Theorem sublt0d

Description: When a subtraction gives a negative result. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	sublt0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	sublt0d.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
Assertion	sublt0d	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) < 0 ↔ 𝐴 < 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		sublt0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		sublt0d.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		0red	⊢ ( 𝜑 → 0 ∈ ℝ )
4	1 2 3	ltsubaddd	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) < 0 ↔ 𝐴 < ( 0 + 𝐵 ) ) )
5	2	recnd	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
6	5	addid2d	⊢ ( 𝜑 → ( 0 + 𝐵 ) = 𝐵 )
7	6	breq2d	⊢ ( 𝜑 → ( 𝐴 < ( 0 + 𝐵 ) ↔ 𝐴 < 𝐵 ) )
8	4 7	bitrd	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) < 0 ↔ 𝐴 < 𝐵 ) )