Theorem addsdid

Description: Distributive law for surreal numbers. Commuted form of part of theorem 7 of Conway p. 19. (Contributed by Scott Fenton, 9-Mar-2025)

	Ref	Expression
Hypotheses	addsdid.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	addsdid.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	addsdid.3	⊢ ( 𝜑 → 𝐶 ∈ No )
Assertion	addsdid	⊢ ( 𝜑 → ( 𝐴 ·s ( 𝐵 +s 𝐶 ) ) = ( ( 𝐴 ·s 𝐵 ) +s ( 𝐴 ·s 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		addsdid.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		addsdid.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		addsdid.3	⊢ ( 𝜑 → 𝐶 ∈ No )
4		addsdi	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 ·s ( 𝐵 +s 𝐶 ) ) = ( ( 𝐴 ·s 𝐵 ) +s ( 𝐴 ·s 𝐶 ) ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( 𝐴 ·s ( 𝐵 +s 𝐶 ) ) = ( ( 𝐴 ·s 𝐵 ) +s ( 𝐴 ·s 𝐶 ) ) )