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Metamath Proof Explorer

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 → A − B ≤ A − C + 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 → A − B ≤ A − C + C − B
5 1 2 3 4 syl3anc ⊢ φ → A − B ≤ A − C + C − B