Metamath Proof Explorer


Theorem leadd2d

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

Ref Expression
Hypotheses leidd.1 ( 𝜑𝐴 ∈ ℝ )
ltnegd.2 ( 𝜑𝐵 ∈ ℝ )
ltadd1d.3 ( 𝜑𝐶 ∈ ℝ )
Assertion leadd2d ( 𝜑 → ( 𝐴𝐵 ↔ ( 𝐶 + 𝐴 ) ≤ ( 𝐶 + 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 leidd.1 ( 𝜑𝐴 ∈ ℝ )
2 ltnegd.2 ( 𝜑𝐵 ∈ ℝ )
3 ltadd1d.3 ( 𝜑𝐶 ∈ ℝ )
4 leadd2 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴𝐵 ↔ ( 𝐶 + 𝐴 ) ≤ ( 𝐶 + 𝐵 ) ) )
5 1 2 3 4 syl3anc ( 𝜑 → ( 𝐴𝐵 ↔ ( 𝐶 + 𝐴 ) ≤ ( 𝐶 + 𝐵 ) ) )