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 ) ) |
| 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 ) ) |