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Metamath Proof Explorer

Theorem exp0

Description: Value of a complex number raised to the zeroth power. Under our definition, 0 ^ 0 = 1 ( 0exp0e1 ), following standard convention, for instance Definition 10-4.1 of Gleason p. 134. (Contributed by NM, 20-May-2004) (Revised by Mario Carneiro, 4-Jun-2014)

Ref Expression
Assertion exp0 ⊢ A ∈ ℂ → A 0 = 1

Proof

Step Hyp Ref Expression
1 0z ⊢ 0 ∈ ℤ
2 expval ⊢ A ∈ ℂ ∧ 0 ∈ ℤ → A 0 = if 0 = 0 1 if 0 < 0 seq 1 × ℕ × A ⁡ 0 1 seq 1 × ℕ × A ⁡ − 0
3 1 2 mpan2 ⊢ A ∈ ℂ → A 0 = if 0 = 0 1 if 0 < 0 seq 1 × ℕ × A ⁡ 0 1 seq 1 × ℕ × A ⁡ − 0
4 eqid ⊢ 0 = 0
5 4 iftruei ⊢ if 0 = 0 1 if 0 < 0 seq 1 × ℕ × A ⁡ 0 1 seq 1 × ℕ × A ⁡ − 0 = 1
6 3 5 eqtrdi ⊢ A ∈ ℂ → A 0 = 1