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Metamath Proof Explorer

Theorem lerec2

Description: Reciprocal swap in a 'less than or equal to' relation. (Contributed by NM, 24-Feb-2005)

Ref Expression
Assertion lerec2 ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → ( 𝐴 ≤ ( 1 / 𝐵 ) ↔ 𝐵 ≤ ( 1 / 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 gt0ne0 ⊢ ( ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) → 𝐵 ≠ 0 )
2 rereccl ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ( 1 / 𝐵 ) ∈ ℝ )
3 1 2 syldan ⊢ ( ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) → ( 1 / 𝐵 ) ∈ ℝ )
4 recgt0 ⊢ ( ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) → 0 < ( 1 / 𝐵 ) )
5 3 4 jca ⊢ ( ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) → ( ( 1 / 𝐵 ) ∈ ℝ ∧ 0 < ( 1 / 𝐵 ) ) )
6 lerec ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( ( 1 / 𝐵 ) ∈ ℝ ∧ 0 < ( 1 / 𝐵 ) ) ) → ( 𝐴 ≤ ( 1 / 𝐵 ) ↔ ( 1 / ( 1 / 𝐵 ) ) ≤ ( 1 / 𝐴 ) ) )
7 5 6 sylan2 ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → ( 𝐴 ≤ ( 1 / 𝐵 ) ↔ ( 1 / ( 1 / 𝐵 ) ) ≤ ( 1 / 𝐴 ) ) )
8 recn ⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℂ )
9 recrec ⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 1 / ( 1 / 𝐵 ) ) = 𝐵 )
10 8 1 9 syl2an2r ⊢ ( ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) → ( 1 / ( 1 / 𝐵 ) ) = 𝐵 )
11 10 adantl ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → ( 1 / ( 1 / 𝐵 ) ) = 𝐵 )
12 11 breq1d ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → ( ( 1 / ( 1 / 𝐵 ) ) ≤ ( 1 / 𝐴 ) ↔ 𝐵 ≤ ( 1 / 𝐴 ) ) )
13 7 12 bitrd ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → ( 𝐴 ≤ ( 1 / 𝐵 ) ↔ 𝐵 ≤ ( 1 / 𝐴 ) ) )