Step |
Hyp |
Ref |
Expression |
1 |
|
2re |
|- 2 e. RR |
2 |
|
elicopnf |
|- ( 2 e. RR -> ( x e. ( 2 [,) +oo ) <-> ( x e. RR /\ 2 <_ x ) ) ) |
3 |
1 2
|
ax-mp |
|- ( x e. ( 2 [,) +oo ) <-> ( x e. RR /\ 2 <_ x ) ) |
4 |
|
chprpcl |
|- ( ( x e. RR /\ 2 <_ x ) -> ( psi ` x ) e. RR+ ) |
5 |
3 4
|
sylbi |
|- ( x e. ( 2 [,) +oo ) -> ( psi ` x ) e. RR+ ) |
6 |
3
|
simplbi |
|- ( x e. ( 2 [,) +oo ) -> x e. RR ) |
7 |
|
0red |
|- ( x e. ( 2 [,) +oo ) -> 0 e. RR ) |
8 |
1
|
a1i |
|- ( x e. ( 2 [,) +oo ) -> 2 e. RR ) |
9 |
|
2pos |
|- 0 < 2 |
10 |
9
|
a1i |
|- ( x e. ( 2 [,) +oo ) -> 0 < 2 ) |
11 |
3
|
simprbi |
|- ( x e. ( 2 [,) +oo ) -> 2 <_ x ) |
12 |
7 8 6 10 11
|
ltletrd |
|- ( x e. ( 2 [,) +oo ) -> 0 < x ) |
13 |
6 12
|
elrpd |
|- ( x e. ( 2 [,) +oo ) -> x e. RR+ ) |
14 |
5 13
|
rpdivcld |
|- ( x e. ( 2 [,) +oo ) -> ( ( psi ` x ) / x ) e. RR+ ) |
15 |
14
|
adantl |
|- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( psi ` x ) / x ) e. RR+ ) |
16 |
|
chtrpcl |
|- ( ( x e. RR /\ 2 <_ x ) -> ( theta ` x ) e. RR+ ) |
17 |
3 16
|
sylbi |
|- ( x e. ( 2 [,) +oo ) -> ( theta ` x ) e. RR+ ) |
18 |
5 17
|
rpdivcld |
|- ( x e. ( 2 [,) +oo ) -> ( ( psi ` x ) / ( theta ` x ) ) e. RR+ ) |
19 |
18
|
adantl |
|- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( psi ` x ) / ( theta ` x ) ) e. RR+ ) |
20 |
13
|
ssriv |
|- ( 2 [,) +oo ) C_ RR+ |
21 |
20
|
a1i |
|- ( T. -> ( 2 [,) +oo ) C_ RR+ ) |
22 |
|
pnt3 |
|- ( x e. RR+ |-> ( ( psi ` x ) / x ) ) ~~>r 1 |
23 |
22
|
a1i |
|- ( T. -> ( x e. RR+ |-> ( ( psi ` x ) / x ) ) ~~>r 1 ) |
24 |
21 23
|
rlimres2 |
|- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( ( psi ` x ) / x ) ) ~~>r 1 ) |
25 |
|
chpchtlim |
|- ( x e. ( 2 [,) +oo ) |-> ( ( psi ` x ) / ( theta ` x ) ) ) ~~>r 1 |
26 |
25
|
a1i |
|- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( ( psi ` x ) / ( theta ` x ) ) ) ~~>r 1 ) |
27 |
|
ax-1ne0 |
|- 1 =/= 0 |
28 |
27
|
a1i |
|- ( T. -> 1 =/= 0 ) |
29 |
19
|
rpne0d |
|- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( psi ` x ) / ( theta ` x ) ) =/= 0 ) |
30 |
15 19 24 26 28 29
|
rlimdiv |
|- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( ( ( psi ` x ) / x ) / ( ( psi ` x ) / ( theta ` x ) ) ) ) ~~>r ( 1 / 1 ) ) |
31 |
|
rpre |
|- ( x e. RR+ -> x e. RR ) |
32 |
|
chpcl |
|- ( x e. RR -> ( psi ` x ) e. RR ) |
33 |
31 32
|
syl |
|- ( x e. RR+ -> ( psi ` x ) e. RR ) |
34 |
33
|
recnd |
|- ( x e. RR+ -> ( psi ` x ) e. CC ) |
35 |
13 34
|
syl |
|- ( x e. ( 2 [,) +oo ) -> ( psi ` x ) e. CC ) |
36 |
13
|
rpcnne0d |
|- ( x e. ( 2 [,) +oo ) -> ( x e. CC /\ x =/= 0 ) ) |
37 |
5
|
rpcnne0d |
|- ( x e. ( 2 [,) +oo ) -> ( ( psi ` x ) e. CC /\ ( psi ` x ) =/= 0 ) ) |
38 |
17
|
rpcnne0d |
|- ( x e. ( 2 [,) +oo ) -> ( ( theta ` x ) e. CC /\ ( theta ` x ) =/= 0 ) ) |
39 |
|
divdivdiv |
|- ( ( ( ( psi ` x ) e. CC /\ ( x e. CC /\ x =/= 0 ) ) /\ ( ( ( psi ` x ) e. CC /\ ( psi ` x ) =/= 0 ) /\ ( ( theta ` x ) e. CC /\ ( theta ` x ) =/= 0 ) ) ) -> ( ( ( psi ` x ) / x ) / ( ( psi ` x ) / ( theta ` x ) ) ) = ( ( ( psi ` x ) x. ( theta ` x ) ) / ( x x. ( psi ` x ) ) ) ) |
