Theorem ltaddspos1d

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

	Ref	Expression
Hypotheses	ltaddspos.1	$$⊢\phi \to A\in \mathrm{No}$$
	ltaddspos.2	$$⊢\phi \to B\in \mathrm{No}$$
Assertion	ltaddspos1d	$$⊢\phi \to \left(0_s{<}_{s}A\leftrightarrow B{<}_s\left(B{+}_{s}A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ltaddspos.1	$$⊢\phi \to A\in \mathrm{No}$$
2		ltaddspos.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	ltadds2d	$$⊢\phi \to \left(0_s{<}_{s}A\leftrightarrow \left(B{+}_s{0}_s\right){<}_s\left(B{+}_{s}A\right)\right)$$
6	2	addsridd	$$⊢\phi \to B{+}_s{0}_s=B$$
7	6	breq1d	$$⊢\phi \to \left(\left(B{+}_s{0}_s\right){<}_s\left(B{+}_{s}A\right)\leftrightarrow B{<}_s\left(B{+}_{s}A\right)\right)$$
8	5 7	bitrd	$$⊢\phi \to \left(0_s{<}_{s}A\leftrightarrow B{<}_s\left(B{+}_{s}A\right)\right)$$