Theorem subsdird

Description: Distribution of surreal multiplication over subtraction. (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	subsdird	|- ( ph -> ( ( A -s B ) x.s C ) = ( ( A x.s C ) -s ( B 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	3 1 2	subsdid	|- ( ph -> ( C x.s ( A -s B ) ) = ( ( C x.s A ) -s ( C x.s B ) ) )
5	1 2	subscld	|- ( ph -> ( A -s B ) e. No )
6	5 3	mulscomd	|- ( ph -> ( ( A -s B ) x.s C ) = ( C x.s ( A -s B ) ) )
7	1 3	mulscomd	|- ( ph -> ( A x.s C ) = ( C x.s A ) )
8	2 3	mulscomd	|- ( ph -> ( B x.s C ) = ( C x.s B ) )
9	7 8	oveq12d	|- ( ph -> ( ( A x.s C ) -s ( B x.s C ) ) = ( ( C x.s A ) -s ( C x.s B ) ) )
10	4 6 9	3eqtr4d	|- ( ph -> ( ( A -s B ) x.s C ) = ( ( A x.s C ) -s ( B x.s C ) ) )