Metamath Proof Explorer
Description: Negative of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016)
|
|
Ref |
Expression |
|
Hypotheses |
leidd.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
|
|
ltnegd.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
|
Assertion |
lenegd |
⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 ↔ - 𝐵 ≤ - 𝐴 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
leidd.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
ltnegd.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
leneg |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ - 𝐵 ≤ - 𝐴 ) ) |
4 |
1 2 3
|
syl2anc |
⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 ↔ - 𝐵 ≤ - 𝐴 ) ) |