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Metamath Proof Explorer

Theorem rpsqrtcl

Description: The square root of a positive real is a positive real. (Contributed by NM, 22-Feb-2008)

		Ref	Expression
	Assertion	rpsqrtcl	$⊢ A \in ℝ^{+} \to \sqrt{A} \in ℝ^{+}$

Step	Hyp	Ref	Expression
1		rpre	$⊢ A \in ℝ^{+} \to A \in ℝ$
2		rpge0	$⊢ A \in ℝ^{+} \to 0 \leq A$
3		resqrtcl	$⊢ (A \in ℝ \land 0 \leq A) \to \sqrt{A} \in ℝ$
4	1 2 3	syl2anc	$⊢ A \in ℝ^{+} \to \sqrt{A} \in ℝ$
5		rpgt0	$⊢ A \in ℝ^{+} \to 0 < A$
6		sqrtgt0	$⊢ (A \in ℝ \land 0 < A) \to 0 < \sqrt{A}$
7	1 5 6	syl2anc	$⊢ A \in ℝ^{+} \to 0 < \sqrt{A}$
8	4 7	elrpd	$⊢ A \in ℝ^{+} \to \sqrt{A} \in ℝ^{+}$