Theorem nndivre

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

	Ref	Expression
Assertion	nndivre	$$⊢\left(A\in ℝ\wedge N\in \mathbb{N}\right)\to \frac{A}{N}\in ℝ$$

Proof

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