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Metamath Proof Explorer

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 )