Metamath Proof Explorer


Theorem cxpexpzd

Description: Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 30-May-2016)

Ref Expression
Hypotheses cxp0d.1 ( 𝜑𝐴 ∈ ℂ )
cxpefd.2 ( 𝜑𝐴 ≠ 0 )
cxpexpzd.3 ( 𝜑𝐵 ∈ ℤ )
Assertion cxpexpzd ( 𝜑 → ( 𝐴𝑐 𝐵 ) = ( 𝐴𝐵 ) )

Proof

Step Hyp Ref Expression
1 cxp0d.1 ( 𝜑𝐴 ∈ ℂ )
2 cxpefd.2 ( 𝜑𝐴 ≠ 0 )
3 cxpexpzd.3 ( 𝜑𝐵 ∈ ℤ )
4 cxpexpz ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℤ ) → ( 𝐴𝑐 𝐵 ) = ( 𝐴𝐵 ) )
5 1 2 3 4 syl3anc ( 𝜑 → ( 𝐴𝑐 𝐵 ) = ( 𝐴𝐵 ) )