Metamath Proof Explorer

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 ⊢ A − B ≤ A − C + 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 → A − B ≤ A − C + C − B
5 1 2 3 4 mp3an ⊢ A − B ≤ A − C + C − B