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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 e. CC -> ( A ^ 1 ) = A )

Proof

Step Hyp Ref Expression
1 1nn 
 |-  1 e. NN
2 expnnval 
 |-  ( ( A e. CC /\ 1 e. NN ) -> ( A ^ 1 ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` 1 ) )
3 1 2 mpan2 
 |-  ( A e. CC -> ( A ^ 1 ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` 1 ) )
4 1z 
 |-  1 e. ZZ
5 seq1 
 |-  ( 1 e. ZZ -> ( seq 1 ( x. , ( NN X. { A } ) ) ` 1 ) = ( ( NN X. { A } ) ` 1 ) )
6 4 5 ax-mp 
 |-  ( seq 1 ( x. , ( NN X. { A } ) ) ` 1 ) = ( ( NN X. { A } ) ` 1 )
7 3 6 eqtrdi 
 |-  ( A e. CC -> ( A ^ 1 ) = ( ( NN X. { A } ) ` 1 ) )
8 fvconst2g 
 |-  ( ( A e. CC /\ 1 e. NN ) -> ( ( NN X. { A } ) ` 1 ) = A )
9 1 8 mpan2 
 |-  ( A e. CC -> ( ( NN X. { A } ) ` 1 ) = A )
10 7 9 eqtrd 
 |-  ( A e. CC -> ( A ^ 1 ) = A )