Metamath Proof Explorer

Theorem leadd1

Description: Addition to both sides of 'less than or equal to'. (Contributed by NM, 18-Oct-1999) (Proof shortened by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion leadd1 ⊢ A ∈ ℝ ∧ B ∈ ℝ ∧ C ∈ ℝ → A ≤ B ↔ A + C ≤ B + C

Proof

Step Hyp Ref Expression
1 ltadd1 ⊢ B ∈ ℝ ∧ A ∈ ℝ ∧ C ∈ ℝ → B < A ↔ B + C < A + C
2 1 3com12 ⊢ A ∈ ℝ ∧ B ∈ ℝ ∧ C ∈ ℝ → B < A ↔ B + C < A + C
3 2 notbid ⊢ A ∈ ℝ ∧ B ∈ ℝ ∧ C ∈ ℝ → ¬ B < A ↔ ¬ B + C < A + C
4 simp1 ⊢ A ∈ ℝ ∧ B ∈ ℝ ∧ C ∈ ℝ → A ∈ ℝ
5 simp2 ⊢ A ∈ ℝ ∧ B ∈ ℝ ∧ C ∈ ℝ → B ∈ ℝ
6 4 5 lenltd ⊢ A ∈ ℝ ∧ B ∈ ℝ ∧ C ∈ ℝ → A ≤ B ↔ ¬ B < A
7 simp3 ⊢ A ∈ ℝ ∧ B ∈ ℝ ∧ C ∈ ℝ → C ∈ ℝ
8 4 7 readdcld ⊢ A ∈ ℝ ∧ B ∈ ℝ ∧ C ∈ ℝ → A + C ∈ ℝ
9 5 7 readdcld ⊢ A ∈ ℝ ∧ B ∈ ℝ ∧ C ∈ ℝ → B + C ∈ ℝ
10 8 9 lenltd ⊢ A ∈ ℝ ∧ B ∈ ℝ ∧ C ∈ ℝ → A + C ≤ B + C ↔ ¬ B + C < A + C
11 3 6 10 3bitr4d ⊢ A ∈ ℝ ∧ B ∈ ℝ ∧ C ∈ ℝ → A ≤ B ↔ A + C ≤ B + C