Theorem ltmulsd

Description: An ordering relationship for surreal multiplication. Compare theorem 8(iii) of Conway p. 19. (Contributed by Scott Fenton, 6-Mar-2025)

	Ref	Expression
Hypotheses	ltmulsd.1	|- ( ph -> A e. No )
	ltmulsd.2	|- ( ph -> B e. No )
	ltmulsd.3	|- ( ph -> C e. No )
	ltmulsd.4	|- ( ph -> D e. No )
	ltmulsd.5	|- ( ph -> A
	ltmulsd.6	|- ( ph -> C
Assertion	ltmulsd	|- ( ph -> ( ( A x.s D ) -s ( A x.s C ) )

Proof

Step	Hyp	Ref	Expression
1		ltmulsd.1	|- ( ph -> A e. No )
2		ltmulsd.2	|- ( ph -> B e. No )
3		ltmulsd.3	|- ( ph -> C e. No )
4		ltmulsd.4	|- ( ph -> D e. No )
5		ltmulsd.5	|- ( ph -> A
6		ltmulsd.6	|- ( ph -> C
7		ltmuls	|- ( ( ( A e. No /\ B e. No ) /\ ( C e. No /\ D e. No ) ) -> ( ( A ( ( A x.s D ) -s ( A x.s C ) )
8	1 2 3 4 7	syl22anc	|- ( ph -> ( ( A ( ( A x.s D ) -s ( A x.s C ) )
9	5 6 8	mp2and	|- ( ph -> ( ( A x.s D ) -s ( A x.s C ) )