Theorem abs3lemd

Description: Lemma involving absolute value of differences. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	abscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	abssubd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	abs3difd.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
	abs3lemd.4	⊢ ( 𝜑 → 𝐷 ∈ ℝ )
	abs3lemd.5	⊢ ( 𝜑 → ( abs ‘ ( 𝐴 − 𝐶 ) ) < ( 𝐷 / 2 ) )
	abs3lemd.6	⊢ ( 𝜑 → ( abs ‘ ( 𝐶 − 𝐵 ) ) < ( 𝐷 / 2 ) )
Assertion	abs3lemd	⊢ ( 𝜑 → ( abs ‘ ( 𝐴 − 𝐵 ) ) < 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		abscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		abssubd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		abs3difd.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		abs3lemd.4	⊢ ( 𝜑 → 𝐷 ∈ ℝ )
5		abs3lemd.5	⊢ ( 𝜑 → ( abs ‘ ( 𝐴 − 𝐶 ) ) < ( 𝐷 / 2 ) )
6		abs3lemd.6	⊢ ( 𝜑 → ( abs ‘ ( 𝐶 − 𝐵 ) ) < ( 𝐷 / 2 ) )
7		abs3lem	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℝ ) ) → ( ( ( abs ‘ ( 𝐴 − 𝐶 ) ) < ( 𝐷 / 2 ) ∧ ( abs ‘ ( 𝐶 − 𝐵 ) ) < ( 𝐷 / 2 ) ) → ( abs ‘ ( 𝐴 − 𝐵 ) ) < 𝐷 ) )
8	1 2 3 4 7	syl22anc	⊢ ( 𝜑 → ( ( ( abs ‘ ( 𝐴 − 𝐶 ) ) < ( 𝐷 / 2 ) ∧ ( abs ‘ ( 𝐶 − 𝐵 ) ) < ( 𝐷 / 2 ) ) → ( abs ‘ ( 𝐴 − 𝐵 ) ) < 𝐷 ) )
9	5 6 8	mp2and	⊢ ( 𝜑 → ( abs ‘ ( 𝐴 − 𝐵 ) ) < 𝐷 )