Theorem div1

Description: A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002) (Proof shortened by Mario Carneiro, 27-May-2016)

	Ref	Expression
Assertion	div1	$$⊢A\in ℂ\to \frac{A}{1}=A$$

Proof

Step	Hyp	Ref	Expression
1		mulid2	$$⊢A\in ℂ\to 1A=A$$
2		ax-1cn	$$⊢1\in ℂ$$
3		ax-1ne0	$$⊢1\ne 0$$
4	2 3	pm3.2i	$$⊢\left(1\in ℂ\wedge 1\ne 0\right)$$
5		divmul	$$⊢\left(A\in ℂ\wedge A\in ℂ\wedge \left(1\in ℂ\wedge 1\ne 0\right)\right)\to \left(\frac{A}{1}=A\leftrightarrow 1A=A\right)$$
6	4 5	mp3an3	$$⊢\left(A\in ℂ\wedge A\in ℂ\right)\to \left(\frac{A}{1}=A\leftrightarrow 1A=A\right)$$
7	6	anidms	$$⊢A\in ℂ\to \left(\frac{A}{1}=A\leftrightarrow 1A=A\right)$$
8	1 7	mpbird	$$⊢A\in ℂ\to \frac{A}{1}=A$$