Metamath Proof Explorer

Theorem fac1

Description: The factorial of 1. (Contributed by NM, 2-Dec-2004) (Revised by Mario Carneiro, 13-Jul-2013)

Ref Expression
Assertion fac1 
|- ( ! ` 1 ) = 1

Proof

Step Hyp Ref Expression
1 1nn 
 |-  1 e. NN
2 facnn 
 |-  ( 1 e. NN -> ( ! ` 1 ) = ( seq 1 ( x. , _I ) ` 1 ) )
3 1 2 ax-mp 
 |-  ( ! ` 1 ) = ( seq 1 ( x. , _I ) ` 1 )
4 1z 
 |-  1 e. ZZ
5 seq1 
 |-  ( 1 e. ZZ -> ( seq 1 ( x. , _I ) ` 1 ) = ( _I ` 1 ) )
6 4 5 ax-mp 
 |-  ( seq 1 ( x. , _I ) ` 1 ) = ( _I ` 1 )
7 fvi 
 |-  ( 1 e. NN -> ( _I ` 1 ) = 1 )
8 1 7 ax-mp 
 |-  ( _I ` 1 ) = 1
9 3 6 8 3eqtri 
 |-  ( ! ` 1 ) = 1