Metamath Proof Explorer


Theorem leadd12dd

Description: Addition to both sides of 'less than or equal to'. (Contributed by Glauco Siliprandi, 5-Apr-2020)

Ref Expression
Hypotheses leadd12dd.a ( 𝜑𝐴 ∈ ℝ )
leadd12dd.b ( 𝜑𝐵 ∈ ℝ )
leadd12dd.c ( 𝜑𝐶 ∈ ℝ )
leadd12dd.d ( 𝜑𝐷 ∈ ℝ )
leadd12dd.ac ( 𝜑𝐴𝐶 )
leadd12dd.bd ( 𝜑𝐵𝐷 )
Assertion leadd12dd ( 𝜑 → ( 𝐴 + 𝐵 ) ≤ ( 𝐶 + 𝐷 ) )

Proof

Step Hyp Ref Expression
1 leadd12dd.a ( 𝜑𝐴 ∈ ℝ )
2 leadd12dd.b ( 𝜑𝐵 ∈ ℝ )
3 leadd12dd.c ( 𝜑𝐶 ∈ ℝ )
4 leadd12dd.d ( 𝜑𝐷 ∈ ℝ )
5 leadd12dd.ac ( 𝜑𝐴𝐶 )
6 leadd12dd.bd ( 𝜑𝐵𝐷 )
7 1 2 readdcld ( 𝜑 → ( 𝐴 + 𝐵 ) ∈ ℝ )
8 3 2 readdcld ( 𝜑 → ( 𝐶 + 𝐵 ) ∈ ℝ )
9 3 4 readdcld ( 𝜑 → ( 𝐶 + 𝐷 ) ∈ ℝ )
10 1 3 2 5 leadd1dd ( 𝜑 → ( 𝐴 + 𝐵 ) ≤ ( 𝐶 + 𝐵 ) )
11 2 4 3 6 leadd2dd ( 𝜑 → ( 𝐶 + 𝐵 ) ≤ ( 𝐶 + 𝐷 ) )
12 7 8 9 10 11 letrd ( 𝜑 → ( 𝐴 + 𝐵 ) ≤ ( 𝐶 + 𝐷 ) )