Theorem sltsub2d

Description: Subtraction from both sides of surreal less-than. (Contributed by Scott Fenton, 5-Feb-2025)

	Ref	Expression
Hypotheses	sltsubd.1	$$⊢\phi \to A\in \mathrm{No}$$
	sltsubd.2	$$⊢\phi \to B\in \mathrm{No}$$
	sltsubd.3	$$⊢\phi \to C\in \mathrm{No}$$
Assertion	sltsub2d	$$⊢\phi \to \left(A{<}_{s}B\leftrightarrow \left(C{-}_{s}B\right){<}_s\left(C{-}_{s}A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		sltsubd.1	$$⊢\phi \to A\in \mathrm{No}$$
2		sltsubd.2	$$⊢\phi \to B\in \mathrm{No}$$
3		sltsubd.3	$$⊢\phi \to C\in \mathrm{No}$$
4		sltsub2	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\wedge C\in \mathrm{No}\right)\to \left(A{<}_{s}B\leftrightarrow \left(C{-}_{s}B\right){<}_s\left(C{-}_{s}A\right)\right)$$
5	1 2 3 4	syl3anc	$$⊢\phi \to \left(A{<}_{s}B\leftrightarrow \left(C{-}_{s}B\right){<}_s\left(C{-}_{s}A\right)\right)$$