Metamath Proof Explorer

Theorem ltneg

Description: Negative of both sides of 'less than'. Theorem I.23 of Apostol p. 20. (Contributed by NM, 27-Aug-1999) (Proof shortened by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion ltneg ⊢ A ∈ ℝ ∧ B ∈ ℝ → A < B ↔ − B < − A

Proof

Step Hyp Ref Expression
1 0re ⊢ 0 ∈ ℝ
2 ltsub2 ⊢ A ∈ ℝ ∧ B ∈ ℝ ∧ 0 ∈ ℝ → A < B ↔ 0 − B < 0 − A
3 1 2 mp3an3 ⊢ A ∈ ℝ ∧ B ∈ ℝ → A < B ↔ 0 − B < 0 − A
4 df-neg ⊢ − B = 0 − B
5 df-neg ⊢ − A = 0 − A
6 4 5 breq12i ⊢ − B < − A ↔ 0 − B < 0 − A
7 3 6 bitr4di ⊢ A ∈ ℝ ∧ B ∈ ℝ → A < B ↔ − B < − A