Theorem nadddid

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

	Ref	Expression
Hypotheses	nadddid.1	⊢ ( 𝜑 → 𝐴 ∈ On )
	nadddid.2	⊢ ( 𝜑 → 𝐵 ∈ On )
	nadddid.3	⊢ ( 𝜑 → 𝐶 ∈ On )
Assertion	nadddid	⊢ ( 𝜑 → ( 𝐴 ·no ( 𝐵 +no 𝐶 ) ) = ( ( 𝐴 ·no 𝐵 ) +no ( 𝐴 ·no 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		nadddid.1	⊢ ( 𝜑 → 𝐴 ∈ On )
2		nadddid.2	⊢ ( 𝜑 → 𝐵 ∈ On )
3		nadddid.3	⊢ ( 𝜑 → 𝐶 ∈ On )
4		nadddi	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ·no ( 𝐵 +no 𝐶 ) ) = ( ( 𝐴 ·no 𝐵 ) +no ( 𝐴 ·no 𝐶 ) ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( 𝐴 ·no ( 𝐵 +no 𝐶 ) ) = ( ( 𝐴 ·no 𝐵 ) +no ( 𝐴 ·no 𝐶 ) ) )