .no ( B +no C ) ) = ( ( A .no B ) +no ( A .no C ) ) ) : No typesetting found for |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( A .no ( B +no C ) ) = ( ( A .no B ) +no ( A .no C ) ) ) with typecode |-
5	1 2 3 4	syl3anc	Could not format ( ph -> ( A .no ( B +no C ) ) = ( ( A .no B ) +no ( A .no C ) ) ) : No typesetting found for |- ( ph -> ( A .no ( B +no C ) ) = ( ( A .no B ) +no ( A .no C ) ) ) with typecode |-