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Theorem chpchtlim 23528
Description: The ψ and  theta functions are asymptotic to each other, so is sufficient to prove either 
theta ( x )  /  x 
~~> r  1 or ψ ( x )  /  x  ~~> r  1 to establish the PNT. (Contributed by Mario Carneiro, 8-Apr-2016.)
Assertion
Ref Expression
chpchtlim  |-  ( x  e.  ( 2 [,) +oo )  |->  ( (ψ `  x )  /  ( theta `  x ) ) )  ~~> r  1

Proof of Theorem chpchtlim
StepHypRef Expression
1 1red 9623 . . 3  |-  ( T. 
->  1  e.  RR )
2 1red 9623 . . . . 5  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  1  e.  RR )
3 2re 10617 . . . . . . . . . . 11  |-  2  e.  RR
4 elicopnf 11632 . . . . . . . . . . 11  |-  ( 2  e.  RR  ->  (
x  e.  ( 2 [,) +oo )  <->  ( x  e.  RR  /\  2  <_  x ) ) )
53, 4ax-mp 5 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  <->  ( x  e.  RR  /\  2  <_  x ) )
65simplbi 460 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR )
76adantl 466 . . . . . . . 8  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  x  e.  RR )
8 0red 9609 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) +oo )  ->  0  e.  RR )
93a1i 11 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) +oo )  ->  2  e.  RR )
10 2pos 10639 . . . . . . . . . . . . 13  |-  0  <  2
1110a1i 11 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) +oo )  ->  0  <  2 )
125simprbi 464 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) +oo )  ->  2  <_  x )
138, 9, 6, 11, 12ltletrd 9753 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  0  < 
x )
146, 13elrpd 11266 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR+ )
1514adantl 466 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  x  e.  RR+ )
1615rpge0d 11272 . . . . . . . 8  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  0  <_  x )
177, 16resqrtcld 13228 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  ( sqr `  x )  e.  RR )
1815relogcld 22872 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  ( log `  x )  e.  RR )
1917, 18remulcld 9636 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( sqr `  x
)  x.  ( log `  x ) )  e.  RR )
2012adantl 466 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  2  <_  x )
21 chtrpcl 23313 . . . . . . 7  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
( theta `  x )  e.  RR+ )
227, 20, 21syl2anc 661 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  ( theta `  x )  e.  RR+ )
2319, 22rerpdivcld 11295 . . . . 5  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
)  e.  RR )
246ssriv 3513 . . . . . 6  |-  ( 2 [,) +oo )  C_  RR
251recnd 9634 . . . . . 6  |-  ( T. 
->  1  e.  CC )
26 rlimconst 13346 . . . . . 6  |-  ( ( ( 2 [,) +oo )  C_  RR  /\  1  e.  CC )  ->  (
x  e.  ( 2 [,) +oo )  |->  1 )  ~~> r  1 )
2724, 25, 26sylancr 663 . . . . 5  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  1 )  ~~> r  1 )
28 ovex 6320 . . . . . . . . 9  |-  ( 2 [,) +oo )  e. 
_V
2928a1i 11 . . . . . . . 8  |-  ( T. 