40 |
35 36 37 38 39
|
syl22anc |
|- ( x e. ( 2 [,) +oo ) -> ( ( ( psi ` x ) / x ) / ( ( psi ` x ) / ( theta ` x ) ) ) = ( ( ( psi ` x ) x. ( theta ` x ) ) / ( x x. ( psi ` x ) ) ) ) |
41 |
6
|
recnd |
|- ( x e. ( 2 [,) +oo ) -> x e. CC ) |
42 |
41 35
|
mulcomd |
|- ( x e. ( 2 [,) +oo ) -> ( x x. ( psi ` x ) ) = ( ( psi ` x ) x. x ) ) |
43 |
42
|
oveq2d |
|- ( x e. ( 2 [,) +oo ) -> ( ( ( psi ` x ) x. ( theta ` x ) ) / ( x x. ( psi ` x ) ) ) = ( ( ( psi ` x ) x. ( theta ` x ) ) / ( ( psi ` x ) x. x ) ) ) |
44 |
|
chtcl |
|- ( x e. RR -> ( theta ` x ) e. RR ) |
45 |
31 44
|
syl |
|- ( x e. RR+ -> ( theta ` x ) e. RR ) |
46 |
45
|
recnd |
|- ( x e. RR+ -> ( theta ` x ) e. CC ) |
47 |
13 46
|
syl |
|- ( x e. ( 2 [,) +oo ) -> ( theta ` x ) e. CC ) |
48 |
|
divcan5 |
|- ( ( ( theta ` x ) e. CC /\ ( x e. CC /\ x =/= 0 ) /\ ( ( psi ` x ) e. CC /\ ( psi ` x ) =/= 0 ) ) -> ( ( ( psi ` x ) x. ( theta ` x ) ) / ( ( psi ` x ) x. x ) ) = ( ( theta ` x ) / x ) ) |
49 |
47 36 37 48
|
syl3anc |
|- ( x e. ( 2 [,) +oo ) -> ( ( ( psi ` x ) x. ( theta ` x ) ) / ( ( psi ` x ) x. x ) ) = ( ( theta ` x ) / x ) ) |
50 |
40 43 49
|
3eqtrd |
|- ( x e. ( 2 [,) +oo ) -> ( ( ( psi ` x ) / x ) / ( ( psi ` x ) / ( theta ` x ) ) ) = ( ( theta ` x ) / x ) ) |
51 |
50
|
mpteq2ia |
|- ( x e. ( 2 [,) +oo ) |-> ( ( ( psi ` x ) / x ) / ( ( psi ` x ) / ( theta ` x ) ) ) ) = ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / x ) ) |
52 |
|
resmpt |
|- ( ( 2 [,) +oo ) C_ RR+ -> ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) |` ( 2 [,) +oo ) ) = ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / x ) ) ) |
53 |
20 52
|
ax-mp |
|- ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) |` ( 2 [,) +oo ) ) = ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / x ) ) |
54 |
51 53
|
eqtr4i |
|- ( x e. ( 2 [,) +oo ) |-> ( ( ( psi ` x ) / x ) / ( ( psi ` x ) / ( theta ` x ) ) ) ) = ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) |` ( 2 [,) +oo ) ) |
55 |
|
1div1e1 |
|- ( 1 / 1 ) = 1 |
56 |
30 54 55
|
3brtr3g |
|- ( T. -> ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) |` ( 2 [,) +oo ) ) ~~>r 1 ) |
57 |
|
rerpdivcl |
|- ( ( ( theta ` x ) e. RR /\ x e. RR+ ) -> ( ( theta ` x ) / x ) e. RR ) |
58 |
45 57
|
mpancom |
|- ( x e. RR+ -> ( ( theta ` x ) / x ) e. RR ) |
59 |
58
|
adantl |
|- ( ( T. /\ x e. RR+ ) -> ( ( theta ` x ) / x ) e. RR ) |
60 |
59
|
recnd |
|- ( ( T. /\ x e. RR+ ) -> ( ( theta ` x ) / x ) e. CC ) |
61 |
60
|
fmpttd |
|- ( T. -> ( x e. RR+ |-> ( ( theta ` x ) / x ) ) : RR+ --> CC ) |
62 |
|
rpssre |
|- RR+ C_ RR |
63 |
62
|
a1i |
|- ( T. -> RR+ C_ RR ) |
64 |
1
|
a1i |
|- ( T. -> 2 e. RR ) |
65 |
61 63 64
|
rlimresb |
|- ( T. -> ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) ~~>r 1 <-> ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) |` ( 2 [,) +oo ) ) ~~>r 1 ) ) |
66 |
56 65
|
mpbird |
|- ( T. -> ( x e. RR+ |-> ( ( theta ` x ) / x ) ) ~~>r 1 ) |
67 |
66
|
mptru |
|- ( x e. RR+ |-> ( ( theta ` x ) / x ) ) ~~>r 1 |