Step |
Hyp |
Ref |
Expression |
1 |
|
rpre |
|- ( x e. RR+ -> x e. RR ) |
2 |
1
|
adantl |
|- ( ( A e. RR+ /\ x e. RR+ ) -> x e. RR ) |
3 |
|
rpge0 |
|- ( x e. RR+ -> 0 <_ x ) |
4 |
3
|
adantl |
|- ( ( A e. RR+ /\ x e. RR+ ) -> 0 <_ x ) |
5 |
|
rpre |
|- ( A e. RR+ -> A e. RR ) |
6 |
5
|
renegcld |
|- ( A e. RR+ -> -u A e. RR ) |
7 |
6
|
adantr |
|- ( ( A e. RR+ /\ x e. RR+ ) -> -u A e. RR ) |
8 |
|
rpcn |
|- ( A e. RR+ -> A e. CC ) |
9 |
|
rpne0 |
|- ( A e. RR+ -> A =/= 0 ) |
10 |
8 9
|
negne0d |
|- ( A e. RR+ -> -u A =/= 0 ) |
11 |
10
|
adantr |
|- ( ( A e. RR+ /\ x e. RR+ ) -> -u A =/= 0 ) |
12 |
7 11
|
rereccld |
|- ( ( A e. RR+ /\ x e. RR+ ) -> ( 1 / -u A ) e. RR ) |
13 |
2 4 12
|
recxpcld |
|- ( ( A e. RR+ /\ x e. RR+ ) -> ( x ^c ( 1 / -u A ) ) e. RR ) |
14 |
|
simprl |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> n e. RR+ ) |
15 |
5
|
ad2antrr |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> A e. RR ) |
16 |
14 15
|
rpcxpcld |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( n ^c A ) e. RR+ ) |
17 |
16
|
rpreccld |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( 1 / ( n ^c A ) ) e. RR+ ) |
18 |
17
|
rprege0d |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( ( 1 / ( n ^c A ) ) e. RR /\ 0 <_ ( 1 / ( n ^c A ) ) ) ) |
19 |
|
absid |
|- ( ( ( 1 / ( n ^c A ) ) e. RR /\ 0 <_ ( 1 / ( n ^c A ) ) ) -> ( abs ` ( 1 / ( n ^c A ) ) ) = ( 1 / ( n ^c A ) ) ) |
20 |
18 19
|
syl |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( abs ` ( 1 / ( n ^c A ) ) ) = ( 1 / ( n ^c A ) ) ) |
21 |
|
simplr |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> x e. RR+ ) |
22 |
|
simprr |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( x ^c ( 1 / -u A ) ) < n ) |
23 |
|
rpreccl |
|- ( A e. RR+ -> ( 1 / A ) e. RR+ ) |
24 |
23
|
ad2antrr |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( 1 / A ) e. RR+ ) |
25 |
24
|
rpcnd |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( 1 / A ) e. CC ) |
26 |
21 25
|
cxprecd |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( ( 1 / x ) ^c ( 1 / A ) ) = ( 1 / ( x ^c ( 1 / A ) ) ) ) |
27 |
|
rpcn |
|- ( x e. RR+ -> x e. CC ) |
28 |
27
|
ad2antlr |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> x e. CC ) |
29 |
|
rpne0 |
|- ( x e. RR+ -> x =/= 0 ) |
30 |
29
|
ad2antlr |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> x =/= 0 ) |
31 |
28 30 25
|
cxpnegd |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( x ^c -u ( 1 / A ) ) = ( 1 / ( x ^c ( 1 / A ) ) ) ) |
32 |
|
1cnd |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> 1 e. CC ) |
33 |
8
|
ad2antrr |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> A e. CC ) |
34 |
9
|
ad2antrr |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> A =/= 0 ) |
35 |
32 33 34
|
divneg2d |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> -u ( 1 / A ) = ( 1 / -u A ) ) |
36 |
35
|
oveq2d |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( x ^c -u ( 1 / A ) ) = ( x ^c ( 1 / -u A ) ) ) |
37 |
26 31 36
|
3eqtr2d |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( ( 1 / x ) ^c ( 1 / A ) ) = ( x ^c ( 1 / -u A ) ) ) |
38 |
33 34
|
recidd |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( A x. ( 1 / A ) ) = 1 ) |
39 |
38
|
