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	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	leadd12dd.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	leadd12dd.c	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
	leadd12dd.d	⊢ ( 𝜑 → 𝐷 ∈ ℝ )
	leadd12dd.ac	⊢ ( 𝜑 → 𝐴 ≤ 𝐶 )
	leadd12dd.bd	⊢ ( 𝜑 → 𝐵 ≤ 𝐷 )
Assertion	leadd12dd	⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) ≤ ( 𝐶 + 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		leadd12dd.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		leadd12dd.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		leadd12dd.c	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
4		leadd12dd.d	⊢ ( 𝜑 → 𝐷 ∈ ℝ )
5		leadd12dd.ac	⊢ ( 𝜑 → 𝐴 ≤ 𝐶 )
6		leadd12dd.bd	⊢ ( 𝜑 → 𝐵 ≤ 𝐷 )
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 ) )