Metamath Proof Explorer

Theorem i4

Description: _i to the fourth power. (Contributed by NM, 31-Jan-2007)

Ref Expression
Assertion i4 
|- ( _i ^ 4 ) = 1

Proof

Step Hyp Ref Expression
1 ax-icn 
 |-  _i e. CC
2 2nn0 
 |-  2 e. NN0
3 expadd 
 |-  ( ( _i e. CC /\ 2 e. NN0 /\ 2 e. NN0 ) -> ( _i ^ ( 2 + 2 ) ) = ( ( _i ^ 2 ) x. ( _i ^ 2 ) ) )
4 1 2 2 3 mp3an 
 |-  ( _i ^ ( 2 + 2 ) ) = ( ( _i ^ 2 ) x. ( _i ^ 2 ) )
5 2p2e4 
 |-  ( 2 + 2 ) = 4
6 5 oveq2i 
 |-  ( _i ^ ( 2 + 2 ) ) = ( _i ^ 4 )
7 i2 
 |-  ( _i ^ 2 ) = -u 1
8 7 7 oveq12i 
 |-  ( ( _i ^ 2 ) x. ( _i ^ 2 ) ) = ( -u 1 x. -u 1 )
9 ax-1cn 
 |-  1 e. CC
10 9 9 mul2negi 
 |-  ( -u 1 x. -u 1 ) = ( 1 x. 1 )
11 1t1e1 
 |-  ( 1 x. 1 ) = 1
12 8 10 11 3eqtri 
 |-  ( ( _i ^ 2 ) x. ( _i ^ 2 ) ) = 1
13 4 6 12 3eqtr3i 
 |-  ( _i ^ 4 ) = 1