Theorem absf

Description: Mapping domain and codomain of the absolute value function. (Contributed by NM, 30-Aug-2007) (Revised by Mario Carneiro, 7-Nov-2013)

	Ref	Expression
Assertion	absf	⊢ abs : ℂ ⟶ ℝ

Proof

Step	Hyp	Ref	Expression
1		df-abs	⊢ abs = ( 𝑥 ∈ ℂ ↦ ( √ ‘ ( 𝑥 · ( ∗ ‘ 𝑥 ) ) ) )
2		absval	⊢ ( 𝑥 ∈ ℂ → ( abs ‘ 𝑥 ) = ( √ ‘ ( 𝑥 · ( ∗ ‘ 𝑥 ) ) ) )
3		abscl	⊢ ( 𝑥 ∈ ℂ → ( abs ‘ 𝑥 ) ∈ ℝ )
4	2 3	eqeltrrd	⊢ ( 𝑥 ∈ ℂ → ( √ ‘ ( 𝑥 · ( ∗ ‘ 𝑥 ) ) ) ∈ ℝ )
5	1 4	fmpti	⊢ abs : ℂ ⟶ ℝ