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Metamath Proof Explorer

Theorem expeq0d

Description: If a positive integer power is zero, then its base is zero. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses expcld.1 ⊢ φ → A ∈ ℂ
expeq0d.2 ⊢ φ → N ∈ ℕ
expeq0d.3 ⊢ φ → A N = 0
Assertion expeq0d ⊢ φ → A = 0

Proof

Step Hyp Ref Expression
1 expcld.1 ⊢ φ → A ∈ ℂ
2 expeq0d.2 ⊢ φ → N ∈ ℕ
3 expeq0d.3 ⊢ φ → A N = 0
4 expeq0 ⊢ A ∈ ℂ ∧ N ∈ ℕ → A N = 0 ↔ A = 0
5 1 2 4 syl2anc ⊢ φ → A N = 0 ↔ A = 0
6 3 5 mpbid ⊢ φ → A = 0