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	$$⊢\phi \to A\in \mathrm{No}$$
	addsdid.2	$$⊢\phi \to B\in \mathrm{No}$$
	addsdid.3	$$⊢\phi \to C\in \mathrm{No}$$
Assertion	addsdird	Could not format assertion : No typesetting found for |- ( ph -> ( ( A +s B ) x.s C ) = ( ( A x.s C ) +s ( B 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	3 1 2	addsdid	Could not format ( ph -> ( C x.s ( A +s B ) ) = ( ( C x.s A ) +s ( C x.s B ) ) ) : No typesetting found for |- ( ph -> ( C x.s ( A +s B ) ) = ( ( C x.s A ) +s ( C x.s B ) ) ) with typecode |-
5	1 2	addscld	Could not format ( ph -> ( A +s B ) e. No ) : No typesetting found for |- ( ph -> ( A +s B ) e. No ) with typecode |-
6	5 3	mulscomd	Could not format ( ph -> ( ( A +s B ) x.s C ) = ( C x.s ( A +s B ) ) ) : No typesetting found for |- ( ph -> ( ( A +s B ) x.s C ) = ( C x.s ( A +s B ) ) ) with typecode |-
7	1 3	mulscomd	Could not format ( ph -> ( A x.s C ) = ( C x.s A ) ) : No typesetting found for |- ( ph -> ( A x.s C ) = ( C x.s A ) ) with typecode |-
8	2 3	mulscomd	Could not format ( ph -> ( B x.s C ) = ( C x.s B ) ) : No typesetting found for |- ( ph -> ( B x.s C ) = ( C x.s B ) ) with typecode |-
9	7 8	oveq12d	Could not format ( ph -> ( ( A x.s C ) +s ( B x.s C ) ) = ( ( C x.s A ) +s ( C x.s B ) ) ) : No typesetting found for |- ( ph -> ( ( A x.s C ) +s ( B x.s C ) ) = ( ( C x.s A ) +s ( C x.s B ) ) ) with typecode |-
10	4 6 9	3eqtr4d	Could not format ( ph -> ( ( A +s B ) x.s C ) = ( ( A x.s C ) +s ( B x.s C ) ) ) : No typesetting found for |- ( ph -> ( ( A +s B ) x.s C ) = ( ( A x.s C ) +s ( B x.s C ) ) ) with typecode |-