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Metamath Proof Explorer

Theorem cjexpd

Description: Complex conjugate of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypotheses recld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
cjexpd.2 ⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
Assertion cjexpd ( 𝜑 → ( ∗ ‘ ( 𝐴 ↑ 𝑁 ) ) = ( ( ∗ ‘ 𝐴 ) ↑ 𝑁 ) )

Proof

Step Hyp Ref Expression
1 recld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2 cjexpd.2 ⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
3 cjexp ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) → ( ∗ ‘ ( 𝐴 ↑ 𝑁 ) ) = ( ( ∗ ‘ 𝐴 ) ↑ 𝑁 ) )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( ∗ ‘ ( 𝐴 ↑ 𝑁 ) ) = ( ( ∗ ‘ 𝐴 ) ↑ 𝑁 ) )