->  ( 2 [,) +oo )  e.  _V )
307, 22rerpdivcld 11295 . . . . . . . 8  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
x  /  ( theta `  x ) )  e.  RR )
31 ovex 6320 . . . . . . . . 9  |-  ( ( ( sqr `  x
)  x.  ( log `  x ) )  /  x )  e.  _V
3231a1i 11 . . . . . . . 8  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( ( sqr `  x
)  x.  ( log `  x ) )  /  x )  e.  _V )
33 eqidd 2468 . . . . . . . 8  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( x  /  ( theta `  x ) ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( x  /  ( theta `  x
) ) ) )
347recnd 9634 . . . . . . . . . . . 12  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  x  e.  CC )
35 cxpsqrt 22948 . . . . . . . . . . . 12  |-  ( x  e.  CC  ->  (
x  ^c  ( 1  /  2 ) )  =  ( sqr `  x ) )
3634, 35syl 16 . . . . . . . . . . 11  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
x  ^c  ( 1  /  2 ) )  =  ( sqr `  x ) )
3736oveq2d 6311 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( log `  x
)  /  ( x  ^c  ( 1  /  2 ) ) )  =  ( ( log `  x )  /  ( sqr `  x
) ) )
3818recnd 9634 . . . . . . . . . . 11  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  ( log `  x )  e.  CC )
3915rpsqrtcld 13222 . . . . . . . . . . . 12  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  ( sqr `  x )  e.  RR+ )
4039rpcnne0d 11277 . . . . . . . . . . 11  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( sqr `  x
)  e.  CC  /\  ( sqr `  x )  =/=  0 ) )
41 divcan5 10258 . . . . . . . . . . 11  |-  ( ( ( log `  x
)  e.  CC  /\  ( ( sqr `  x
)  e.  CC  /\  ( sqr `  x )  =/=  0 )  /\  ( ( sqr `  x
)  e.  CC  /\  ( sqr `  x )  =/=  0 ) )  ->  ( ( ( sqr `  x )  x.  ( log `  x
) )  /  (
( sqr `  x
)  x.  ( sqr `  x ) ) )  =  ( ( log `  x )  /  ( sqr `  x ) ) )
4238, 40, 40, 41syl3anc 1228 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( ( sqr `  x
)  x.  ( log `  x ) )  / 
( ( sqr `  x
)  x.  ( sqr `  x ) ) )  =  ( ( log `  x )  /  ( sqr `  x ) ) )
43 remsqsqrt 13069 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  0  <_  x )  -> 
( ( sqr `  x
)  x.  ( sqr `  x ) )  =  x )
447, 16, 43syl2anc 661 . . . . . . . . . . 11  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( sqr `  x
)  x.  ( sqr `  x ) )  =  x )
4544oveq2d 6311 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( ( sqr `  x
)  x.  ( log `  x ) )  / 
( ( sqr `  x
)  x.  ( sqr `  x ) ) )  =  ( ( ( sqr `  x )  x.  ( log `  x
) )  /  x
) )
4637, 42, 453eqtr2d 2514 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( log `  x
)  /  ( x  ^c  ( 1  /  2 ) ) )  =  ( ( ( sqr `  x
)  x.  ( log `  x ) )  /  x ) )
4746mpteq2dva 4539 . . . . . . . 8  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( log `  x
)  /  ( x  ^c  ( 1  /  2 ) ) ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( ( ( sqr `  x
)  x.  ( log `  x ) )  /  x ) ) )
4829, 30, 32, 33, 47offval2 6551 . . . . . . 7  |-  ( T. 
->  ( ( x  e.  ( 2 [,) +oo )  |->  ( x  / 
( theta `  x )
) )  oF  x.  ( x  e.  ( 2 [,) +oo )  |->  ( ( log `  x )  /  (
x  ^c  ( 1  /  2 ) ) ) ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( ( x  /  ( theta `  x
) )  x.  (
( ( sqr `  x
)  x.  ( log `  x ) )  /  x ) ) ) )
4915rpne0d 11273 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  x  =/=  0 )
5022rpcnne0d 11277 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  e.  CC  /\  ( theta `  x )  =/=  0
) )
5119recnd 9634 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( sqr `  x
)  x.  ( log `  x ) )  e.  CC )
52 dmdcan 10266 . . . . . . . . 9  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( ( theta `  x )  e.  CC  /\  ( theta `  x )  =/=  0 )  /\  (
( sqr `  x
)  x.  ( log `  x ) )  e.  CC )  ->  (
( x  /  ( theta `  x ) )  x.  ( ( ( sqr `  x )  x.  ( log `  x
) )  /  x
) )  =  ( ( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) )
5334, 49, 50, 51, 52syl211anc 1234 . . . . . . . 8  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( x  /  ( theta `  x ) )  x.  ( ( ( sqr `  x )  x.  ( log `  x
) )  /  x
) )  =  ( ( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) )
5453mpteq2dva 4539 . . . . . . 7  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( x  / 
( theta `  x )
)  x.  ( ( ( sqr `  x
)  x.  ( log `  x ) )  /  x ) ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( ( ( sqr `  x )  x.  ( log `  x
) )  /  ( theta `  x ) ) ) )
5548, 54eqtrd 2508 . . . . . 6  |-  ( T. 
