Theorem ltmulnegs2d

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

	Ref	Expression
Hypotheses	ltmulnegs.1	$$⊢\phi \to A\in \mathrm{No}$$
	ltmulnegs.2	$$⊢\phi \to B\in \mathrm{No}$$
	ltmulnegs.3	$$⊢\phi \to C\in \mathrm{No}$$
	ltmulnegs.4	$$⊢\phi \to C{<}_s{0}_s$$
Assertion	ltmulnegs2d	$$⊢\phi \to \left(A{<}_{s}B\leftrightarrow \left(C{\cdot }_{s}B\right){<}_s\left(C{\cdot }_{s}A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ltmulnegs.1	$$⊢\phi \to A\in \mathrm{No}$$
2		ltmulnegs.2	$$⊢\phi \to B\in \mathrm{No}$$
3		ltmulnegs.3	$$⊢\phi \to C\in \mathrm{No}$$
4		ltmulnegs.4	$$⊢\phi \to C{<}_s{0}_s$$
5	1 2 3 4	ltmulnegs1d	$$⊢\phi \to \left(A{<}_{s}B\leftrightarrow \left(B{\cdot }_{s}C\right){<}_s\left(A{\cdot }_{s}C\right)\right)$$
6	2 3	mulscomd	$$⊢\phi \to B{\cdot }_{s}C=C{\cdot }_{s}B$$
7	1 3	mulscomd	$$⊢\phi \to A{\cdot }_{s}C=C{\cdot }_{s}A$$
8	6 7	breq12d	$$⊢\phi \to \left(\left(B{\cdot }_{s}C\right){<}_s\left(A{\cdot }_{s}C\right)\leftrightarrow \left(C{\cdot }_{s}B\right){<}_s\left(C{\cdot }_{s}A\right)\right)$$
9	5 8	bitrd	$$⊢\phi \to \left(A{<}_{s}B\leftrightarrow \left(C{\cdot }_{s}B\right){<}_s\left(C{\cdot }_{s}A\right)\right)$$