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Metamath Proof Explorer

Theorem ltdiv23ii

Description: Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999)

Ref Expression
Hypotheses ltplus1.1 ⊢ 𝐴 ∈ ℝ
prodgt0.2 ⊢ 𝐵 ∈ ℝ
ltmul1.3 ⊢ 𝐶 ∈ ℝ
ltdiv23i.4 ⊢ 0 < 𝐵
ltdiv23i.5 ⊢ 0 < 𝐶
Assertion ltdiv23ii ( ( 𝐴 / 𝐵 ) < 𝐶 ↔ ( 𝐴 / 𝐶 ) < 𝐵 )

Proof

Step Hyp Ref Expression
1 ltplus1.1 ⊢ 𝐴 ∈ ℝ
2 prodgt0.2 ⊢ 𝐵 ∈ ℝ
3 ltmul1.3 ⊢ 𝐶 ∈ ℝ
4 ltdiv23i.4 ⊢ 0 < 𝐵
5 ltdiv23i.5 ⊢ 0 < 𝐶
6 1 2 3 ltdiv23i ⊢ ( ( 0 < 𝐵 ∧ 0 < 𝐶 ) → ( ( 𝐴 / 𝐵 ) < 𝐶 ↔ ( 𝐴 / 𝐶 ) < 𝐵 ) )
7 4 5 6 mp2an ⊢ ( ( 𝐴 / 𝐵 ) < 𝐶 ↔ ( 𝐴 / 𝐶 ) < 𝐵 )