Theorem abs3difd

Description: Absolute value of differences around common element. (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 )
Assertion	abs3difd	⊢ ( 𝜑 → ( abs ‘ ( 𝐴 − 𝐵 ) ) ≤ ( ( abs ‘ ( 𝐴 − 𝐶 ) ) + ( abs ‘ ( 𝐶 − 𝐵 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		abscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		abssubd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		abs3difd.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		abs3dif	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( abs ‘ ( 𝐴 − 𝐵 ) ) ≤ ( ( abs ‘ ( 𝐴 − 𝐶 ) ) + ( abs ‘ ( 𝐶 − 𝐵 ) ) ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( abs ‘ ( 𝐴 − 𝐵 ) ) ≤ ( ( abs ‘ ( 𝐴 − 𝐶 ) ) + ( abs ‘ ( 𝐶 − 𝐵 ) ) ) )