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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 ( 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 ) ) )