->  ( ( x  e.  ( 2 [,) +oo )  |->  ( x  / 
( theta `  x )
) )  oF  x.  ( x  e.  ( 2 [,) +oo )  |->  ( ( log `  x )  /  (
x  ^c  ( 1  /  2 ) ) ) ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( ( ( sqr `  x )  x.  ( log `  x
) )  /  ( theta `  x ) ) ) )
56 chto1lb 23527 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  |->  ( x  /  ( theta `  x
) ) )  e.  O(1)
5714ssriv 3513 . . . . . . . . 9  |-  ( 2 [,) +oo )  C_  RR+
5857a1i 11 . . . . . . . 8  |-  ( T. 
->  ( 2 [,) +oo )  C_  RR+ )
59 1rp 11236 . . . . . . . . . . 11  |-  1  e.  RR+
60 rphalfcl 11256 . . . . . . . . . . 11  |-  ( 1  e.  RR+  ->  ( 1  /  2 )  e.  RR+ )
6159, 60ax-mp 5 . . . . . . . . . 10  |-  ( 1  /  2 )  e.  RR+
62 cxploglim 23171 . . . . . . . . . 10  |-  ( ( 1  /  2 )  e.  RR+  ->  ( x  e.  RR+  |->  ( ( log `  x )  /  ( x  ^c  ( 1  / 
2 ) ) ) )  ~~> r  0 )
6361, 62ax-mp 5 . . . . . . . . 9  |-  ( x  e.  RR+  |->  ( ( log `  x )  /  ( x  ^c  ( 1  / 
2 ) ) ) )  ~~> r  0
6463a1i 11 . . . . . . . 8  |-  ( T. 
->  ( x  e.  RR+  |->  ( ( log `  x
)  /  ( x  ^c  ( 1  /  2 ) ) ) )  ~~> r  0 )
6558, 64rlimres2 13363 . . . . . . 7  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( log `  x
)  /  ( x  ^c  ( 1  /  2 ) ) ) )  ~~> r  0 )
66 o1rlimmul 13420 . . . . . . 7  |-  ( ( ( x  e.  ( 2 [,) +oo )  |->  ( x  /  ( theta `  x ) ) )  e.  O(1)  /\  (
x  e.  ( 2 [,) +oo )  |->  ( ( log `  x
)  /  ( x  ^c  ( 1  /  2 ) ) ) )  ~~> r  0 )  ->  ( (
x  e.  ( 2 [,) +oo )  |->  ( x  /  ( theta `  x ) ) )  oF  x.  (
x  e.  ( 2 [,) +oo )  |->  ( ( log `  x
)  /  ( x  ^c  ( 1  /  2 ) ) ) ) )  ~~> r  0 )
6756, 65, 66sylancr 663 . . . . . 6  |-  ( T. 
