Description: Negative of both sides of 'less than or equal to'. (Contributed by NM, 12-Sep-1999) (Proof shortened by Mario Carneiro, 27-May-2016)
Ref | Expression | ||
---|---|---|---|
Assertion | leneg | ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ - 𝐵 ≤ - 𝐴 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re | ⊢ 0 ∈ ℝ | |
2 | lesub2 | ⊢ ( ( 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 ) ) |