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 ‘ ( 𝐴 − 𝐵 ) )

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 ‘ ( 𝐴 − 𝐵 ) ) )
4	1 2 3	mp2an	⊢ ( abs ‘ ( ( abs ‘ 𝐴 ) − ( abs ‘ 𝐵 ) ) ) ≤ ( abs ‘ ( 𝐴 − 𝐵 ) )