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Metamath Proof Explorer

Theorem ltrecii

Description: The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999)

Ref Expression
Hypotheses ltplus1.1 ⊢ 𝐴 ∈ ℝ
prodgt0.2 ⊢ 𝐵 ∈ ℝ
ltreci.3 ⊢ 0 < 𝐴
ltreci.4 ⊢ 0 < 𝐵
Assertion ltrecii ( 𝐴 < 𝐵 ↔ ( 1 / 𝐵 ) < ( 1 / 𝐴 ) )

Proof

Step Hyp Ref Expression
1 ltplus1.1 ⊢ 𝐴 ∈ ℝ
2 prodgt0.2 ⊢ 𝐵 ∈ ℝ
3 ltreci.3 ⊢ 0 < 𝐴
4 ltreci.4 ⊢ 0 < 𝐵
5 1 2 ltreci ⊢ ( ( 0 < 𝐴 ∧ 0 < 𝐵 ) → ( 𝐴 < 𝐵 ↔ ( 1 / 𝐵 ) < ( 1 / 𝐴 ) ) )
6 3 4 5 mp2an ⊢ ( 𝐴 < 𝐵 ↔ ( 1 / 𝐵 ) < ( 1 / 𝐴 ) )