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Metamath Proof Explorer

Theorem abslei

Description: Absolute value and 'less than or equal to' relation. (Contributed by NM, 6-Apr-2005)

	Ref	Expression
Hypotheses	sqrtthi.1	$⊢ A \in ℝ$
	sqr11.1	$⊢ B \in ℝ$
Assertion	abslei	$⊢ \|A\| \leq B \leftrightarrow (- B \leq A \land A \leq B)$

Step	Hyp	Ref	Expression
1		sqrtthi.1	$⊢ A \in ℝ$
2		sqr11.1	$⊢ B \in ℝ$
3		absle	$⊢ (A \in ℝ \land B \in ℝ) \to (\|A\| \leq B \leftrightarrow (- B \leq A \land A \leq B))$
4	1 2 3	mp2an	$⊢ \|A\| \leq B \leftrightarrow (- B \leq A \land A \leq B)$