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Metamath Proof Explorer

Theorem nndivre

Description: The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008)

		Ref	Expression
	Assertion	nndivre	$⊢ (A \in ℝ \land N \in ℕ) \to \frac{A}{N} \in ℝ$

Step	Hyp	Ref	Expression
1		nnre	$⊢ N \in ℕ \to N \in ℝ$
2		nnne0	$⊢ N \in ℕ \to N \neq 0$
3	1 2	jca	$⊢ N \in ℕ \to (N \in ℝ \land N \neq 0)$
4		redivcl	$⊢ (A \in ℝ \land N \in ℝ \land N \neq 0) \to \frac{A}{N} \in ℝ$
5	4	3expb	$⊢ (A \in ℝ \land (N \in ℝ \land N \neq 0)) \to \frac{A}{N} \in ℝ$
6	3 5	sylan2	$⊢ (A \in ℝ \land N \in ℕ) \to \frac{A}{N} \in ℝ$