Theorem posdifsd

Description: Comparison of two surreals whose difference is positive. (Contributed by Scott Fenton, 10-Mar-2025)

	Ref	Expression
Hypotheses	posdifsd.1	$$⊢\phi \to A\in \mathrm{No}$$
	posdifsd.2	$$⊢\phi \to B\in \mathrm{No}$$
Assertion	posdifsd	$$⊢\phi \to \left(A{<}_{s}B\leftrightarrow 0_s{<}_s\left(B{-}_{s}A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		posdifsd.1	$$⊢\phi \to A\in \mathrm{No}$$
2		posdifsd.2	$$⊢\phi \to B\in \mathrm{No}$$
3		0no	$$⊢0_s\in \mathrm{No}$$
4	3	a1i	$$⊢\phi \to 0_s\in \mathrm{No}$$
5	2 1	subscld	$$⊢\phi \to B{-}_{s}A\in \mathrm{No}$$
6	4 5 1	ltadds1d	$$⊢\phi \to \left(0_s{<}_s\left(B{-}_{s}A\right)\leftrightarrow \left(0_s{+}_{s}A\right){<}_s\left(\left(B{-}_{s}A\right){+}_{s}A\right)\right)$$
7		addslid	$$⊢A\in \mathrm{No}\to 0_s{+}_{s}A=A$$
8	1 7	syl	$$⊢\phi \to 0_s{+}_{s}A=A$$
9		npcans	$$⊢\left(B\in \mathrm{No}\wedge A\in \mathrm{No}\right)\to \left(B{-}_{s}A\right){+}_{s}A=B$$
10	2 1 9	syl2anc	$$⊢\phi \to \left(B{-}_{s}A\right){+}_{s}A=B$$
11	8 10	breq12d	$$⊢\phi \to \left(\left(0_s{+}_{s}A\right){<}_s\left(\left(B{-}_{s}A\right){+}_{s}A\right)\leftrightarrow A{<}_{s}B\right)$$
12	6 11	bitr2d	$$⊢\phi \to \left(A{<}_{s}B\leftrightarrow 0_s{<}_s\left(B{-}_{s}A\right)\right)$$