Theorem absidm

Description: The absolute value function is idempotent. (Contributed by NM, 20-Nov-2004)

	Ref	Expression
Assertion	absidm	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ ( abs ‘ 𝐴 ) ) = ( abs ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		abscl	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ 𝐴 ) ∈ ℝ )
2		absge0	⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( abs ‘ 𝐴 ) )
3		absid	⊢ ( ( ( abs ‘ 𝐴 ) ∈ ℝ ∧ 0 ≤ ( abs ‘ 𝐴 ) ) → ( abs ‘ ( abs ‘ 𝐴 ) ) = ( abs ‘ 𝐴 ) )
4	1 2 3	syl2anc	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ ( abs ‘ 𝐴 ) ) = ( abs ‘ 𝐴 ) )