Theorem sltsubsub2bd

Description: Equivalence for the surreal less-than relationship between differences. (Contributed by Scott Fenton, 21-Feb-2025)

	Ref	Expression
Hypotheses	sltsubsubbd.1	$$⊢\phi \to A\in \mathrm{No}$$
	sltsubsubbd.2	$$⊢\phi \to B\in \mathrm{No}$$
	sltsubsubbd.3	$$⊢\phi \to C\in \mathrm{No}$$
	sltsubsubbd.4	$$⊢\phi \to D\in \mathrm{No}$$
Assertion	sltsubsub2bd	$$⊢\phi \to \left(\left(A{-}_{s}B\right){<}_s\left(C{-}_{s}D\right)\leftrightarrow \left(D{-}_{s}C\right){<}_s\left(B{-}_{s}A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		sltsubsubbd.1	$$⊢\phi \to A\in \mathrm{No}$$
2		sltsubsubbd.2	$$⊢\phi \to B\in \mathrm{No}$$
3		sltsubsubbd.3	$$⊢\phi \to C\in \mathrm{No}$$
4		sltsubsubbd.4	$$⊢\phi \to D\in \mathrm{No}$$
5	4 3	subscld	$$⊢\phi \to D{-}_{s}C\in \mathrm{No}$$
6	2 1	subscld	$$⊢\phi \to B{-}_{s}A\in \mathrm{No}$$
7	5 6	sltnegd	$$⊢\phi \to \left(\left(D{-}_{s}C\right){<}_s\left(B{-}_{s}A\right)\leftrightarrow {+}_s\left(B{-}_{s}A\right){<}_s{+}_s\left(D{-}_{s}C\right)\right)$$
8	2 1	negsubsdi2d	$$⊢\phi \to {+}_s\left(B{-}_{s}A\right)=A{-}_{s}B$$
9	4 3	negsubsdi2d	$$⊢\phi \to {+}_s\left(D{-}_{s}C\right)=C{-}_{s}D$$
10	8 9	breq12d	$$⊢\phi \to \left({+}_s\left(B{-}_{s}A\right){<}_s{+}_s\left(D{-}_{s}C\right)\leftrightarrow \left(A{-}_{s}B\right){<}_s\left(C{-}_{s}D\right)\right)$$
11	7 10	bitr2d	$$⊢\phi \to \left(\left(A{-}_{s}B\right){<}_s\left(C{-}_{s}D\right)\leftrightarrow \left(D{-}_{s}C\right){<}_s\left(B{-}_{s}A\right)\right)$$