| Step |
Hyp |
Ref |
Expression |
| 1 |
|
1nn0 |
|- 1 e. NN0 |
| 2 |
|
prmoval |
|- ( 1 e. NN0 -> ( #p ` 1 ) = prod_ k e. ( 1 ... 1 ) if ( k e. Prime , k , 1 ) ) |
| 3 |
1 2
|
ax-mp |
|- ( #p ` 1 ) = prod_ k e. ( 1 ... 1 ) if ( k e. Prime , k , 1 ) |
| 4 |
|
1z |
|- 1 e. ZZ |
| 5 |
|
ax-1cn |
|- 1 e. CC |
| 6 |
|
1nprm |
|- -. 1 e. Prime |
| 7 |
|
eleq1 |
|- ( k = 1 -> ( k e. Prime <-> 1 e. Prime ) ) |
| 8 |
6 7
|
mtbiri |
|- ( k = 1 -> -. k e. Prime ) |
| 9 |
8
|
iffalsed |
|- ( k = 1 -> if ( k e. Prime , k , 1 ) = 1 ) |
| 10 |
9
|
fprod1 |
|- ( ( 1 e. ZZ /\ 1 e. CC ) -> prod_ k e. ( 1 ... 1 ) if ( k e. Prime , k , 1 ) = 1 ) |
| 11 |
4 5 10
|
mp2an |
|- prod_ k e. ( 1 ... 1 ) if ( k e. Prime , k , 1 ) = 1 |
| 12 |
3 11
|
eqtri |
|- ( #p ` 1 ) = 1 |