Theorem ltaddspos2d

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

	Ref	Expression
Hypotheses	ltaddspos.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	ltaddspos.2	⊢ ( 𝜑 → 𝐵 ∈ No )
Assertion	ltaddspos2d	⊢ ( 𝜑 → ( 0s <s 𝐴 ↔ 𝐵 <s ( 𝐴 +s 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltaddspos.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		ltaddspos.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		0no	⊢ 0s ∈ No
4	3	a1i	⊢ ( 𝜑 → 0s ∈ No )
5	4 1 2	ltadds1d	⊢ ( 𝜑 → ( 0s <s 𝐴 ↔ ( 0s +s 𝐵 ) <s ( 𝐴 +s 𝐵 ) ) )
6		addslid	⊢ ( 𝐵 ∈ No → ( 0s +s 𝐵 ) = 𝐵 )
7	2 6	syl	⊢ ( 𝜑 → ( 0s +s 𝐵 ) = 𝐵 )
8	7	breq1d	⊢ ( 𝜑 → ( ( 0s +s 𝐵 ) <s ( 𝐴 +s 𝐵 ) ↔ 𝐵 <s ( 𝐴 +s 𝐵 ) ) )
9	5 8	bitrd	⊢ ( 𝜑 → ( 0s <s 𝐴 ↔ 𝐵 <s ( 𝐴 +s 𝐵 ) ) )