Theorem sltsubaddd

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

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

Proof

Step	Hyp	Ref	Expression
1		sltsubadd.1	$$⊢\phi \to A\in \mathrm{No}$$
2		sltsubadd.2	$$⊢\phi \to B\in \mathrm{No}$$
3		sltsubadd.3	$$⊢\phi \to C\in \mathrm{No}$$
4	1 2	subscld	$$⊢\phi \to A{-}_{s}B\in \mathrm{No}$$
5	4 3 2	sltadd1d	$$⊢\phi \to \left(\left(A{-}_{s}B\right){<}_{s}C\leftrightarrow \left(\left(A{-}_{s}B\right){+}_{s}B\right){<}_s\left(C{+}_{s}B\right)\right)$$
6		npcans	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to \left(A{-}_{s}B\right){+}_{s}B=A$$
7	1 2 6	syl2anc	$$⊢\phi \to \left(A{-}_{s}B\right){+}_{s}B=A$$
8	7	breq1d	$$⊢\phi \to \left(\left(\left(A{-}_{s}B\right){+}_{s}B\right){<}_s\left(C{+}_{s}B\right)\leftrightarrow A{<}_s\left(C{+}_{s}B\right)\right)$$
9	5 8	bitrd	$$⊢\phi \to \left(\left(A{-}_{s}B\right){<}_{s}C\leftrightarrow A{<}_s\left(C{+}_{s}B\right)\right)$$