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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 ) ) )