Theorem absge0

Description: Absolute value is nonnegative. (Contributed by NM, 20-Nov-2004) (Revised by Mario Carneiro, 29-May-2016)

	Ref	Expression
Assertion	absge0	⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( abs ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		cjmulrcl	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ∈ ℝ )
2		cjmulge0	⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) )
3		sqrtge0	⊢ ( ( ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ∈ ℝ ∧ 0 ≤ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) → 0 ≤ ( √ ‘ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) )
4	1 2 3	syl2anc	⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( √ ‘ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) )
5		absval	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ 𝐴 ) = ( √ ‘ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) )
6	4 5	breqtrrd	⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( abs ‘ 𝐴 ) )