Theorem logcxp0

Description: Logarithm of a complex power. Generalization of logcxp . (Contributed by AV, 22-May-2020)

	Ref	Expression
Assertion	logcxp0	⊢ ( ( 𝐴 ∈ ( ℂ ∖ { 0 } ) ∧ 𝐵 ∈ ℂ ∧ ( 𝐵 · ( log ‘ 𝐴 ) ) ∈ ran log ) → ( log ‘ ( 𝐴 ↑𝑐 𝐵 ) ) = ( 𝐵 · ( log ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		eldifi	⊢ ( 𝐴 ∈ ( ℂ ∖ { 0 } ) → 𝐴 ∈ ℂ )
2	1	3ad2ant1	⊢ ( ( 𝐴 ∈ ( ℂ ∖ { 0 } ) ∧ 𝐵 ∈ ℂ ∧ ( 𝐵 · ( log ‘ 𝐴 ) ) ∈ ran log ) → 𝐴 ∈ ℂ )
3		eldifsni	⊢ ( 𝐴 ∈ ( ℂ ∖ { 0 } ) → 𝐴 ≠ 0 )
4	3	3ad2ant1	⊢ ( ( 𝐴 ∈ ( ℂ ∖ { 0 } ) ∧ 𝐵 ∈ ℂ ∧ ( 𝐵 · ( log ‘ 𝐴 ) ) ∈ ran log ) → 𝐴 ≠ 0 )
5		simp2	⊢ ( ( 𝐴 ∈ ( ℂ ∖ { 0 } ) ∧ 𝐵 ∈ ℂ ∧ ( 𝐵 · ( log ‘ 𝐴 ) ) ∈ ran log ) → 𝐵 ∈ ℂ )
6	2 4 5	cxpefd	⊢ ( ( 𝐴 ∈ ( ℂ ∖ { 0 } ) ∧ 𝐵 ∈ ℂ ∧ ( 𝐵 · ( log ‘ 𝐴 ) ) ∈ ran log ) → ( 𝐴 ↑𝑐 𝐵 ) = ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) )
7	6	fveq2d	⊢ ( ( 𝐴 ∈ ( ℂ ∖ { 0 } ) ∧ 𝐵 ∈ ℂ ∧ ( 𝐵 · ( log ‘ 𝐴 ) ) ∈ ran log ) → ( log ‘ ( 𝐴 ↑𝑐 𝐵 ) ) = ( log ‘ ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) ) )
8		logef	⊢ ( ( 𝐵 · ( log ‘ 𝐴 ) ) ∈ ran log → ( log ‘ ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) ) = ( 𝐵 · ( log ‘ 𝐴 ) ) )
9	8	3ad2ant3	⊢ ( ( 𝐴 ∈ ( ℂ ∖ { 0 } ) ∧ 𝐵 ∈ ℂ ∧ ( 𝐵 · ( log ‘ 𝐴 ) ) ∈ ran log ) → ( log ‘ ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) ) = ( 𝐵 · ( log ‘ 𝐴 ) ) )
10	7 9	eqtrd	⊢ ( ( 𝐴 ∈ ( ℂ ∖ { 0 } ) ∧ 𝐵 ∈ ℂ ∧ ( 𝐵 · ( log ‘ 𝐴 ) ) ∈ ran log ) → ( log ‘ ( 𝐴 ↑𝑐 𝐵 ) ) = ( 𝐵 · ( log ‘ 𝐴 ) ) )