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Metamath Proof Explorer

Theorem lereci

Description: The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999)

Ref Expression
Hypotheses ltplus1.1 ⊢ A ∈ ℝ
prodgt0.2 ⊢ B ∈ ℝ
Assertion lereci ⊢ 0 < A ∧ 0 < B → A ≤ B ↔ 1 B ≤ 1 A

Proof

Step Hyp Ref Expression
1 ltplus1.1 ⊢ A ∈ ℝ
2 prodgt0.2 ⊢ B ∈ ℝ
3 lerec ⊢ A ∈ ℝ ∧ 0 < A ∧ B ∈ ℝ ∧ 0 < B → A ≤ B ↔ 1 B ≤ 1 A
4 2 3 mpanr1 ⊢ A ∈ ℝ ∧ 0 < A ∧ 0 < B → A ≤ B ↔ 1 B ≤ 1 A
5 1 4 mpanl1 ⊢ 0 < A ∧ 0 < B → A ≤ B ↔ 1 B ≤ 1 A