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Metamath Proof Explorer

Theorem cxpval

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

Ref Expression
Assertion cxpval ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) = if ( 𝐴 = 0 , if ( 𝐵 = 0 , 1 , 0 ) , ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 simpl ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → 𝑥 = 𝐴 )
2 1 eqeq1d ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝑥 = 0 ↔ 𝐴 = 0 ) )
3 simpr ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → 𝑦 = 𝐵 )
4 3 eqeq1d ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝑦 = 0 ↔ 𝐵 = 0 ) )
5 4 ifbid ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → if ( 𝑦 = 0 , 1 , 0 ) = if ( 𝐵 = 0 , 1 , 0 ) )
6 1 fveq2d ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( log ‘ 𝑥 ) = ( log ‘ 𝐴 ) )
7 3 6 oveq12d ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝑦 · ( log ‘ 𝑥 ) ) = ( 𝐵 · ( log ‘ 𝐴 ) ) )
8 7 fveq2d ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( exp ‘ ( 𝑦 · ( log ‘ 𝑥 ) ) ) = ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) )
9 2 5 8 ifbieq12d ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → if ( 𝑥 = 0 , if ( 𝑦 = 0 , 1 , 0 ) , ( exp ‘ ( 𝑦 · ( log ‘ 𝑥 ) ) ) ) = if ( 𝐴 = 0 , if ( 𝐵 = 0 , 1 , 0 ) , ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) ) )
10 df-cxp ⊢ ↑𝑐 = ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ if ( 𝑥 = 0 , if ( 𝑦 = 0 , 1 , 0 ) , ( exp ‘ ( 𝑦 · ( log ‘ 𝑥 ) ) ) ) )
11 ax-1cn ⊢ 1 ∈ ℂ
12 0cn ⊢ 0 ∈ ℂ
13 11 12 ifcli ⊢ if ( 𝐵 = 0 , 1 , 0 ) ∈ ℂ
14 13 elexi ⊢ if ( 𝐵 = 0 , 1 , 0 ) ∈ V
15 fvex ⊢ ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) ∈ V
16 14 15 ifex ⊢ if ( 𝐴 = 0 , if ( 𝐵 = 0 , 1 , 0 ) , ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) ) ∈ V
17 9 10 16 ovmpoa ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) = if ( 𝐴 = 0 , if ( 𝐵 = 0 , 1 , 0 ) , ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) ) )