Theorem sltmulneg1d

Description: Multiplication of both sides of surreal less-than by a negative number. (Contributed by Scott Fenton, 14-Mar-2025)

	Ref	Expression
Hypotheses	sltmulneg.1	$$⊢\phi \to A\in \mathrm{No}$$
	sltmulneg.2	$$⊢\phi \to B\in \mathrm{No}$$
	sltmulneg.3	$$⊢\phi \to C\in \mathrm{No}$$
	sltmulneg.4	No typesetting found for |- ( ph -> C
Assertion	sltmulneg1d	Could not format assertion : No typesetting found for |- ( ph -> ( A ( B x.s C )

Proof

Step	Hyp	Ref	Expression
1		sltmulneg.1	$$⊢\phi \to A\in \mathrm{No}$$
2		sltmulneg.2	$$⊢\phi \to B\in \mathrm{No}$$
3		sltmulneg.3	$$⊢\phi \to C\in \mathrm{No}$$
4		sltmulneg.4	Could not format ( ph -> C C
5	1 3	mulnegs2d	Could not format ( ph -> ( A x.s ( -us ` C ) ) = ( -us ` ( A x.s C ) ) ) : No typesetting found for |- ( ph -> ( A x.s ( -us ` C ) ) = ( -us ` ( A x.s C ) ) ) with typecode |-
6	2 3	mulnegs2d	Could not format ( ph -> ( B x.s ( -us ` C ) ) = ( -us ` ( B x.s C ) ) ) : No typesetting found for |- ( ph -> ( B x.s ( -us ` C ) ) = ( -us ` ( B x.s C ) ) ) with typecode |-
7	5 6	breq12d	Could not format ( ph -> ( ( A x.s ( -us ` C ) ) ( -us ` ( A x.s C ) ) ( ( A x.s ( -us ` C ) ) ( -us ` ( A x.s C ) )
8	3	negscld	Could not format ( ph -> ( -us ` C ) e. No ) : No typesetting found for |- ( ph -> ( -us ` C ) e. No ) with typecode |-
9		negs0s	Could not format ( -us ` 0s ) = 0s : No typesetting found for |- ( -us ` 0s ) = 0s with typecode |-
10		0sno	Could not format 0s e. No : No typesetting found for |- 0s e. No with typecode |-
11	10	a1i	Could not format ( ph -> 0s e. No ) : No typesetting found for |- ( ph -> 0s e. No ) with typecode |-
12	3 11	sltnegd	Could not format ( ph -> ( C ( -us ` 0s ) ( C ( -us ` 0s )
13	4 12	mpbid	Could not format ( ph -> ( -us ` 0s ) ( -us ` 0s )
14	9 13	eqbrtrrid	Could not format ( ph -> 0s 0s
15	1 2 8 14	sltmul1d	Could not format ( ph -> ( A ( A x.s ( -us ` C ) ) ( A ( A x.s ( -us ` C ) )
16	2 3	mulscld	Could not format ( ph -> ( B x.s C ) e. No ) : No typesetting found for |- ( ph -> ( B x.s C ) e. No ) with typecode |-
17	1 3	mulscld	Could not format ( ph -> ( A x.s C ) e. No ) : No typesetting found for |- ( ph -> ( A x.s C ) e. No ) with typecode |-
18	16 17	sltnegd	Could not format ( ph -> ( ( B x.s C ) ( -us ` ( A x.s C ) ) ( ( B x.s C ) ( -us ` ( A x.s C ) )
19	7 15 18	3bitr4d	Could not format ( ph -> ( A ( B x.s C ) ( A ( B x.s C )