Metamath Proof Explorer


Theorem exp0

Description: Value of a complex number raised to the 0th power. Note that under our definition, 0 ^ 0 = 1 , following the convention used by Gleason. Part of 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 )