Theorem leabs

Description: A real number is less than or equal to its absolute value. (Contributed by NM, 27-Feb-2005)

	Ref	Expression
Assertion	leabs	$$⊢A\in ℝ\to A\le \left|A\right|$$

Proof

Step	Hyp	Ref	Expression
1		0red	$$⊢A\in ℝ\to 0\in ℝ$$
2		id	$$⊢A\in ℝ\to A\in ℝ$$
3		absid	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to \left|A\right|=A$$
4		eqcom	$$⊢\left|A\right|=A\leftrightarrow A=\left|A\right|$$
5		eqle	$$⊢\left(A\in ℝ\wedge A=\left|A\right|\right)\to A\le \left|A\right|$$
6	4 5	sylan2b	$$⊢\left(A\in ℝ\wedge \left|A\right|=A\right)\to A\le \left|A\right|$$
7	3 6	syldan	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to A\le \left|A\right|$$
8		recn	$$⊢A\in ℝ\to A\in ℂ$$
9		absge0	$$⊢A\in ℂ\to 0\le \left|A\right|$$
10	8 9	syl	$$⊢A\in ℝ\to 0\le \left|A\right|$$
11		abscl	$$⊢A\in ℂ\to \left|A\right|\in ℝ$$
12	8 11	syl	$$⊢A\in ℝ\to \left|A\right|\in ℝ$$
13		0re	$$⊢0\in ℝ$$
14		letr	$$⊢\left(A\in ℝ\wedge 0\in ℝ\wedge \left|A\right|\in ℝ\right)\to \left(\left(A\le 0\wedge 0\le \left|A\right|\right)\to A\le \left|A\right|\right)$$
15	13 14	mp3an2	$$⊢\left(A\in ℝ\wedge \left|A\right|\in ℝ\right)\to \left(\left(A\le 0\wedge 0\le \left|A\right|\right)\to A\le \left|A\right|\right)$$
16	12 15	mpdan	$$⊢A\in ℝ\to \left(\left(A\le 0\wedge 0\le \left|A\right|\right)\to A\le \left|A\right|\right)$$
17	10 16	mpan2d	$$⊢A\in ℝ\to \left(A\le 0\to A\le \left|A\right|\right)$$
18	17	imp	$$⊢\left(A\in ℝ\wedge A\le 0\right)\to A\le \left|A\right|$$
19	1 2 7 18	lecasei	$$⊢A\in ℝ\to A\le \left|A\right|$$