Metamath Proof Explorer


Theorem abs2difabsi

Description: Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007)

Ref Expression
Hypotheses abs2difabsi.1 𝐴 ∈ ℂ
abs2difabsi.2 𝐵 ∈ ℂ
Assertion abs2difabsi ( abs ‘ ( ( abs ‘ 𝐴 ) − ( abs ‘ 𝐵 ) ) ) ≤ ( abs ‘ ( 𝐴𝐵 ) )

Proof

Step Hyp Ref Expression
1 abs2difabsi.1 𝐴 ∈ ℂ
2 abs2difabsi.2 𝐵 ∈ ℂ
3 abs2difabs ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( abs ‘ ( ( abs ‘ 𝐴 ) − ( abs ‘ 𝐵 ) ) ) ≤ ( abs ‘ ( 𝐴𝐵 ) ) )
4 1 2 3 mp2an ( abs ‘ ( ( abs ‘ 𝐴 ) − ( abs ‘ 𝐵 ) ) ) ≤ ( abs ‘ ( 𝐴𝐵 ) )