Theorem diveq1

Description: Equality in terms of unit ratio. (Contributed by Stefan O'Rear, 27-Aug-2015)

	Ref	Expression
Assertion	diveq1	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to \left(\frac{A}{B}=1\leftrightarrow A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		ax-1cn	$$⊢1\in ℂ$$
2		divmul2	$$⊢\left(A\in ℂ\wedge 1\in ℂ\wedge \left(B\in ℂ\wedge B\ne 0\right)\right)\to \left(\frac{A}{B}=1\leftrightarrow A=B\cdot 1\right)$$
3	1 2	mp3an2	$$⊢\left(A\in ℂ\wedge \left(B\in ℂ\wedge B\ne 0\right)\right)\to \left(\frac{A}{B}=1\leftrightarrow A=B\cdot 1\right)$$
4	3	3impb	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to \left(\frac{A}{B}=1\leftrightarrow A=B\cdot 1\right)$$
5		simp2	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to B\in ℂ$$
6	5	mulridd	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to B\cdot 1=B$$
7	6	eqeq2d	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to \left(A=B\cdot 1\leftrightarrow A=B\right)$$
8	4 7	bitrd	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to \left(\frac{A}{B}=1\leftrightarrow A=B\right)$$