->  ( ( x  e.  ( 2 [,) +oo )  |->  ( x  / 
( theta `  x )
) )  oF  x.  ( x  e.  ( 2 [,) +oo )  |->  ( ( log `  x )  /  (
x  ^c  ( 1  /  2 ) ) ) ) )  ~~> r  0 )
6855, 67eqbrtrrd 4475 . . . . 5  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( ( sqr `  x )  x.  ( log `  x ) )  /  ( theta `  x
) ) )  ~~> r  0 )
692, 23, 27, 68rlimadd 13444 . . . 4  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( 1  +  ( ( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) ) )  ~~> r  ( 1  +  0 ) )
70 1p0e1 10660 . . . 4  |-  ( 1  +  0 )  =  1
7169, 70syl6breq 4492 . . 3  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( 1  +  ( ( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) ) )  ~~> r  1 )
72 1re 9607 . . . 4  |-  1  e.  RR
73 readdcl 9587 . . . 4  |-  ( ( 1  e.  RR  /\  ( ( ( sqr `  x )  x.  ( log `  x ) )  /  ( theta `  x
) )  e.  RR )  ->  ( 1  +  ( ( ( sqr `  x )  x.  ( log `  x ) )  /  ( theta `  x
) ) )  e.  RR )
7472, 23, 73sylancr 663 . . 3  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
1  +  ( ( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) )  e.  RR )
75 chpcl 23262 . . . . 5  |-  ( x  e.  RR  ->  (ψ `  x )  e.  RR )
767, 75syl 16 . . . 4  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (ψ `  x )  e.  RR )
7776, 22rerpdivcld 11295 . . 3  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
(ψ `  x )  /  ( theta `  x
) )  e.  RR )
78 chtcl 23247 . . . . . . . 8  |-  ( x  e.  RR  ->  ( theta `  x )  e.  RR )
797, 78syl 16 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  ( theta `  x )  e.  RR )
8079, 19readdcld 9635 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  +  ( ( sqr `  x )  x.  ( log `  x ) ) )  e.  RR )
813a1i 11 . . . . . . . 8  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  2  e.  RR )
82 1le2 10761 . . . . . . . . 9  |-  1  <_  2
8382a1i 11 . . . . . . . 8  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  1  <_  2 )
842, 81, 7, 83, 20letrd 9750 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  1  <_  x )
85 chpub 23359 . . . . . . 7  |-  ( ( x  e.  RR  /\  1  <_  x )  -> 
(ψ `  x )  <_  ( ( theta `  x
)  +  ( ( sqr `  x )  x.  ( log `  x
) ) ) )
867, 84, 85syl2anc 661 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (ψ `  x )  <_  (
( theta `  x )  +  ( ( sqr `  x )  x.  ( log `  x ) ) ) )
8776, 80, 22, 86lediv1dd 11322 . . . . 5  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
(ψ `  x )  /  ( theta `  x
) )  <_  (
( ( theta `  x
)  +  ( ( sqr `  x )  x.  ( log `  x
) ) )  / 
( theta `  x )
) )
8822rpcnd 11270 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  ( theta `  x )  e.  CC )
89 divdir 10242 . . . . . . 7  |-  ( ( ( theta `  x )  e.  CC  /\  ( ( sqr `  x )  x.  ( log `  x
) )  e.  CC  /\  ( ( theta `  x
)  e.  CC  /\  ( theta `  x )  =/=  0 ) )  -> 
( ( ( theta `  x )  +  ( ( sqr `  x
)  x.  ( log `  x ) ) )  /  ( theta `  x
) )  =  ( ( ( theta `  x
)  /  ( theta `  x ) )  +  ( ( ( sqr `  x )  x.  ( log `  x ) )  /  ( theta `  x
) ) ) )
9088, 51, 50, 89syl3anc 1228 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( ( theta `  x
)  +  ( ( sqr `  x )  x.  ( log `  x
) ) )  / 
( theta `  x )
)  =  ( ( ( theta `  x )  /  ( theta `  x
) )  +  ( ( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) ) )
91 divid 10246 . . . . . . . 8  |-  ( ( ( theta `  x )  e.  CC  /\  ( theta `  x )  =/=  0
)  ->  ( ( theta `  x )  / 
( theta `  x )
)  =  1 )
9250, 91syl 16 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  ( theta `  x
) )  =  1 )
9392oveq1d 6310 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( ( theta `  x
)  /  ( theta `  x ) )  +  ( ( ( sqr `  x )  x.  ( log `  x ) )  /  ( theta `  x
) ) )  =  ( 1  +  ( ( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) ) )
9490, 93eqtrd 2508 . . . . 5  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( ( theta `  x
