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	|- ( ph -> A e. No )
	addsdid.2	|- ( ph -> B e. No )
	addsdid.3	|- ( ph -> C e. No )
Assertion	addsdid	|- ( ph -> ( A x.s ( B +s C ) ) = ( ( A x.s B ) +s ( A x.s C ) ) )

Proof

Step	Hyp	Ref	Expression
1		addsdid.1	|- ( ph -> A e. No )
2		addsdid.2	|- ( ph -> B e. No )
3		addsdid.3	|- ( ph -> C e. No )
4		addsdi	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( A x.s ( B +s C ) ) = ( ( A x.s B ) +s ( A x.s C ) ) )
5	1 2 3 4	syl3anc	|- ( ph -> ( A x.s ( B +s C ) ) = ( ( A x.s B ) +s ( A x.s C ) ) )