Theorem ltsubsposd

Description: Subtraction of a positive number decreases the sum. (Contributed by Scott Fenton, 15-Apr-2025)

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

Proof

Step	Hyp	Ref	Expression
1		ltsubspos.1	$$⊢\phi \to A\in \mathrm{No}$$
2		ltsubspos.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	4 1 2	ltsubs2d	$$⊢\phi \to \left(0_s{<}_{s}A\leftrightarrow \left(B{-}_{s}A\right){<}_s\left(B{-}_s{0}_s\right)\right)$$
6		subsid1	$$⊢B\in \mathrm{No}\to B{-}_s{0}_s=B$$
7	2 6	syl	$$⊢\phi \to B{-}_s{0}_s=B$$
8	7	breq2d	$$⊢\phi \to \left(\left(B{-}_{s}A\right){<}_s\left(B{-}_s{0}_s\right)\leftrightarrow \left(B{-}_{s}A\right){<}_{s}B\right)$$
9	5 8	bitrd	$$⊢\phi \to \left(0_s{<}_{s}A\leftrightarrow \left(B{-}_{s}A\right){<}_{s}B\right)$$