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Metamath Proof Explorer

Theorem lediv2ad

Description: Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses rpred.1 ⊢ φ → A ∈ ℝ +
rpaddcld.1 ⊢ φ → B ∈ ℝ +
lediv2ad.3 ⊢ φ → C ∈ ℝ
lediv2ad.4 ⊢ φ → 0 ≤ C
lediv2ad.5 ⊢ φ → A ≤ B
Assertion lediv2ad ⊢ φ → C B ≤ C A

Proof

Step Hyp Ref Expression
1 rpred.1 ⊢ φ → A ∈ ℝ +
2 rpaddcld.1 ⊢ φ → B ∈ ℝ +
3 lediv2ad.3 ⊢ φ → C ∈ ℝ
4 lediv2ad.4 ⊢ φ → 0 ≤ C
5 lediv2ad.5 ⊢ φ → A ≤ B
6 1 rpregt0d ⊢ φ → A ∈ ℝ ∧ 0 < A
7 2 rpregt0d ⊢ φ → B ∈ ℝ ∧ 0 < B
8 3 4 jca ⊢ φ → C ∈ ℝ ∧ 0 ≤ C
9 lediv2a ⊢ A ∈ ℝ ∧ 0 < A ∧ B ∈ ℝ ∧ 0 < B ∧ C ∈ ℝ ∧ 0 ≤ C ∧ A ≤ B → C B ≤ C A
10 6 7 8 5 9 syl31anc ⊢ φ → C B ≤ C A