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Metamath Proof Explorer

Theorem 0cxp

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

Ref Expression
Assertion 0cxp ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 0 ↑𝑐 𝐴 ) = 0 )

Proof

Step Hyp Ref Expression
1 0cn ⊢ 0 ∈ ℂ
2 cxpval ⊢ ( ( 0 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( 0 ↑𝑐 𝐴 ) = if ( 0 = 0 , if ( 𝐴 = 0 , 1 , 0 ) , ( exp ‘ ( 𝐴 · ( log ‘ 0 ) ) ) ) )
3 1 2 mpan ⊢ ( 𝐴 ∈ ℂ → ( 0 ↑𝑐 𝐴 ) = if ( 0 = 0 , if ( 𝐴 = 0 , 1 , 0 ) , ( exp ‘ ( 𝐴 · ( log ‘ 0 ) ) ) ) )
4 eqid ⊢ 0 = 0
5 4 iftruei ⊢ if ( 0 = 0 , if ( 𝐴 = 0 , 1 , 0 ) , ( exp ‘ ( 𝐴 · ( log ‘ 0 ) ) ) ) = if ( 𝐴 = 0 , 1 , 0 )
6 3 5 eqtrdi ⊢ ( 𝐴 ∈ ℂ → ( 0 ↑𝑐 𝐴 ) = if ( 𝐴 = 0 , 1 , 0 ) )
7 ifnefalse ⊢ ( 𝐴 ≠ 0 → if ( 𝐴 = 0 , 1 , 0 ) = 0 )
8 6 7 sylan9eq ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 0 ↑𝑐 𝐴 ) = 0 )