Theorem sltaddpos2d

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

	Ref	Expression
Hypotheses	sltaddpos.1	$$⊢\phi \to A\in \mathrm{No}$$
	sltaddpos.2	$$⊢\phi \to B\in \mathrm{No}$$
Assertion	sltaddpos2d	$$⊢\phi \to \left(0_s{<}_{s}A\leftrightarrow B{<}_s\left(A{+}_{s}B\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		sltaddpos.1	$$⊢\phi \to A\in \mathrm{No}$$
2		sltaddpos.2	$$⊢\phi \to B\in \mathrm{No}$$
3		0sno	$$⊢0_s\in \mathrm{No}$$
4	3	a1i	$$⊢\phi \to 0_s\in \mathrm{No}$$
5	4 1 2	sltadd1d	$$⊢\phi \to \left(0_s{<}_{s}A\leftrightarrow \left(0_s{+}_{s}B\right){<}_s\left(A{+}_{s}B\right)\right)$$
6		addslid	$$⊢B\in \mathrm{No}\to 0_s{+}_{s}B=B$$
7	2 6	syl	$$⊢\phi \to 0_s{+}_{s}B=B$$
8	7	breq1d	$$⊢\phi \to \left(\left(0_s{+}_{s}B\right){<}_s\left(A{+}_{s}B\right)\leftrightarrow B{<}_s\left(A{+}_{s}B\right)\right)$$
9	5 8	bitrd	$$⊢\phi \to \left(0_s{<}_{s}A\leftrightarrow B{<}_s\left(A{+}_{s}B\right)\right)$$