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	$$⊢\phi \to A\in \mathrm{No}$$
	addsdid.2	$$⊢\phi \to B\in \mathrm{No}$$
	addsdid.3	$$⊢\phi \to C\in \mathrm{No}$$
Assertion	addsdid	Could not format assertion : No typesetting found for |- ( ph -> ( A x.s ( B +s C ) ) = ( ( A x.s B ) +s ( A x.s C ) ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		addsdid.1	$$⊢\phi \to A\in \mathrm{No}$$
2		addsdid.2	$$⊢\phi \to B\in \mathrm{No}$$
3		addsdid.3	$$⊢\phi \to C\in \mathrm{No}$$
4		addsdi	Could not format ( ( A e. No /\ B e. No /\ C e. No ) -> ( A x.s ( B +s C ) ) = ( ( A x.s B ) +s ( A x.s C ) ) ) : No typesetting found for |- ( ( A e. No /\ B e. No /\ C e. No ) -> ( A x.s ( B +s C ) ) = ( ( A x.s B ) +s ( A x.s C ) ) ) with typecode |-
5	1 2 3 4	syl3anc	Could not format ( ph -> ( A x.s ( B +s C ) ) = ( ( A x.s B ) +s ( A x.s C ) ) ) : No typesetting found for |- ( ph -> ( A x.s ( B +s C ) ) = ( ( A x.s B ) +s ( A x.s C ) ) ) with typecode |-