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Metamath Proof Explorer

Theorem cxpef

Description: Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014)

Ref Expression
Assertion cxpef ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) = ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) )

Proof

Step Hyp Ref Expression
1 cxpval ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) = if ( 𝐴 = 0 , if ( 𝐵 = 0 , 1 , 0 ) , ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) ) )
2 1 3adant2 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) = if ( 𝐴 = 0 , if ( 𝐵 = 0 , 1 , 0 ) , ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) ) )
3 simp2 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → 𝐴 ≠ 0 )
4 3 neneqd ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ¬ 𝐴 = 0 )
5 4 iffalsed ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → if ( 𝐴 = 0 , if ( 𝐵 = 0 , 1 , 0 ) , ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) ) = ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) )
6 2 5 eqtrd ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) = ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) )