Step |
Hyp |
Ref |
Expression |
1 |
|
prmnn |
|- ( P e. Prime -> P e. NN ) |
2 |
|
nnexpcl |
|- ( ( P e. NN /\ K e. NN0 ) -> ( P ^ K ) e. NN ) |
3 |
1 2
|
sylan |
|- ( ( P e. Prime /\ K e. NN0 ) -> ( P ^ K ) e. NN ) |
4 |
|
0sgm |
|- ( ( P ^ K ) e. NN -> ( 0 sigma ( P ^ K ) ) = ( # ` { x e. NN | x || ( P ^ K ) } ) ) |
5 |
3 4
|
syl |
|- ( ( P e. Prime /\ K e. NN0 ) -> ( 0 sigma ( P ^ K ) ) = ( # ` { x e. NN | x || ( P ^ K ) } ) ) |
6 |
|
fzfid |
|- ( ( P e. Prime /\ K e. NN0 ) -> ( 0 ... K ) e. Fin ) |
7 |
|
eqid |
|- ( n e. ( 0 ... K ) |-> ( P ^ n ) ) = ( n e. ( 0 ... K ) |-> ( P ^ n ) ) |
8 |
7
|
dvdsppwf1o |
|- ( ( P e. Prime /\ K e. NN0 ) -> ( n e. ( 0 ... K ) |-> ( P ^ n ) ) : ( 0 ... K ) -1-1-onto-> { x e. NN | x || ( P ^ K ) } ) |
9 |
6 8
|
hasheqf1od |
|- ( ( P e. Prime /\ K e. NN0 ) -> ( # ` ( 0 ... K ) ) = ( # ` { x e. NN | x || ( P ^ K ) } ) ) |
10 |
5 9
|
eqtr4d |
|- ( ( P e. Prime /\ K e. NN0 ) -> ( 0 sigma ( P ^ K ) ) = ( # ` ( 0 ... K ) ) ) |
11 |
|
simpr |
|- ( ( P e. Prime /\ K e. NN0 ) -> K e. NN0 ) |
12 |
|
nn0uz |
|- NN0 = ( ZZ>= ` 0 ) |
13 |
11 12
|
eleqtrdi |
|- ( ( P e. Prime /\ K e. NN0 ) -> K e. ( ZZ>= ` 0 ) ) |
14 |
|
hashfz |
|- ( K e. ( ZZ>= ` 0 ) -> ( # ` ( 0 ... K ) ) = ( ( K - 0 ) + 1 ) ) |
15 |
13 14
|
syl |
|- ( ( P e. Prime /\ K e. NN0 ) -> ( # ` ( 0 ... K ) ) = ( ( K - 0 ) + 1 ) ) |
16 |
|
nn0cn |
|- ( K e. NN0 -> K e. CC ) |
17 |
16
|
adantl |
|- ( ( P e. Prime /\ K e. NN0 ) -> K e. CC ) |
18 |
17
|
subid1d |
|- ( ( P e. Prime /\ K e. NN0 ) -> ( K - 0 ) = K ) |
19 |
18
|
oveq1d |
|- ( ( P e. Prime /\ K e. NN0 ) -> ( ( K - 0 ) + 1 ) = ( K + 1 ) ) |
20 |
10 15 19
|
3eqtrd |
|- ( ( P e. Prime /\ K e. NN0 ) -> ( 0 sigma ( P ^ K ) ) = ( K + 1 ) ) |