Theorem absid

Description: A nonnegative number is its own absolute value. (Contributed by NM, 11-Oct-1999) (Revised by Mario Carneiro, 29-May-2016)

	Ref	Expression
Assertion	absid	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to \left|A\right|=A$$

Proof

Step	Hyp	Ref	Expression
1		simpl	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to A\in ℝ$$
2	1	recnd	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to A\in ℂ$$
3		absval	$$⊢A\in ℂ\to \left|A\right|=\sqrt{A\bar{A}}$$
4	2 3	syl	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to \left|A\right|=\sqrt{A\bar{A}}$$
5	1	cjred	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to \bar{A}=A$$
6	5	oveq2d	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to A\bar{A}=AA$$
7	2	sqvald	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to A^2=AA$$
8	6 7	eqtr4d	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to A\bar{A}=A^2$$
9	8	fveq2d	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to \sqrt{A\bar{A}}=\sqrt{A^2}$$
10		sqrtsq	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to \sqrt{A^2}=A$$
11	4 9 10	3eqtrd	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to \left|A\right|=A$$