Theorem logccne0

Description: The logarithm isn't 0 if its argument isn't 0 or 1. (Contributed by David A. Wheeler, 17-Jul-2017)

	Ref	Expression
Assertion	logccne0	$$⊢\left(A\in ℂ\wedge A\ne 0\wedge A\ne 1\right)\to \log A\ne 0$$

Proof

Step	Hyp	Ref	Expression
1		simp3	$$⊢\left(A\in ℂ\wedge A\ne 0\wedge A\ne 1\right)\to A\ne 1$$
2	1	neneqd	$$⊢\left(A\in ℂ\wedge A\ne 0\wedge A\ne 1\right)\to \neg A=1$$
3		logeq0im1	$$⊢\left(A\in ℂ\wedge A\ne 0\wedge \log A=0\right)\to A=1$$
4	3	3expia	$$⊢\left(A\in ℂ\wedge A\ne 0\right)\to \left(\log A=0\to A=1\right)$$
5	4	3adant3	$$⊢\left(A\in ℂ\wedge A\ne 0\wedge A\ne 1\right)\to \left(\log A=0\to A=1\right)$$
6	2 5	mtod	$$⊢\left(A\in ℂ\wedge A\ne 0\wedge A\ne 1\right)\to \neg \log A=0$$
7	6	neqned	$$⊢\left(A\in ℂ\wedge A\ne 0\wedge A\ne 1\right)\to \log A\ne 0$$