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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 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
cxpefd.2 ⊢ ( 𝜑 → 𝐴 ≠ 0 )
cxpexpzd.3 ⊢ ( 𝜑 → 𝐵 ∈ ℤ )
Assertion cxpexpzd ( 𝜑 → ( 𝐴 ↑𝑐 𝐵 ) = ( 𝐴 ↑ 𝐵 ) )

Proof

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