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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 ∈ ℕ 0
Assertion cjexpd ⊢ φ → A N ‾ = A ‾ N

Proof

Step Hyp Ref Expression
1 recld.1 ⊢ φ → A ∈ ℂ
2 cjexpd.2 ⊢ φ → N ∈ ℕ 0
3 cjexp ⊢ A ∈ ℂ ∧ N ∈ ℕ 0 → A N ‾ = A ‾ N
4 1 2 3 syl2anc ⊢ φ → A N ‾ = A ‾ N