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	Could not format assertion : No typesetting found for |- ( ( ( 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	Could not format 0s e. No : No typesetting found for |- 0s e. No with typecode |-
2	1 1	pm3.2i	Could not format ( 0s e. No /\ 0s e. No ) : No typesetting found for |- ( 0s e. No /\ 0s e. No ) with typecode |-
3		mulsprop	Could not format ( ( ( 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 ) ) ( ( 0s x.s 0s ) e. No /\ ( ( A ( ( A x.s D ) -s ( A x.s C ) )
4	2 3	mp3an1	Could not format ( ( ( 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 ) ) ( ( 0s x.s 0s ) e. No /\ ( ( A ( ( A x.s D ) -s ( A x.s C ) )
5	4	simprd	Could not format ( ( ( A e. No /\ B e. No ) /\ ( C e. No /\ D e. No ) ) -> ( ( A ( ( A x.s D ) -s ( A x.s C ) ) ( ( A ( ( A x.s D ) -s ( A x.s C ) )