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	syl6eq	⊢ ( 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 )