Step |
Hyp |
Ref |
Expression |
1 |
|
exp11nnd.1 |
|- ( ph -> A e. RR+ ) |
2 |
|
exp11nnd.2 |
|- ( ph -> B e. RR+ ) |
3 |
|
exp11nnd.3 |
|- ( ph -> N e. NN ) |
4 |
|
exp11nnd.4 |
|- ( ph -> ( A ^ N ) = ( B ^ N ) ) |
5 |
1
|
rpred |
|- ( ph -> A e. RR ) |
6 |
3
|
nnnn0d |
|- ( ph -> N e. NN0 ) |
7 |
5 6
|
reexpcld |
|- ( ph -> ( A ^ N ) e. RR ) |
8 |
2
|
rpred |
|- ( ph -> B e. RR ) |
9 |
8 6
|
reexpcld |
|- ( ph -> ( B ^ N ) e. RR ) |
10 |
7 9
|
lttri3d |
|- ( ph -> ( ( A ^ N ) = ( B ^ N ) <-> ( -. ( A ^ N ) < ( B ^ N ) /\ -. ( B ^ N ) < ( A ^ N ) ) ) ) |
11 |
4 10
|
mpbid |
|- ( ph -> ( -. ( A ^ N ) < ( B ^ N ) /\ -. ( B ^ N ) < ( A ^ N ) ) ) |
12 |
1 2 3
|
ltexp1d |
|- ( ph -> ( A < B <-> ( A ^ N ) < ( B ^ N ) ) ) |
13 |
12
|
notbid |
|- ( ph -> ( -. A < B <-> -. ( A ^ N ) < ( B ^ N ) ) ) |
14 |
2 1 3
|
ltexp1d |
|- ( ph -> ( B < A <-> ( B ^ N ) < ( A ^ N ) ) ) |
15 |
14
|
notbid |
|- ( ph -> ( -. B < A <-> -. ( B ^ N ) < ( A ^ N ) ) ) |
16 |
13 15
|
anbi12d |
|- ( ph -> ( ( -. A < B /\ -. B < A ) <-> ( -. ( A ^ N ) < ( B ^ N ) /\ -. ( B ^ N ) < ( A ^ N ) ) ) ) |
17 |
11 16
|
mpbird |
|- ( ph -> ( -. A < B /\ -. B < A ) ) |
18 |
5 8
|
lttri3d |
|- ( ph -> ( A = B <-> ( -. A < B /\ -. B < A ) ) ) |
19 |
17 18
|
mpbird |
|- ( ph -> A = B ) |