Step |
Hyp |
Ref |
Expression |
1 |
|
oveq1 |
|- ( a = b -> ( a ^ N ) = ( b ^ N ) ) |
2 |
|
oveq1 |
|- ( a = A -> ( a ^ N ) = ( A ^ N ) ) |
3 |
|
oveq1 |
|- ( a = B -> ( a ^ N ) = ( B ^ N ) ) |
4 |
|
rpssre |
|- RR+ C_ RR |
5 |
|
rpre |
|- ( a e. RR+ -> a e. RR ) |
6 |
|
nnnn0 |
|- ( N e. NN -> N e. NN0 ) |
7 |
|
reexpcl |
|- ( ( a e. RR /\ N e. NN0 ) -> ( a ^ N ) e. RR ) |
8 |
5 6 7
|
syl2anr |
|- ( ( N e. NN /\ a e. RR+ ) -> ( a ^ N ) e. RR ) |
9 |
|
simplrl |
|- ( ( ( N e. NN /\ ( a e. RR+ /\ b e. RR+ ) ) /\ a < b ) -> a e. RR+ ) |
10 |
9
|
rpred |
|- ( ( ( N e. NN /\ ( a e. RR+ /\ b e. RR+ ) ) /\ a < b ) -> a e. RR ) |
11 |
|
simplrr |
|- ( ( ( N e. NN /\ ( a e. RR+ /\ b e. RR+ ) ) /\ a < b ) -> b e. RR+ ) |
12 |
11
|
rpred |
|- ( ( ( N e. NN /\ ( a e. RR+ /\ b e. RR+ ) ) /\ a < b ) -> b e. RR ) |
13 |
9
|
rpge0d |
|- ( ( ( N e. NN /\ ( a e. RR+ /\ b e. RR+ ) ) /\ a < b ) -> 0 <_ a ) |
14 |
|
simpr |
|- ( ( ( N e. NN /\ ( a e. RR+ /\ b e. RR+ ) ) /\ a < b ) -> a < b ) |
15 |
|
simpll |
|- ( ( ( N e. NN /\ ( a e. RR+ /\ b e. RR+ ) ) /\ a < b ) -> N e. NN ) |
16 |
|
expmordi |
|- ( ( ( a e. RR /\ b e. RR ) /\ ( 0 <_ a /\ a < b ) /\ N e. NN ) -> ( a ^ N ) < ( b ^ N ) ) |
17 |
10 12 13 14 15 16
|
syl221anc |
|- ( ( ( N e. NN /\ ( a e. RR+ /\ b e. RR+ ) ) /\ a < b ) -> ( a ^ N ) < ( b ^ N ) ) |
18 |
17
|
ex |
|- ( ( N e. NN /\ ( a e. RR+ /\ b e. RR+ ) ) -> ( a < b -> ( a ^ N ) < ( b ^ N ) ) ) |
19 |
1 2 3 4 8 18
|
ltord1 |
|- ( ( N e. NN /\ ( A e. RR+ /\ B e. RR+ ) ) -> ( A < B <-> ( A ^ N ) < ( B ^ N ) ) ) |
20 |
19
|
3impb |
|- ( ( N e. NN /\ A e. RR+ /\ B e. RR+ ) -> ( A < B <-> ( A ^ N ) < ( B ^ N ) ) ) |