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Metamath Proof Explorer

Theorem 0cxpd

Description: Value of the complex power function when the first argument is zero. (Contributed by Mario Carneiro, 30-May-2016)

Ref Expression
Hypotheses cxp0d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
cxpefd.2 ⊢ ( 𝜑 → 𝐴 ≠ 0 )
Assertion 0cxpd ( 𝜑 → ( 0 ↑𝑐 𝐴 ) = 0 )

Proof

Step Hyp Ref Expression
1 cxp0d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2 cxpefd.2 ⊢ ( 𝜑 → 𝐴 ≠ 0 )
3 0cxp ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 0 ↑𝑐 𝐴 ) = 0 )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( 0 ↑𝑐 𝐴 ) = 0 )