Theorem addsdird

Description: Distributive law for surreal numbers. 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	addsdird	⊢ ( 𝜑 → ( ( 𝐴 +s 𝐵 ) ·s 𝐶 ) = ( ( 𝐴 ·s 𝐶 ) +s ( 𝐵 ·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	3 1 2	addsdid	⊢ ( 𝜑 → ( 𝐶 ·s ( 𝐴 +s 𝐵 ) ) = ( ( 𝐶 ·s 𝐴 ) +s ( 𝐶 ·s 𝐵 ) ) )
5	1 2	addscld	⊢ ( 𝜑 → ( 𝐴 +s 𝐵 ) ∈ No )
6	5 3	mulscomd	⊢ ( 𝜑 → ( ( 𝐴 +s 𝐵 ) ·s 𝐶 ) = ( 𝐶 ·s ( 𝐴 +s 𝐵 ) ) )
7	1 3	mulscomd	⊢ ( 𝜑 → ( 𝐴 ·s 𝐶 ) = ( 𝐶 ·s 𝐴 ) )
8	2 3	mulscomd	⊢ ( 𝜑 → ( 𝐵 ·s 𝐶 ) = ( 𝐶 ·s 𝐵 ) )
9	7 8	oveq12d	⊢ ( 𝜑 → ( ( 𝐴 ·s 𝐶 ) +s ( 𝐵 ·s 𝐶 ) ) = ( ( 𝐶 ·s 𝐴 ) +s ( 𝐶 ·s 𝐵 ) ) )
10	4 6 9	3eqtr4d	⊢ ( 𝜑 → ( ( 𝐴 +s 𝐵 ) ·s 𝐶 ) = ( ( 𝐴 ·s 𝐶 ) +s ( 𝐵 ·s 𝐶 ) ) )