Theorem subsdird

Description: Distribution of surreal multiplication over subtraction. (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	subsdird	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	subsdid	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	subscld	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 |-