Theorem abs2difi

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

	Ref	Expression
Hypotheses	abs2difi.1	⊢ 𝐴 ∈ ℂ
	abs2difi.2	⊢ 𝐵 ∈ ℂ
Assertion	abs2difi	⊢ ( ( abs ‘ 𝐴 ) − ( abs ‘ 𝐵 ) ) ≤ ( abs ‘ ( 𝐴 − 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		abs2difi.1	⊢ 𝐴 ∈ ℂ
2		abs2difi.2	⊢ 𝐵 ∈ ℂ
3		abs2dif	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( abs ‘ 𝐴 ) − ( abs ‘ 𝐵 ) ) ≤ ( abs ‘ ( 𝐴 − 𝐵 ) ) )
4	1 2 3	mp2an	⊢ ( ( abs ‘ 𝐴 ) − ( abs ‘ 𝐵 ) ) ≤ ( abs ‘ ( 𝐴 − 𝐵 ) )