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 Description: Addition to both sides of 'less than or equal to'. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 18-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
leadd1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴𝐵 ↔ (𝐴 + 𝐶) ≤ (𝐵 + 𝐶)))

StepHypRef Expression
1 ltadd1 8203 . . . 4 ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 < 𝐴 ↔ (𝐵 + 𝐶) < (𝐴 + 𝐶)))
213com12 1185 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 < 𝐴 ↔ (𝐵 + 𝐶) < (𝐴 + 𝐶)))
32notbid 656 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (¬ 𝐵 < 𝐴 ↔ ¬ (𝐵 + 𝐶) < (𝐴 + 𝐶)))
4 simp1 981 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℝ)
5 simp2 982 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℝ)
64, 5lenltd 7892 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))
7 simp3 983 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℝ)
84, 7readdcld 7807 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 + 𝐶) ∈ ℝ)
95, 7readdcld 7807 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 + 𝐶) ∈ ℝ)
108, 9lenltd 7892 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐶) ≤ (𝐵 + 𝐶) ↔ ¬ (𝐵 + 𝐶) < (𝐴 + 𝐶)))
113, 6, 103bitr4d 219 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴𝐵 ↔ (𝐴 + 𝐶) ≤ (𝐵 + 𝐶)))