Theorem absgt0

Description: The absolute value of a nonzero number is positive. (Contributed by NM, 1-Oct-1999) (Proof shortened by Mario Carneiro, 29-May-2016)

	Ref	Expression
Assertion	absgt0	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ≠ 0 ↔ 0 < ( abs ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		0red	⊢ ( 𝐴 ∈ ℂ → 0 ∈ ℝ )
2		abscl	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ 𝐴 ) ∈ ℝ )
3		absge0	⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( abs ‘ 𝐴 ) )
4	1 2 3	leltned	⊢ ( 𝐴 ∈ ℂ → ( 0 < ( abs ‘ 𝐴 ) ↔ ( abs ‘ 𝐴 ) ≠ 0 ) )
5		abs00	⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ 𝐴 ) = 0 ↔ 𝐴 = 0 ) )
6	5	necon3bid	⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ 𝐴 ) ≠ 0 ↔ 𝐴 ≠ 0 ) )
7	4 6	bitr2d	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ≠ 0 ↔ 0 < ( abs ‘ 𝐴 ) ) )