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Metamath Proof Explorer

Theorem expp1zd

Description: Value of a nonzero complex number raised to an integer power plus one. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses expcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
sqrecd.1 ⊢ ( 𝜑 → 𝐴 ≠ 0 )
expclzd.3 ⊢ ( 𝜑 → 𝑁 ∈ ℤ )
Assertion expp1zd ( 𝜑 → ( 𝐴 ↑ ( 𝑁 + 1 ) ) = ( ( 𝐴 ↑ 𝑁 ) · 𝐴 ) )

Proof

Step Hyp Ref Expression
1 expcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2 sqrecd.1 ⊢ ( 𝜑 → 𝐴 ≠ 0 )
3 expclzd.3 ⊢ ( 𝜑 → 𝑁 ∈ ℤ )
4 expp1z ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ ( 𝑁 + 1 ) ) = ( ( 𝐴 ↑ 𝑁 ) · 𝐴 ) )
5 1 2 3 4 syl3anc ⊢ ( 𝜑 → ( 𝐴 ↑ ( 𝑁 + 1 ) ) = ( ( 𝐴 ↑ 𝑁 ) · 𝐴 ) )