Theorem xdivcl

Description: Closure law for the extended division. (Contributed by Thierry Arnoux, 15-Mar-2017)

	Ref	Expression
Assertion	xdivcl	$$⊢\left(A\in {ℝ}^{*}\wedge B\in ℝ\wedge B\ne 0\right)\to A{÷}_{𝑒}B\in {ℝ}^{*}$$

Proof

Step	Hyp	Ref	Expression
1		simp1	$$⊢\left(A\in {ℝ}^{*}\wedge B\in ℝ\wedge B\ne 0\right)\to A\in {ℝ}^{*}$$
2		simp2	$$⊢\left(A\in {ℝ}^{*}\wedge B\in ℝ\wedge B\ne 0\right)\to B\in ℝ$$
3		simp3	$$⊢\left(A\in {ℝ}^{*}\wedge B\in ℝ\wedge B\ne 0\right)\to B\ne 0$$
4	1 2 3	xdivcld	$$⊢\left(A\in {ℝ}^{*}\wedge B\in ℝ\wedge B\ne 0\right)\to A{÷}_{𝑒}B\in {ℝ}^{*}$$