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Metamath Proof Explorer

Theorem exp1

Description: Value of a complex number raised to the first power. (Contributed by NM, 20-Oct-2004) (Revised by Mario Carneiro, 2-Jul-2013)

Ref Expression
Assertion exp1 ( 𝐴 ∈ ℂ → ( 𝐴 ↑ 1 ) = 𝐴 )

Proof

Step Hyp Ref Expression
1 1nn ⊢ 1 ∈ ℕ
2 expnnval ⊢ ( ( 𝐴 ∈ ℂ ∧ 1 ∈ ℕ ) → ( 𝐴 ↑ 1 ) = ( seq 1 ( · , ( ℕ × { 𝐴 } ) ) ‘ 1 ) )
3 1 2 mpan2 ⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ↑ 1 ) = ( seq 1 ( · , ( ℕ × { 𝐴 } ) ) ‘ 1 ) )
4 1z ⊢ 1 ∈ ℤ
5 seq1 ⊢ ( 1 ∈ ℤ → ( seq 1 ( · , ( ℕ × { 𝐴 } ) ) ‘ 1 ) = ( ( ℕ × { 𝐴 } ) ‘ 1 ) )
6 4 5 ax-mp ⊢ ( seq 1 ( · , ( ℕ × { 𝐴 } ) ) ‘ 1 ) = ( ( ℕ × { 𝐴 } ) ‘ 1 )
7 3 6 eqtrdi ⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ↑ 1 ) = ( ( ℕ × { 𝐴 } ) ‘ 1 ) )
8 fvconst2g ⊢ ( ( 𝐴 ∈ ℂ ∧ 1 ∈ ℕ ) → ( ( ℕ × { 𝐴 } ) ‘ 1 ) = 𝐴 )
9 1 8 mpan2 ⊢ ( 𝐴 ∈ ℂ → ( ( ℕ × { 𝐴 } ) ‘ 1 ) = 𝐴 )
10 7 9 eqtrd ⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ↑ 1 ) = 𝐴 )