Theorem sltmul

Description: An ordering relationship for surreal multiplication. Compare theorem 8(iii) of Conway p. 19. (Contributed by Scott Fenton, 5-Mar-2025)

	Ref	Expression
Assertion	sltmul	|- ( ( ( A e. No /\ B e. No ) /\ ( C e. No /\ D e. No ) ) -> ( ( A ( ( A x.s D ) -s ( A x.s C ) )

Proof

Step	Hyp	Ref	Expression
1		0sno	|- 0s e. No
2	1 1	pm3.2i	|- ( 0s e. No /\ 0s e. No )
3		mulsprop	|- ( ( ( 0s e. No /\ 0s e. No ) /\ ( A e. No /\ B e. No ) /\ ( C e. No /\ D e. No ) ) -> ( ( 0s x.s 0s ) e. No /\ ( ( A ( ( A x.s D ) -s ( A x.s C ) )
4	2 3	mp3an1	|- ( ( ( A e. No /\ B e. No ) /\ ( C e. No /\ D e. No ) ) -> ( ( 0s x.s 0s ) e. No /\ ( ( A ( ( A x.s D ) -s ( A x.s C ) )
5	4	simprd	|- ( ( ( A e. No /\ B e. No ) /\ ( C e. No /\ D e. No ) ) -> ( ( A ( ( A x.s D ) -s ( A x.s C ) )