Theorem expeq0d

Description: Positive integer exponentiation is 0 iff its mantissa is 0. (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 )