Theorem abs3difi

Description: Absolute value of differences around common element. (Contributed by NM, 2-Oct-1999)

	Ref	Expression
Hypotheses	absvalsqi.1	⊢ 𝐴 ∈ ℂ
	abssub.2	⊢ 𝐵 ∈ ℂ
	abs3dif.3	⊢ 𝐶 ∈ ℂ
Assertion	abs3difi	⊢ ( abs ‘ ( 𝐴 − 𝐵 ) ) ≤ ( ( abs ‘ ( 𝐴 − 𝐶 ) ) + ( abs ‘ ( 𝐶 − 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		absvalsqi.1	⊢ 𝐴 ∈ ℂ
2		abssub.2	⊢ 𝐵 ∈ ℂ
3		abs3dif.3	⊢ 𝐶 ∈ ℂ
4		abs3dif	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( abs ‘ ( 𝐴 − 𝐵 ) ) ≤ ( ( abs ‘ ( 𝐴 − 𝐶 ) ) + ( abs ‘ ( 𝐶 − 𝐵 ) ) ) )
5	1 2 3 4	mp3an	⊢ ( abs ‘ ( 𝐴 − 𝐵 ) ) ≤ ( ( abs ‘ ( 𝐴 − 𝐶 ) ) + ( abs ‘ ( 𝐶 − 𝐵 ) ) )