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Metamath Proof Explorer

Theorem explog

Description: Exponentiation of a nonzero complex number to an integer power. (Contributed by Paul Chapman, 21-Apr-2008)

Ref Expression
Assertion explog ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ 𝑁 ) = ( exp ‘ ( 𝑁 · ( log ‘ 𝐴 ) ) ) )

Proof

Step Hyp Ref Expression
1 logcl ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( log ‘ 𝐴 ) ∈ ℂ )
2 efexp ⊢ ( ( ( log ‘ 𝐴 ) ∈ ℂ ∧ 𝑁 ∈ ℤ ) → ( exp ‘ ( 𝑁 · ( log ‘ 𝐴 ) ) ) = ( ( exp ‘ ( log ‘ 𝐴 ) ) ↑ 𝑁 ) )
3 1 2 stoic3 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( exp ‘ ( 𝑁 · ( log ‘ 𝐴 ) ) ) = ( ( exp ‘ ( log ‘ 𝐴 ) ) ↑ 𝑁 ) )
4 eflog ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( exp ‘ ( log ‘ 𝐴 ) ) = 𝐴 )
5 4 3adant3 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( exp ‘ ( log ‘ 𝐴 ) ) = 𝐴 )
6 5 oveq1d ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( ( exp ‘ ( log ‘ 𝐴 ) ) ↑ 𝑁 ) = ( 𝐴 ↑ 𝑁 ) )
7 3 6 eqtr2d ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ 𝑁 ) = ( exp ‘ ( 𝑁 · ( log ‘ 𝐴 ) ) ) )