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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 )