)  +  ( ( sqr `  x )  x.  ( log `  x
) ) )  / 
( theta `  x )
)  =  ( 1  +  ( ( ( sqr `  x )  x.  ( log `  x
) )  /  ( theta `  x ) ) ) )
9587, 94breqtrd 4477 . . . 4  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
(ψ `  x )  /  ( theta `  x
) )  <_  (
1  +  ( ( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) ) )
9695adantrr 716 . . 3  |-  ( ( T.  /\  ( x  e.  ( 2 [,) +oo )  /\  1  <_  x ) )  -> 
( (ψ `  x
)  /  ( theta `  x ) )  <_ 
( 1  +  ( ( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) ) )
9788mulid2d 9626 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
1  x.  ( theta `  x ) )  =  ( theta `  x )
)
98 chtlepsi 23345 . . . . . . 7  |-  ( x  e.  RR  ->  ( theta `  x )  <_ 
(ψ `  x )
)
997, 98syl 16 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  ( theta `  x )  <_ 
(ψ `  x )
)
10097, 99eqbrtrd 4473 . . . . 5  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
1  x.  ( theta `  x ) )  <_ 
(ψ `  x )
)
1012, 76, 22lemuldivd 11313 . . . . 5  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( 1  x.  ( theta `  x ) )  <_  (ψ `  x
)  <->  1  <_  (
(ψ `  x )  /  ( theta `  x
) ) ) )
102100, 101mpbid 210 . . . 4  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  1  <_  ( (ψ `  x
)  /  ( theta `  x ) ) )
103102adantrr 716 . . 3  |-  ( ( T.  /\  ( x  e.  ( 2 [,) +oo )  /\  1  <_  x ) )  -> 
1  <_  ( (ψ `  x )  /  ( theta `  x ) ) )
1041, 1, 71, 74, 77, 96, 103rlimsqz2 13452 . 2  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( (ψ `  x
)  /  ( theta `  x ) ) )  ~~> r  1 )
105104trud 1388 1  |-  ( x  e.  ( 2 [,) +oo )  |->  ( (ψ `  x )  /  ( theta `  x ) ) )  ~~> r  1
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379   T. wtru 1380    e. wcel 1767    =/= wne 2662   _Vcvv 3118    C_ wss 3481   class class class wbr 4453    |-> cmpt 4511   ` cfv 5594  (class class class)co 6295    oFcof 6533   CCcc 9502   RRcr 9503   0cc0 9504   1c1 9505    + caddc 9507    x. cmul 9509   +oocpnf 9637    < clt 9640    <_ cle 9641    / cdiv 10218   2c2 10597   RR+crp 11232   [,)cico 11543   sqrcsqrt 13045    ~~> r crli 13287   O(1)co1 13288   logclog 22806    ^c ccxp 22807   thetaccht 23228  ψcchp 23230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-ioc 11546  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-fac 12334  df-bc 12361  df-hash 12386  df-shft 12879  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-limsup 13273  df-clim 13290  df-rlim 13291  df-o1 13292  df-lo1 13293  df-sum 13488  df-ef 13681  df-e 13682  df-sin 13683  df-cos 13684  df-pi 13686  df-dvds 13864  df-gcd 14020  df-prm 14093  df-pc 14236  df-struct 14508  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-mulr 14585  df-starv 14586  df-sca 14587  df-vsca 14588  df-ip 14589  df-tset 14590  df-ple 14591  df-ds 14593  df-unif 14594  df-hom 14595  df-cco 14596  df-rest 14694  df-topn 14695  df-0g 14713  df-gsum 14714  df-topgen 14715  df-pt 14716  df-prds 14719  df-xrs 14773  df-qtop 14778  df-imas 14779  df-xps 14781  df-mre 14857  df-mrc 14858  df-acs 14860  df-mgm 15745  df-sgrp 15784  df-mnd 15794  df-submnd 15839  df-mulg 15931  df-cntz 16226  df-cmn 16671  df-psmet 18279  df-xmet 18280  df-met 18281  df-bl 18282  df-mopn 18283  df-fbas 18284  df-fg 18285  df-cnfld 18289  df-top 19266  df-bases 19268  df-topon 19269  df-topsp 19270  df-cld 19386  df-ntr 19387  df-cls 19388  df-nei 19465  df-lp 19503  df-perf 19504  df-cn 19594  df-cnp 19595  df-haus 19682  df-tx 19929  df-hmeo 20122  df-fil 20213  df-fm 20305  df-flim 20306  df-flf 20307  df-xms 20689  df-ms 20690  df-tms 20691  df-cncf 21248  df-limc 22136  df-dv 22137  df-log 22808  df-cxp 22809  df-cht 23234  df-vma 23235  df-chp 23236  df-ppi 23237
This theorem is referenced by:  chpo1ub  23529  pnt2  23662
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