Metamath Proof Explorer


Theorem leneg

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 ) → ( 𝐴𝐵 ↔ - 𝐵 ≤ - 𝐴 ) )

Proof

Step Hyp Ref Expression
1 0re 0 ∈ ℝ
2 lesub2 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 ∈ ℝ ) → ( 𝐴𝐵 ↔ ( 0 − 𝐵 ) ≤ ( 0 − 𝐴 ) ) )
3 1 2 mp3an3 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴𝐵 ↔ ( 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 ) → ( 𝐴𝐵 ↔ - 𝐵 ≤ - 𝐴 ) )