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Theorem chpchtlim 22733
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 9406 . . 3  |-  ( T. 
->  1  e.  RR )
2 1red 9406 . . . . 5  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  1  e.  RR )
3 2re 10396 . . . . . . . . . . 11  |-  2  e.  RR
4 elicopnf 11390 . . . . . . . . . . 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 9392 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) +oo )  ->  0  e.  RR )
93a1i 11 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) +oo )  ->  2  e.  RR )
10 2pos 10418 . . . . . . . . . . . . 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 9536 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  0  < 
x )
146, 13elrpd 11030 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR+ )
1514adantl 466 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  x  e.  RR+ )
1615rpge0d 11036 . . . . . . . 8  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  0  <_  x )
177, 16resqrcld 12909 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  ( sqr `  x )  e.  RR )
1815relogcld 22077 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  ( log `  x )  e.  RR )
1917, 18remulcld 9419 . . . . . 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 22518 . . . . . . 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 11059 . . . . 5  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
)  e.  RR )
246ssriv 3365 . . . . . 6  |-  ( 2 [,) +oo )  C_  RR
251recnd 9417 . . . . . 6  |-  ( T. 
->  1  e.  CC )
26 rlimconst 13027 . . . . . 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 6121 . . . . . . . . 9  |-  ( 2 [,) +oo )  e. 
_V
2928a1i 11 . . . . . . . 8  |-  ( T. 
->  ( 2 [,) +oo )  e.  _V )
307, 22rerpdivcld 11059 . . . . . . . 8  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
x  /  ( theta `  x ) )  e.  RR )
31 ovex 6121 . . . . . . . . 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 2444 . . . . . . . 8  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( x  /  ( theta `  x ) ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( x  /  ( theta `  x
) ) ) )
347recnd 9417 . . . . . . . . . . . 12  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  x  e.  CC )
35 cxpsqr 22153 . . . . . . . . . . . 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 6112 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( log `  x
)  /  ( x  ^c  ( 1  /  2 ) ) )  =  ( ( log `  x )  /  ( sqr `  x
) ) )
3818recnd 9417 . . . . . . . . . . 11  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  ( log `  x )  e.  CC )
3915rpsqrcld 12903 . . . . . . . . . . . 12  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  ( sqr `  x )  e.  RR+ )
4039rpcnne0d 11041 . . . . . . . . . . 11  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( sqr `  x
)  e.  CC  /\  ( sqr `  x )  =/=  0 ) )
41 divcan5 10038 . . . . . . . . . . 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 1218 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( ( sqr `  x
)  x.  ( log `  x ) )  / 
( ( sqr `  x
)  x.  ( sqr `  x ) ) )  =  ( ( log `  x )  /  ( sqr `  x ) ) )
43 remsqsqr 12751 . . . . . . . . . . . 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 6112 . . . . . . . . . 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 2481 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( log `  x
)  /  ( x  ^c  ( 1  /  2 ) ) )  =  ( ( ( sqr `  x
)  x.  ( log `  x ) )  /  x ) )
4746mpteq2dva 4383 . . . . . . . 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 6341 . . . . . . 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 11037 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  x  =/=  0 )
5022rpcnne0d 11041 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  e.  CC  /\  ( theta `  x )  =/=  0
) )
5119recnd 9417 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( sqr `  x
)  x.  ( log `  x ) )  e.  CC )
52 dmdcan 10046 . . . . . . . . 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 1224 . . . . . . . 8  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( x  /  ( theta `  x ) )  x.  ( ( ( sqr `  x )  x.  ( log `  x
) )  /  x
) )  =  ( ( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) )
5453mpteq2dva 4383 . . . . . . 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 2475 . . . . . 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 22732 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  |->  ( x  /  ( theta `  x
) ) )  e.  O(1)
5714ssriv 3365 . . . . . . . . 9  |-  ( 2 [,) +oo )  C_  RR+
5857a1i 11 . . . . . . . 8  |-  ( T. 
