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