Theorem xdiv0

Description: Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016)

	Ref	Expression
Assertion	xdiv0	$$⊢\left(A\in ℝ\wedge A\ne 0\right)\to 0{÷}_{𝑒}A=0$$

Proof

Step	Hyp	Ref	Expression
1		0re	$$⊢0\in ℝ$$
2		rexdiv	$$⊢\left(0\in ℝ\wedge A\in ℝ\wedge A\ne 0\right)\to 0{÷}_{𝑒}A=\frac{0}{A}$$
3	1 2	mp3an1	$$⊢\left(A\in ℝ\wedge A\ne 0\right)\to 0{÷}_{𝑒}A=\frac{0}{A}$$
4		recn	$$⊢A\in ℝ\to A\in ℂ$$
5		div0	$$⊢\left(A\in ℂ\wedge A\ne 0\right)\to \frac{0}{A}=0$$
6	4 5	sylan	$$⊢\left(A\in ℝ\wedge A\ne 0\right)\to \frac{0}{A}=0$$
7	3 6	eqtrd	$$⊢\left(A\in ℝ\wedge A\ne 0\right)\to 0{÷}_{𝑒}A=0$$