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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 )