| Step |
Hyp |
Ref |
Expression |
| 1 |
|
fveq2 |
|- ( ( Lam ` A ) = 0 -> ( exp ` ( Lam ` A ) ) = ( exp ` 0 ) ) |
| 2 |
|
ef0 |
|- ( exp ` 0 ) = 1 |
| 3 |
1 2
|
eqtrdi |
|- ( ( Lam ` A ) = 0 -> ( exp ` ( Lam ` A ) ) = 1 ) |
| 4 |
3
|
eleq1d |
|- ( ( Lam ` A ) = 0 -> ( ( exp ` ( Lam ` A ) ) e. NN <-> 1 e. NN ) ) |
| 5 |
|
isppw2 |
|- ( A e. NN -> ( ( Lam ` A ) =/= 0 <-> E. p e. Prime E. k e. NN A = ( p ^ k ) ) ) |
| 6 |
|
vmappw |
|- ( ( p e. Prime /\ k e. NN ) -> ( Lam ` ( p ^ k ) ) = ( log ` p ) ) |
| 7 |
6
|
fveq2d |
|- ( ( p e. Prime /\ k e. NN ) -> ( exp ` ( Lam ` ( p ^ k ) ) ) = ( exp ` ( log ` p ) ) ) |
| 8 |
|
prmnn |
|- ( p e. Prime -> p e. NN ) |
| 9 |
8
|
nnrpd |
|- ( p e. Prime -> p e. RR+ ) |
| 10 |
9
|
reeflogd |
|- ( p e. Prime -> ( exp ` ( log ` p ) ) = p ) |
| 11 |
10 8
|
eqeltrd |
|- ( p e. Prime -> ( exp ` ( log ` p ) ) e. NN ) |
| 12 |
11
|
adantr |
|- ( ( p e. Prime /\ k e. NN ) -> ( exp ` ( log ` p ) ) e. NN ) |
| 13 |
7 12
|
eqeltrd |
|- ( ( p e. Prime /\ k e. NN ) -> ( exp ` ( Lam ` ( p ^ k ) ) ) e. NN ) |
| 14 |
|
fveq2 |
|- ( A = ( p ^ k ) -> ( Lam ` A ) = ( Lam ` ( p ^ k ) ) ) |
| 15 |
14
|
fveq2d |
|- ( A = ( p ^ k ) -> ( exp ` ( Lam ` A ) ) = ( exp ` ( Lam ` ( p ^ k ) ) ) ) |
| 16 |
15
|
eleq1d |
|- ( A = ( p ^ k ) -> ( ( exp ` ( Lam ` A ) ) e. NN <-> ( exp ` ( Lam ` ( p ^ k ) ) ) e. NN ) ) |
| 17 |
13 16
|
syl5ibrcom |
|- ( ( p e. Prime /\ k e. NN ) -> ( A = ( p ^ k ) -> ( exp ` ( Lam ` A ) ) e. NN ) ) |
| 18 |
17
|
rexlimivv |
|- ( E. p e. Prime E. k e. NN A = ( p ^ k ) -> ( exp ` ( Lam ` A ) ) e. NN ) |
| 19 |
5 18
|
biimtrdi |
|- ( A e. NN -> ( ( Lam ` A ) =/= 0 -> ( exp ` ( Lam ` A ) ) e. NN ) ) |
| 20 |
19
|
imp |
|- ( ( A e. NN /\ ( Lam ` A ) =/= 0 ) -> ( exp ` ( Lam ` A ) ) e. NN ) |
| 21 |
|
1nn |
|- 1 e. NN |
| 22 |
21
|
a1i |
|- ( A e. NN -> 1 e. NN ) |
| 23 |
4 20 22
|
pm2.61ne |
|- ( A e. NN -> ( exp ` ( Lam ` A ) ) e. NN ) |