Theorem int-ineq1stprincd

Description: FirstPrincipleOfInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020)

	Ref	Expression
Hypotheses	int-ineq1stprincd.1	$$⊢\phi \to A\in ℝ$$
	int-ineq1stprincd.2	$$⊢\phi \to B\in ℝ$$
	int-ineq1stprincd.3	$$⊢\phi \to C\in ℝ$$
	int-ineq1stprincd.4	$$⊢\phi \to D\in ℝ$$
	int-ineq1stprincd.5	$$⊢\phi \to B\le A$$
	int-ineq1stprincd.6	$$⊢\phi \to D\le C$$
Assertion	int-ineq1stprincd	$$⊢\phi \to B+D\le A+C$$

Proof

Step	Hyp	Ref	Expression
1		int-ineq1stprincd.1	$$⊢\phi \to A\in ℝ$$
2		int-ineq1stprincd.2	$$⊢\phi \to B\in ℝ$$
3		int-ineq1stprincd.3	$$⊢\phi \to C\in ℝ$$
4		int-ineq1stprincd.4	$$⊢\phi \to D\in ℝ$$
5		int-ineq1stprincd.5	$$⊢\phi \to B\le A$$
6		int-ineq1stprincd.6	$$⊢\phi \to D\le C$$
7	2 4 1 3 5 6	le2addd	$$⊢\phi \to B+D\le A+C$$