Theorem int-ineq1stprincd

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

	Ref	Expression
Hypotheses	int-ineq1stprincd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	int-ineq1stprincd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	int-ineq1stprincd.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
	int-ineq1stprincd.4	⊢ ( 𝜑 → 𝐷 ∈ ℝ )
	int-ineq1stprincd.5	⊢ ( 𝜑 → 𝐵 ≤ 𝐴 )
	int-ineq1stprincd.6	⊢ ( 𝜑 → 𝐷 ≤ 𝐶 )
Assertion	int-ineq1stprincd	⊢ ( 𝜑 → ( 𝐵 + 𝐷 ) ≤ ( 𝐴 + 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		int-ineq1stprincd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		int-ineq1stprincd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		int-ineq1stprincd.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
4		int-ineq1stprincd.4	⊢ ( 𝜑 → 𝐷 ∈ ℝ )
5		int-ineq1stprincd.5	⊢ ( 𝜑 → 𝐵 ≤ 𝐴 )
6		int-ineq1stprincd.6	⊢ ( 𝜑 → 𝐷 ≤ 𝐶 )
7	2 4 1 3 5 6	le2addd	⊢ ( 𝜑 → ( 𝐵 + 𝐷 ) ≤ ( 𝐴 + 𝐶 ) )