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Metamath Proof Explorer

Theorem lerec

Description: The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 3-Oct-1999) (Proof shortened by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion lerec ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → ( 𝐴 ≤ 𝐵 ↔ ( 1 / 𝐵 ) ≤ ( 1 / 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 ltrec ⊢ ( ( ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ∧ ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ) → ( 𝐵 < 𝐴 ↔ ( 1 / 𝐴 ) < ( 1 / 𝐵 ) ) )
2 1 ancoms ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → ( 𝐵 < 𝐴 ↔ ( 1 / 𝐴 ) < ( 1 / 𝐵 ) ) )
3 2 notbid ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → ( ¬ 𝐵 < 𝐴 ↔ ¬ ( 1 / 𝐴 ) < ( 1 / 𝐵 ) ) )
4 simpll ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → 𝐴 ∈ ℝ )
5 simprl ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → 𝐵 ∈ ℝ )
6 4 5 lenltd ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → ( 𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴 ) )
7 simprr ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → 0 < 𝐵 )
8 7 gt0ne0d ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → 𝐵 ≠ 0 )
9 5 8 rereccld ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → ( 1 / 𝐵 ) ∈ ℝ )
10 simplr ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → 0 < 𝐴 )
11 10 gt0ne0d ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → 𝐴 ≠ 0 )
12 4 11 rereccld ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → ( 1 / 𝐴 ) ∈ ℝ )
13 9 12 lenltd ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → ( ( 1 / 𝐵 ) ≤ ( 1 / 𝐴 ) ↔ ¬ ( 1 / 𝐴 ) < ( 1 / 𝐵 ) ) )
14 3 6 13 3bitr4d ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → ( 𝐴 ≤ 𝐵 ↔ ( 1 / 𝐵 ) ≤ ( 1 / 𝐴 ) ) )