Theorem subsdid

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	subsdid	|- ( 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	2 3	subscld	|- ( ph -> ( B -s C ) e. No )
5	1 3 4	addsdid	|- ( ph -> ( A x.s ( C +s ( B -s C ) ) ) = ( ( A x.s C ) +s ( A x.s ( B -s C ) ) ) )
6		pncan3s	|- ( ( C e. No /\ B e. No ) -> ( C +s ( B -s C ) ) = B )
7	3 2 6	syl2anc	|- ( ph -> ( C +s ( B -s C ) ) = B )
8	7	oveq2d	|- ( ph -> ( A x.s ( C +s ( B -s C ) ) ) = ( A x.s B ) )
9	5 8	eqtr3d	|- ( ph -> ( ( A x.s C ) +s ( A x.s ( B -s C ) ) ) = ( A x.s B ) )
10	1 2	mulscld	|- ( ph -> ( A x.s B ) e. No )
11	1 3	mulscld	|- ( ph -> ( A x.s C ) e. No )
12	1 4	mulscld	|- ( ph -> ( A x.s ( B -s C ) ) e. No )
13	10 11 12	subaddsd	|- ( ph -> ( ( ( A x.s B ) -s ( A x.s C ) ) = ( A x.s ( B -s C ) ) <-> ( ( A x.s C ) +s ( A x.s ( B -s C ) ) ) = ( A x.s B ) ) )
14	9 13	mpbird	|- ( ph -> ( ( A x.s B ) -s ( A x.s C ) ) = ( A x.s ( B -s C ) ) )
15	14	eqcomd	|- ( ph -> ( A x.s ( B -s C ) ) = ( ( A x.s B ) -s ( A x.s C ) ) )