This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem lemuldiv

Description: 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006)

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

Proof

Step Hyp Ref Expression
1 ltdivmul2 ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐵 / 𝐶 ) < 𝐴 ↔ 𝐵 < ( 𝐴 · 𝐶 ) ) )
2 1 3com12 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐵 / 𝐶 ) < 𝐴 ↔ 𝐵 < ( 𝐴 · 𝐶 ) ) )
3 2 notbid ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ¬ ( 𝐵 / 𝐶 ) < 𝐴 ↔ ¬ 𝐵 < ( 𝐴 · 𝐶 ) ) )
4 simp1 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → 𝐴 ∈ ℝ )
5 gt0ne0 ⊢ ( ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) → 𝐶 ≠ 0 )
6 5 3adant1 ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) → 𝐶 ≠ 0 )
7 redivcl ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐶 ≠ 0 ) → ( 𝐵 / 𝐶 ) ∈ ℝ )
8 6 7 syld3an3 ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) → ( 𝐵 / 𝐶 ) ∈ ℝ )
9 8 3expb ⊢ ( ( 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( 𝐵 / 𝐶 ) ∈ ℝ )
10 9 3adant1 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( 𝐵 / 𝐶 ) ∈ ℝ )
11 4 10 lenltd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( 𝐴 ≤ ( 𝐵 / 𝐶 ) ↔ ¬ ( 𝐵 / 𝐶 ) < 𝐴 ) )
12 remulcl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 · 𝐶 ) ∈ ℝ )
13 12 3adant2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 · 𝐶 ) ∈ ℝ )
14 simp2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐵 ∈ ℝ )
15 13 14 lenltd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 · 𝐶 ) ≤ 𝐵 ↔ ¬ 𝐵 < ( 𝐴 · 𝐶 ) ) )
16 15 3adant3r ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐴 · 𝐶 ) ≤ 𝐵 ↔ ¬ 𝐵 < ( 𝐴 · 𝐶 ) ) )
17 3 11 16 3bitr4rd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐴 · 𝐶 ) ≤ 𝐵 ↔ 𝐴 ≤ ( 𝐵 / 𝐶 ) ) )