Step |
Hyp |
Ref |
Expression |
1 |
|
simpl |
|- ( ( A e. RR /\ 1 < A ) -> A e. RR ) |
2 |
|
0lt1 |
|- 0 < 1 |
3 |
|
0re |
|- 0 e. RR |
4 |
|
1re |
|- 1 e. RR |
5 |
|
lttr |
|- ( ( 0 e. RR /\ 1 e. RR /\ A e. RR ) -> ( ( 0 < 1 /\ 1 < A ) -> 0 < A ) ) |
6 |
3 4 5
|
mp3an12 |
|- ( A e. RR -> ( ( 0 < 1 /\ 1 < A ) -> 0 < A ) ) |
7 |
2 6
|
mpani |
|- ( A e. RR -> ( 1 < A -> 0 < A ) ) |
8 |
7
|
imp |
|- ( ( A e. RR /\ 1 < A ) -> 0 < A ) |
9 |
1 8
|
elrpd |
|- ( ( A e. RR /\ 1 < A ) -> A e. RR+ ) |
10 |
9
|
3ad2ant1 |
|- ( ( ( A e. RR /\ 1 < A ) /\ N e. ZZ /\ B e. RR+ ) -> A e. RR+ ) |
11 |
|
simp2 |
|- ( ( ( A e. RR /\ 1 < A ) /\ N e. ZZ /\ B e. RR+ ) -> N e. ZZ ) |
12 |
|
reexplog |
|- ( ( A e. RR+ /\ N e. ZZ ) -> ( A ^ N ) = ( exp ` ( N x. ( log ` A ) ) ) ) |
13 |
10 11 12
|
syl2anc |
|- ( ( ( A e. RR /\ 1 < A ) /\ N e. ZZ /\ B e. RR+ ) -> ( A ^ N ) = ( exp ` ( N x. ( log ` A ) ) ) ) |
14 |
|
reeflog |
|- ( B e. RR+ -> ( exp ` ( log ` B ) ) = B ) |
15 |
14
|
3ad2ant3 |
|- ( ( ( A e. RR /\ 1 < A ) /\ N e. ZZ /\ B e. RR+ ) -> ( exp ` ( log ` B ) ) = B ) |
16 |
15
|
eqcomd |
|- ( ( ( A e. RR /\ 1 < A ) /\ N e. ZZ /\ B e. RR+ ) -> B = ( exp ` ( log ` B ) ) ) |
17 |
13 16
|
breq12d |
|- ( ( ( A e. RR /\ 1 < A ) /\ N e. ZZ /\ B e. RR+ ) -> ( ( A ^ N ) <_ B <-> ( exp ` ( N x. ( log ` A ) ) ) <_ ( exp ` ( log ` B ) ) ) ) |
18 |
|
zre |
|- ( N e. ZZ -> N e. RR ) |
19 |
18
|
3ad2ant2 |
|- ( ( ( A e. RR /\ 1 < A ) /\ N e. ZZ /\ B e. RR+ ) -> N e. RR ) |
20 |
|
rplogcl |
|- ( ( A e. RR /\ 1 < A ) -> ( log ` A ) e. RR+ ) |
21 |
20
|
3ad2ant1 |
|- ( ( ( A e. RR /\ 1 < A ) /\ N e. ZZ /\ B e. RR+ ) -> ( log ` A ) e. RR+ ) |
22 |
21
|
rpred |
|- ( ( ( A e. RR /\ 1 < A ) /\ N e. ZZ /\ B e. RR+ ) -> ( log ` A ) e. RR ) |
23 |
19 22
|
remulcld |
|- ( ( ( A e. RR /\ 1 < A ) /\ N e. ZZ /\ B e. RR+ ) -> ( N x. ( log ` A ) ) e. RR ) |
24 |
|
relogcl |
|- ( B e. RR+ -> ( log ` B ) e. RR ) |
25 |
24
|
3ad2ant3 |
|- ( ( ( A e. RR /\ 1 < A ) /\ N e. ZZ /\ B e. RR+ ) -> ( log ` B ) e. RR ) |
26 |
|
efle |
|- ( ( ( N x. ( log ` A ) ) e. RR /\ ( log ` B ) e. RR ) -> ( ( N x. ( log ` A ) ) <_ ( log ` B ) <-> ( exp ` ( N x. ( log ` A ) ) ) <_ ( exp ` ( log ` B ) ) ) ) |
27 |
23 25 26
|
syl2anc |
|- ( ( ( A e. RR /\ 1 < A ) /\ N e. ZZ /\ B e. RR+ ) -> ( ( N x. ( log ` A ) ) <_ ( log ` B ) <-> ( exp ` ( N x. ( log ` A ) ) ) <_ ( exp ` ( log ` B ) ) ) ) |
28 |
19 25 21
|
lemuldivd |
|- ( ( ( A e. RR /\ 1 < A ) /\ N e. ZZ /\ B e. RR+ ) -> ( ( N x. ( log ` A ) ) <_ ( log ` B ) <-> N <_ ( ( log ` B ) / ( log ` A ) ) ) ) |
29 |
25 21
|
rerpdivcld |
|- ( ( ( A e. RR /\ 1 < A ) /\ N e. ZZ /\ B e. RR+ ) -> ( ( log ` B ) / ( log ` A ) ) e. RR ) |
30 |
|
flge |
|- ( ( ( ( log ` B ) / ( log ` A ) ) e. RR /\ N e. ZZ ) -> ( N <_ ( ( log ` B ) / ( log ` A ) ) <-> N <_ ( |_ ` ( ( log ` B ) / ( log ` A ) ) ) ) ) |
31 |
29 11 30
|
syl2anc |
|- ( ( ( A e. RR /\ 1 < A ) /\ N e. ZZ /\ B e. RR+ ) -> ( N <_ ( ( log ` B ) / ( log ` A ) ) <-> N <_ ( |_ ` ( ( log ` B ) / ( log ` A ) ) ) ) ) |
32 |
28 31
|
bitrd |
|- ( ( ( A e. RR /\ 1 < A ) /\ N e. ZZ /\ B e. RR+ ) -> ( ( N x. ( log ` A ) ) <_ ( log ` B ) <-> N <_ ( |_ ` ( ( log ` B ) / ( log ` A ) ) ) ) ) |
33 |
17 27 32
|
3bitr2d |
|- ( ( ( A e. RR /\ 1 < A ) /\ N e. ZZ /\ B e. RR+ ) -> ( ( A ^ N ) <_ B <-> N <_ ( |_ ` ( ( log ` B ) / ( log ` A ) ) ) ) ) |