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 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1 ) → ( log ‘ 𝐴 ) ≠ 0 )

Proof

Step	Hyp	Ref	Expression
1		simp3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1 ) → 𝐴 ≠ 1 )
2	1	neneqd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1 ) → ¬ 𝐴 = 1 )
3		logeq0im1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ ( log ‘ 𝐴 ) = 0 ) → 𝐴 = 1 )
4	3	3expia	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ( log ‘ 𝐴 ) = 0 → 𝐴 = 1 ) )
5	4	3adant3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1 ) → ( ( log ‘ 𝐴 ) = 0 → 𝐴 = 1 ) )
6	2 5	mtod	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1 ) → ¬ ( log ‘ 𝐴 ) = 0 )
7	6	neqned	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1 ) → ( log ‘ 𝐴 ) ≠ 0 )