Theorem le2subd

Description: Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	leidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	ltnegd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	ltadd1d.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
	lt2addd.4	⊢ ( 𝜑 → 𝐷 ∈ ℝ )
	le2addd.5	⊢ ( 𝜑 → 𝐴 ≤ 𝐶 )
	le2addd.6	⊢ ( 𝜑 → 𝐵 ≤ 𝐷 )
Assertion	le2subd	⊢ ( 𝜑 → ( 𝐴 − 𝐷 ) ≤ ( 𝐶 − 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		leidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		ltnegd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		ltadd1d.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
4		lt2addd.4	⊢ ( 𝜑 → 𝐷 ∈ ℝ )
5		le2addd.5	⊢ ( 𝜑 → 𝐴 ≤ 𝐶 )
6		le2addd.6	⊢ ( 𝜑 → 𝐵 ≤ 𝐷 )
7		le2sub	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐷 ∈ ℝ ) ∧ ( 𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ) → ( ( 𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐷 ) → ( 𝐴 − 𝐷 ) ≤ ( 𝐶 − 𝐵 ) ) )
8	1 4 3 2 7	syl22anc	⊢ ( 𝜑 → ( ( 𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐷 ) → ( 𝐴 − 𝐷 ) ≤ ( 𝐶 − 𝐵 ) ) )
9	5 6 8	mp2and	⊢ ( 𝜑 → ( 𝐴 − 𝐷 ) ≤ ( 𝐶 − 𝐵 ) )