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Metamath Proof Explorer

Theorem ltmuldiv

Description: 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999) (Proof shortened by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion ltmuldiv ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐴 · 𝐶 ) < 𝐵 ↔ 𝐴 < ( 𝐵 / 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 simp1 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → 𝐴 ∈ ℝ )
2 simp3l ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → 𝐶 ∈ ℝ )
3 1 2 remulcld ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( 𝐴 · 𝐶 ) ∈ ℝ )
4 ltdiv1 ⊢ ( ( ( 𝐴 · 𝐶 ) ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐴 · 𝐶 ) < 𝐵 ↔ ( ( 𝐴 · 𝐶 ) / 𝐶 ) < ( 𝐵 / 𝐶 ) ) )
5 3 4 syld3an1 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐴 · 𝐶 ) < 𝐵 ↔ ( ( 𝐴 · 𝐶 ) / 𝐶 ) < ( 𝐵 / 𝐶 ) ) )
6 1 recnd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → 𝐴 ∈ ℂ )
7 2 recnd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → 𝐶 ∈ ℂ )
8 simp3r ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → 0 < 𝐶 )
9 8 gt0ne0d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → 𝐶 ≠ 0 )
10 6 7 9 divcan4d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐴 · 𝐶 ) / 𝐶 ) = 𝐴 )
11 10 breq1d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( ( 𝐴 · 𝐶 ) / 𝐶 ) < ( 𝐵 / 𝐶 ) ↔ 𝐴 < ( 𝐵 / 𝐶 ) ) )
12 5 11 bitrd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐴 · 𝐶 ) < 𝐵 ↔ 𝐴 < ( 𝐵 / 𝐶 ) ) )