Theorem nadddird

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	nadddird	|- ( ph -> ( ( A +no B ) .no C ) = ( ( A .no C ) +no ( B .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	3 1 2	nadddid	|- ( ph -> ( C .no ( A +no B ) ) = ( ( C .no A ) +no ( C .no B ) ) )
5	1 2	naddcld	|- ( ph -> ( A +no B ) e. On )
6	5 3	nmulcomd	|- ( ph -> ( ( A +no B ) .no C ) = ( C .no ( A +no B ) ) )
7	1 3	nmulcomd	|- ( ph -> ( A .no C ) = ( C .no A ) )
8	2 3	nmulcomd	|- ( ph -> ( B .no C ) = ( C .no B ) )
9	7 8	oveq12d	|- ( ph -> ( ( A .no C ) +no ( B .no C ) ) = ( ( C .no A ) +no ( C .no B ) ) )
10	4 6 9	3eqtr4d	|- ( ph -> ( ( A +no B ) .no C ) = ( ( A .no C ) +no ( B .no C ) ) )