Theorem lesubsd

Description: Swap subtrahends in a surreal inequality. (Contributed by Scott Fenton, 29-Jan-2026)

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

Proof

Step	Hyp	Ref	Expression
1		lesubsd.1	$$⊢\phi \to A\in \mathrm{No}$$
2		lesubsd.2	$$⊢\phi \to B\in \mathrm{No}$$
3		lesubsd.3	$$⊢\phi \to C\in \mathrm{No}$$
4		npcans	$$⊢\left(B\in \mathrm{No}\wedge A\in \mathrm{No}\right)\to \left(B{-}_{s}A\right){+}_{s}A=B$$
5	2 1 4	syl2anc	$$⊢\phi \to \left(B{-}_{s}A\right){+}_{s}A=B$$
6	2 1	subscld	$$⊢\phi \to B{-}_{s}A\in \mathrm{No}$$
7	1 6	addscomd	$$⊢\phi \to A{+}_s\left(B{-}_{s}A\right)=\left(B{-}_{s}A\right){+}_{s}A$$
8		npcans	$$⊢\left(B\in \mathrm{No}\wedge C\in \mathrm{No}\right)\to \left(B{-}_{s}C\right){+}_{s}C=B$$
9	2 3 8	syl2anc	$$⊢\phi \to \left(B{-}_{s}C\right){+}_{s}C=B$$
10	5 7 9	3eqtr4rd	$$⊢\phi \to \left(B{-}_{s}C\right){+}_{s}C=A{+}_s\left(B{-}_{s}A\right)$$
11	10	breq2d	$$⊢\phi \to \left(\left(A{+}_{s}C\right){\le }_s\left(\left(B{-}_{s}C\right){+}_{s}C\right)\leftrightarrow \left(A{+}_{s}C\right){\le }_s\left(A{+}_s\left(B{-}_{s}A\right)\right)\right)$$
12	2 3	subscld	$$⊢\phi \to B{-}_{s}C\in \mathrm{No}$$
13	1 12 3	leadds1d	$$⊢\phi \to \left(A{\le }_s\left(B{-}_{s}C\right)\leftrightarrow \left(A{+}_{s}C\right){\le }_s\left(\left(B{-}_{s}C\right){+}_{s}C\right)\right)$$
14	3 6 1	leadds2d	$$⊢\phi \to \left(C{\le }_s\left(B{-}_{s}A\right)\leftrightarrow \left(A{+}_{s}C\right){\le }_s\left(A{+}_s\left(B{-}_{s}A\right)\right)\right)$$
15	11 13 14	3bitr4d	$$⊢\phi \to \left(A{\le }_s\left(B{-}_{s}C\right)\leftrightarrow C{\le }_s\left(B{-}_{s}A\right)\right)$$