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	$$⊢\mathrm{abs}:ℂ⟶ℝ$$

Proof

Step	Hyp	Ref	Expression
1		df-abs	$$⊢\mathrm{abs}=\left(x\in ℂ⟼\sqrt{x\bar{x}}\right)$$
2		absval	$$⊢x\in ℂ\to \left|x\right|=\sqrt{x\bar{x}}$$
3		abscl	$$⊢x\in ℂ\to \left|x\right|\in ℝ$$
4	2 3	eqeltrrd	$$⊢x\in ℂ\to \sqrt{x\bar{x}}\in ℝ$$
5	1 4	fmpti	$$⊢\mathrm{abs}:ℂ⟶ℝ$$