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	|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( log ` A ) =/= 0 )

Proof

Step	Hyp	Ref	Expression
1		simp3	|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> A =/= 1 )
2	1	neneqd	|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> -. A = 1 )
3		logeq0im1	|- ( ( A e. CC /\ A =/= 0 /\ ( log ` A ) = 0 ) -> A = 1 )
4	3	3expia	|- ( ( A e. CC /\ A =/= 0 ) -> ( ( log ` A ) = 0 -> A = 1 ) )
5	4	3adant3	|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( log ` A ) = 0 -> A = 1 ) )
6	2 5	mtod	|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> -. ( log ` A ) = 0 )
7	6	neqned	|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( log ` A ) =/= 0 )