Theorem divne1d

Description: If two complex numbers are unequal, their quotient is not one. Contrapositive of diveq1d . (Contributed by David Moews, 28-Feb-2017)

	Ref	Expression
Hypotheses	div1d.1	$$⊢\phi \to A\in ℂ$$
	divcld.2	$$⊢\phi \to B\in ℂ$$
	divcld.3	$$⊢\phi \to B\ne 0$$
	divne1d.4	$$⊢\phi \to A\ne B$$
Assertion	divne1d	$$⊢\phi \to \frac{A}{B}\ne 1$$

Proof

Step	Hyp	Ref	Expression
1		div1d.1	$$⊢\phi \to A\in ℂ$$
2		divcld.2	$$⊢\phi \to B\in ℂ$$
3		divcld.3	$$⊢\phi \to B\ne 0$$
4		divne1d.4	$$⊢\phi \to A\ne B$$
5	1 2 3	diveq1ad	$$⊢\phi \to \left(\frac{A}{B}=1\leftrightarrow A=B\right)$$
6	5	necon3bid	$$⊢\phi \to \left(\frac{A}{B}\ne 1\leftrightarrow A\ne B\right)$$
7	4 6	mpbird	$$⊢\phi \to \frac{A}{B}\ne 1$$