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 A 1 = A

Proof

Step Hyp Ref Expression
1 1nn 1
2 expnnval A 1 A 1 = seq 1 × × A 1
3 1 2 mpan2 A A 1 = seq 1 × × A 1
4 1z 1
5 seq1 1 seq 1 × × A 1 = × A 1
6 4 5 ax-mp seq 1 × × A 1 = × A 1
7 3 6 eqtrdi A A 1 = × A 1
8 fvconst2g A 1 × A 1 = A
9 1 8 mpan2 A × A 1 = A
10 7 9 eqtrd A A 1 = A