Theorem sltmuld

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

Proof

Step	Hyp	Ref	Expression
1		sltmuld.1	|- ( ph -> A e. No )
2		sltmuld.2	|- ( ph -> B e. No )
3		sltmuld.3	|- ( ph -> C e. No )
4		sltmuld.4	|- ( ph -> D e. No )
5		sltmuld.5	|- ( ph -> A
6		sltmuld.6	|- ( ph -> C
7		sltmul	|- ( ( ( 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 ) )