Theorem logm1

Description: The natural logarithm of negative 1 . (Contributed by Paul Chapman, 21-Apr-2008) (Revised by Mario Carneiro, 13-May-2014)

	Ref	Expression
Assertion	logm1	⊢ ( log ‘ - 1 ) = ( i · π )

Proof

Step	Hyp	Ref	Expression
1		1rp	⊢ 1 ∈ ℝ+
2		logneg	⊢ ( 1 ∈ ℝ+ → ( log ‘ - 1 ) = ( ( log ‘ 1 ) + ( i · π ) ) )
3	1 2	ax-mp	⊢ ( log ‘ - 1 ) = ( ( log ‘ 1 ) + ( i · π ) )
4		log1	⊢ ( log ‘ 1 ) = 0
5	4	oveq1i	⊢ ( ( log ‘ 1 ) + ( i · π ) ) = ( 0 + ( i · π ) )
6		ax-icn	⊢ i ∈ ℂ
7		picn	⊢ π ∈ ℂ
8	6 7	mulcli	⊢ ( i · π ) ∈ ℂ
9	8	addlidi	⊢ ( 0 + ( i · π ) ) = ( i · π )
10	3 5 9	3eqtri	⊢ ( log ‘ - 1 ) = ( i · π )