Theorem ltsubadds2d

Description: Surreal less-than relationship between subtraction and addition. (Contributed by Scott Fenton, 27-Feb-2025)

	Ref	Expression
Hypotheses	ltsubadds.1	$$⊢\phi \to A\in \mathrm{No}$$
	ltsubadds.2	$$⊢\phi \to B\in \mathrm{No}$$
	ltsubadds.3	$$⊢\phi \to C\in \mathrm{No}$$
Assertion	ltsubadds2d	$$⊢\phi \to \left(\left(A{-}_{s}B\right){<}_{s}C\leftrightarrow A{<}_s\left(B{+}_{s}C\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ltsubadds.1	$$⊢\phi \to A\in \mathrm{No}$$
2		ltsubadds.2	$$⊢\phi \to B\in \mathrm{No}$$
3		ltsubadds.3	$$⊢\phi \to C\in \mathrm{No}$$
4	1 2 3	ltsubaddsd	$$⊢\phi \to \left(\left(A{-}_{s}B\right){<}_{s}C\leftrightarrow A{<}_s\left(C{+}_{s}B\right)\right)$$
5	2 3	addscomd	$$⊢\phi \to B{+}_{s}C=C{+}_{s}B$$
6	5	breq2d	$$⊢\phi \to \left(A{<}_s\left(B{+}_{s}C\right)\leftrightarrow A{<}_s\left(C{+}_{s}B\right)\right)$$
7	4 6	bitr4d	$$⊢\phi \to \left(\left(A{-}_{s}B\right){<}_{s}C\leftrightarrow A{<}_s\left(B{+}_{s}C\right)\right)$$