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Metamath Proof Explorer

Theorem sqn0rp

Description: The square of a nonzero real is a positive real. (Contributed by AV, 5-Mar-2023)

		Ref	Expression
	Assertion	sqn0rp	$⊢ (A \in ℝ \land A \neq 0) \to A^{2} \in ℝ^{+}$

Step	Hyp	Ref	Expression
1		resqcl	$⊢ A \in ℝ \to A^{2} \in ℝ$
2	1	adantr	$⊢ (A \in ℝ \land A \neq 0) \to A^{2} \in ℝ$
3		sqgt0	$⊢ (A \in ℝ \land A \neq 0) \to 0 < A^{2}$
4	2 3	elrpd	$⊢ (A \in ℝ \land A \neq 0) \to A^{2} \in ℝ^{+}$