Theorem ltmuls2d

Description: Multiplication of both sides of surreal less-than by a positive number. (Contributed by Scott Fenton, 10-Mar-2025)

	Ref	Expression
Hypotheses	ltmuls12d.1	$$⊢\phi \to A\in \mathrm{No}$$
	ltmuls12d.2	$$⊢\phi \to B\in \mathrm{No}$$
	ltmuls12d.3	$$⊢\phi \to C\in \mathrm{No}$$
	ltmuls12d.4	$$⊢\phi \to 0_s{<}_{s}C$$
Assertion	ltmuls2d	$$⊢\phi \to \left(A{<}_{s}B\leftrightarrow \left(C{\cdot }_{s}A\right){<}_s\left(C{\cdot }_{s}B\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ltmuls12d.1	$$⊢\phi \to A\in \mathrm{No}$$
2		ltmuls12d.2	$$⊢\phi \to B\in \mathrm{No}$$
3		ltmuls12d.3	$$⊢\phi \to C\in \mathrm{No}$$
4		ltmuls12d.4	$$⊢\phi \to 0_s{<}_{s}C$$
5		ltmuls2	$$⊢\left(\left(C\in \mathrm{No}\wedge 0_s{<}_{s}C\right)\wedge A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to \left(A{<}_{s}B\leftrightarrow \left(C{\cdot }_{s}A\right){<}_s\left(C{\cdot }_{s}B\right)\right)$$
6	3 4 1 2 5	syl211anc	$$⊢\phi \to \left(A{<}_{s}B\leftrightarrow \left(C{\cdot }_{s}A\right){<}_s\left(C{\cdot }_{s}B\right)\right)$$