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	$$⊢\phi \to A\in ℝ$$
	leadd12dd.b	$$⊢\phi \to B\in ℝ$$
	leadd12dd.c	$$⊢\phi \to C\in ℝ$$
	leadd12dd.d	$$⊢\phi \to D\in ℝ$$
	leadd12dd.ac	$$⊢\phi \to A\le C$$
	leadd12dd.bd	$$⊢\phi \to B\le D$$
Assertion	leadd12dd	$$⊢\phi \to A+B\le C+D$$

Proof

Step	Hyp	Ref	Expression
1		leadd12dd.a	$$⊢\phi \to A\in ℝ$$
2		leadd12dd.b	$$⊢\phi \to B\in ℝ$$
3		leadd12dd.c	$$⊢\phi \to C\in ℝ$$
4		leadd12dd.d	$$⊢\phi \to D\in ℝ$$
5		leadd12dd.ac	$$⊢\phi \to A\le C$$
6		leadd12dd.bd	$$⊢\phi \to B\le D$$
7	1 2	readdcld	$$⊢\phi \to A+B\in ℝ$$
8	3 2	readdcld	$$⊢\phi \to C+B\in ℝ$$
9	3 4	readdcld	$$⊢\phi \to C+D\in ℝ$$
10	1 3 2 5	leadd1dd	$$⊢\phi \to A+B\le C+B$$
11	2 4 3 6	leadd2dd	$$⊢\phi \to C+B\le C+D$$
12	7 8 9 10 11	letrd	$$⊢\phi \to A+B\le C+D$$