oveq2d |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( n ^c ( A x. ( 1 / A ) ) ) = ( n ^c 1 ) ) |
40 |
14 15 25
|
cxpmuld |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( n ^c ( A x. ( 1 / A ) ) ) = ( ( n ^c A ) ^c ( 1 / A ) ) ) |
41 |
14
|
rpcnd |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> n e. CC ) |
42 |
41
|
cxp1d |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( n ^c 1 ) = n ) |
43 |
39 40 42
|
3eqtr3d |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( ( n ^c A ) ^c ( 1 / A ) ) = n ) |
44 |
22 37 43
|
3brtr4d |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( ( 1 / x ) ^c ( 1 / A ) ) < ( ( n ^c A ) ^c ( 1 / A ) ) ) |
45 |
|
rpreccl |
|- ( x e. RR+ -> ( 1 / x ) e. RR+ ) |
46 |
45
|
ad2antlr |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( 1 / x ) e. RR+ ) |
47 |
46
|
rpred |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( 1 / x ) e. RR ) |
48 |
46
|
rpge0d |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> 0 <_ ( 1 / x ) ) |
49 |
16
|
rpred |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( n ^c A ) e. RR ) |
50 |
16
|
rpge0d |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> 0 <_ ( n ^c A ) ) |
51 |
47 48 49 50 24
|
cxplt2d |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( ( 1 / x ) < ( n ^c A ) <-> ( ( 1 / x ) ^c ( 1 / A ) ) < ( ( n ^c A ) ^c ( 1 / A ) ) ) ) |
52 |
44 51
|
mpbird |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( 1 / x ) < ( n ^c A ) ) |
53 |
21 16 52
|
ltrec1d |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( 1 / ( n ^c A ) ) < x ) |
54 |
20 53
|
eqbrtrd |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( abs ` ( 1 / ( n ^c A ) ) ) < x ) |
55 |
54
|
expr |
|- ( ( ( A e. RR+ /\ x e. RR+ ) /\ n e. RR+ ) -> ( ( x ^c ( 1 / -u A ) ) < n -> ( abs ` ( 1 / ( n ^c A ) ) ) < x ) ) |
56 |
55
|
ralrimiva |
|- ( ( A e. RR+ /\ x e. RR+ ) -> A. n e. RR+ ( ( x ^c ( 1 / -u A ) ) < n -> ( abs ` ( 1 / ( n ^c A ) ) ) < x ) ) |
57 |
|
breq1 |
|- ( y = ( x ^c ( 1 / -u A ) ) -> ( y < n <-> ( x ^c ( 1 / -u A ) ) < n ) ) |
58 |
57
|
rspceaimv |
|- ( ( ( x ^c ( 1 / -u A ) ) e. RR /\ A. n e. RR+ ( ( x ^c ( 1 / -u A ) ) < n -> ( abs ` ( 1 / ( n ^c A ) ) ) < x ) ) -> E. y e. RR A. n e. RR+ ( y < n -> ( abs ` ( 1 / ( n ^c A ) ) ) < x ) ) |
59 |
13 56 58
|
syl2anc |
|- ( ( A e. RR+ /\ x e. RR+ ) -> E. y e. RR A. n e. RR+ ( y < n -> ( abs ` ( 1 / ( n ^c A ) ) ) < x ) ) |
60 |
59
|
ralrimiva |
|- ( A e. RR+ -> A. x e. RR+ E. y e. RR A. n e. RR+ ( y < n -> ( abs ` ( 1 / ( n ^c A ) ) ) < x ) ) |
61 |
|
id |
|- ( n e. RR+ -> n e. RR+ ) |
62 |
|
rpcxpcl |
|- ( ( n e. RR+ /\ A e. RR ) -> ( n ^c A ) e. RR+ ) |
63 |
61 5 62
|
syl2anr |
|- ( ( A e. RR+ /\ n e. RR+ ) -> ( n ^c A ) e. RR+ ) |
64 |
63
|
rpreccld |
|- ( ( A e. RR+ /\ n e. RR+ ) -> ( 1 / ( n ^c A ) ) e. RR+ ) |
65 |
64
|
rpcnd |
|- ( ( A e. RR+ /\ n e. RR+ ) -> ( 1 / ( n ^c A ) ) e. CC ) |
66 |
65
|
ralrimiva |
|- ( A e. RR+ -> A. n e. RR+ ( 1 / ( n ^c A ) ) e. CC ) |
67 |
|
rpssre |
|- RR+ C_ RR |
68 |
67
|
a1i |
|- ( A e. RR+ -> RR+ C_ RR ) |
69 |
66 68
|
rlim0lt |
|- ( A e. RR+ -> ( ( n e. RR+ |-> ( 1 / ( n ^c A ) ) ) ~~>r 0 <-> A. x e. RR+ E. y e. RR A. n e. RR+ ( y < n -> ( abs ` ( 1 / ( n ^c A ) ) ) < x ) ) ) |
70 |
60 69
|
mpbird |
|- ( A e. RR+ -> ( n e. RR+ |-> ( 1 / ( n ^c A ) ) ) ~~>r 0 ) |