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 ) ) )