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	$$⊢\phi \to A\in \mathrm{No}$$
	sltmuld.2	$$⊢\phi \to B\in \mathrm{No}$$
	sltmuld.3	$$⊢\phi \to C\in \mathrm{No}$$
	sltmuld.4	$$⊢\phi \to D\in \mathrm{No}$$
	sltmuld.5	$$⊢\phi \to A{<}_{s}B$$
	sltmuld.6	$$⊢\phi \to C{<}_{s}D$$
Assertion	sltmuld	Could not format assertion : No typesetting found for |- ( ph -> ( ( A x.s D ) -s ( A x.s C ) )

Proof

Step	Hyp	Ref	Expression
1		sltmuld.1	$$⊢\phi \to A\in \mathrm{No}$$
2		sltmuld.2	$$⊢\phi \to B\in \mathrm{No}$$
3		sltmuld.3	$$⊢\phi \to C\in \mathrm{No}$$
4		sltmuld.4	$$⊢\phi \to D\in \mathrm{No}$$
5		sltmuld.5	$$⊢\phi \to A{<}_{s}B$$
6		sltmuld.6	$$⊢\phi \to C{<}_{s}D$$
7		sltmul	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 ) )
8	1 2 3 4 7	syl22anc	Could not format ( ph -> ( ( A ( ( A x.s D ) -s ( A x.s C ) ) ( ( A ( ( A x.s D ) -s ( A x.s C ) )
9	5 6 8	mp2and	Could not format ( ph -> ( ( A x.s D ) -s ( A x.s C ) ) ( ( A x.s D ) -s ( A x.s C ) )