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Metamath Proof Explorer

Theorem ltdiv23i

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
Assertion ltdiv23i ⊢ 0 < B ∧ 0 < C → 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 ltdiv23 ⊢ A ∈ ℝ ∧ B ∈ ℝ ∧ 0 < B ∧ C ∈ ℝ ∧ 0 < C → A B < C ↔ A C < B
5 1 4 mp3an1 ⊢ B ∈ ℝ ∧ 0 < B ∧ C ∈ ℝ ∧ 0 < C → A B < C ↔ A C < B
6 2 5 mpanl1 ⊢ 0 < B ∧ C ∈ ℝ ∧ 0 < C → A B < C ↔ A C < B
7 3 6 mpanr1 ⊢ 0 < B ∧ 0 < C → A B < C ↔ A C < B