Metamath Proof Explorer


Theorem expeq0d

Description: Positive integer exponentiation is 0 iff its base is 0. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses expcld.1 ( 𝜑𝐴 ∈ ℂ )
expeq0d.2 ( 𝜑𝑁 ∈ ℕ )
expeq0d.3 ( 𝜑 → ( 𝐴𝑁 ) = 0 )
Assertion expeq0d ( 𝜑𝐴 = 0 )

Proof

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