Metamath Proof Explorer


Theorem ltsubposd

Description: Subtracting a positive number from another number decreases it. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses leidd.1 ( 𝜑𝐴 ∈ ℝ )
ltnegd.2 ( 𝜑𝐵 ∈ ℝ )
Assertion ltsubposd ( 𝜑 → ( 0 < 𝐴 ↔ ( 𝐵𝐴 ) < 𝐵 ) )

Proof

Step Hyp Ref Expression
1 leidd.1 ( 𝜑𝐴 ∈ ℝ )
2 ltnegd.2 ( 𝜑𝐵 ∈ ℝ )
3 ltsubpos ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 < 𝐴 ↔ ( 𝐵𝐴 ) < 𝐵 ) )
4 1 2 3 syl2anc ( 𝜑 → ( 0 < 𝐴 ↔ ( 𝐵𝐴 ) < 𝐵 ) )