Theorem ltsubsposd

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

	Ref	Expression
Hypotheses	ltsubspos.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	ltsubspos.2	⊢ ( 𝜑 → 𝐵 ∈ No )
Assertion	ltsubsposd	⊢ ( 𝜑 → ( 0s <s 𝐴 ↔ ( 𝐵 -s 𝐴 ) <s 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ltsubspos.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		ltsubspos.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		0no	⊢ 0s ∈ No
4	3	a1i	⊢ ( 𝜑 → 0s ∈ No )
5	4 1 2	ltsubs2d	⊢ ( 𝜑 → ( 0s <s 𝐴 ↔ ( 𝐵 -s 𝐴 ) <s ( 𝐵 -s 0s ) ) )
6		subsid1	⊢ ( 𝐵 ∈ No → ( 𝐵 -s 0s ) = 𝐵 )
7	2 6	syl	⊢ ( 𝜑 → ( 𝐵 -s 0s ) = 𝐵 )
8	7	breq2d	⊢ ( 𝜑 → ( ( 𝐵 -s 𝐴 ) <s ( 𝐵 -s 0s ) ↔ ( 𝐵 -s 𝐴 ) <s 𝐵 ) )
9	5 8	bitrd	⊢ ( 𝜑 → ( 0s <s 𝐴 ↔ ( 𝐵 -s 𝐴 ) <s 𝐵 ) )