->  ( 2 [,) +oo )  C_  RR+ )
59 1rp 11000 . . . . . . . . . . 11  |-  1  e.  RR+
60 rphalfcl 11020 . . . . . . . . . . 11  |-  ( 1  e.  RR+  ->  ( 1  /  2 )  e.  RR+ )
6159, 60ax-mp 5 . . . . . . . . . 10  |-  ( 1  /  2 )  e.  RR+
62 cxploglim 22376 . . . . . . . . . 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 13044 . . . . . . 7  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( log `  x
)  /  ( x  ^c  ( 1  /  2 ) ) ) )  ~~> r  0 )
66 o1rlimmul 13101 . . . . . . 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 4319 . . . . 5  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( ( sqr `  x )  x.  ( log `  x ) )  /  ( theta `  x
) ) )  ~~> r  0 )
692, 23, 27, 68rlimadd 13125 . . . 4  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( 1  +  ( ( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) ) )  ~~> r  ( 1  +  0 ) )
70 1p0e1 10439 . . . 4  |-  ( 1  +  0 )  =  1
7169, 70syl6breq 4336 . . 3  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( 1  +  ( ( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) ) )  ~~> r  1 )
72 1re 9390 . . . 4  |-  1  e.  RR
73 readdcl 9370 . . . 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 22467 . . . . 5  |-  ( x  e.  RR  ->  (ψ `  x )  e.  RR )
767, 75syl 16 . . . 4  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (ψ `  x )  e.  RR )
7776, 22rerpdivcld 11059 . . 3  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
(ψ `  x )  /  ( theta `  x
) )  e.  RR )
78 chtcl 22452 . . . . . . . 8  |-  ( x  e.  RR  ->  ( theta `  x )  e.  RR )
797, 78syl 16 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  ( theta `  x )  e.  RR )
80 readdcl 9370 . . . . . . 7  |-  ( ( ( theta `  x )  e.  RR  /\  ( ( sqr `  x )  x.  ( log `  x
) )  e.  RR )  ->  ( ( theta `  x )  +  ( ( sqr `  x
)  x.  ( log `  x ) ) )  e.  RR )
8179, 19, 80syl2anc 661 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  +  ( ( sqr `  x )  x.  ( log `  x ) ) )  e.  RR )
823a1i 11 . . . . . . . 8  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  2  e.  RR )
83 1le2 10540 . . . . . . . . 9  |-  1  <_  2
8483a1i 11 . . . . . . . 8  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  1  <_  2 )
852, 82, 7, 84, 20letrd 9533 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  1  <_  x )
86 chpub 22564 . . . . . . 7  |-  ( ( x  e.  RR  /\  1  <_  x )  -> 
(ψ `  x )  <_  ( ( theta `  x
)  +  ( ( sqr `  x )  x.  ( log `  x
) ) ) )
877, 85, 86syl2anc 661 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (ψ `  x )  <_  (
( theta `  x )  +  ( ( sqr `  x )  x.  ( log `  x ) ) ) )
8876, 81, 22, 87lediv1dd 11086 . . . . 5  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
(ψ `  x )  /  ( theta `  x
) )  <_  (
( ( theta `  x
)  +  ( ( sqr `  x )  x.  ( log `  x
) ) )  / 
( theta `  x )
) )
8922rpcnd 11034 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  ( theta `  x )  e.  CC )
90 divdir 10022 . . . . . . 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
) ) ) )
9189, 51, 50, 90syl3anc 1218 . . . . . 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 )
) ) )
92 divid 10026 . . . . . . . 8  |-  ( ( ( theta `  x )  e.  CC  /\  ( theta `  x )  =/=  0
)  ->  ( ( theta `  x )  / 
( theta `  x )
)  =  1 )
9350, 92syl 16 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  ( theta `  x
) )  =  1 )
9493oveq1d 6111 . . . . . 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 )
) ) )
9591, 94eqtrd 2475 . . . . 5  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( ( theta `  x
)  +  ( ( sqr `  x )  x.  ( log `  x
) ) )  / 
( theta `  x )
)  =  ( 1  +  ( ( ( sqr `  x )  x.  ( log `  x
) )  /  ( theta `  x ) ) ) )
9688, 95breqtrd 4321 . . . 4  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
(ψ `  x )  /  ( theta `  x
) )  <_  (
1  +  ( ( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) ) )
9796adantrr 716 . . 3  |-  ( ( T.  /\  ( x  e.  ( 2 [,) +oo )  /\  1  <_  x ) )  -> 
( (ψ `  x
)  /  ( theta `  x ) )  <_ 
( 1  +  ( ( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) ) )
9889mulid2d 9409 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
1  x.  ( theta `  x ) )  =  ( theta `  x )
)
99 chtlepsi 22550 . . . . . . 7  |-  ( x  e.  RR  ->  ( theta `  x )  <_ 
(ψ `  x )
)
1007, 99syl 16 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  ( theta `  x )  <_ 
(ψ `  x )
)
10198, 100eqbrtrd 4317 . . . . 5  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
1  x.  ( theta `  x ) )  <_ 
(ψ `  x )
)
1022, 76, 22lemuldivd 11077 . . . . 5  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( 1  x.  ( theta `  x ) )  <_  (ψ `  x
)  <->  1  <_  (
(ψ `  x )  /  ( theta `  x
) ) ) )
103101, 102mpbid 210 . . . 4  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  1  <_  ( (ψ `  x
)  /  ( theta `  x ) ) )
104103adantrr 716 . . 3  |-  ( ( T.  /\  ( x  e.  ( 2 [,) +oo )  /\  1  <_  x ) )  -> 
1  <_  ( (ψ `  x )  /  ( theta `  x ) ) )
1051, 1, 71, 74, 77, 97, 104rlimsqz2 13133 . 2  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( (ψ `  x
)  /  ( theta `  x ) ) )  ~~> r  1 )
106105trud 1378 1  |-  ( x  e.  ( 2 [,) +oo )  |->  ( (ψ `  x )  /  ( theta `  x ) ) )  ~~> r  1
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369   T. wtru 1370    e. wcel 1756    =/= wne 2611   _Vcvv 2977    C_ wss 3333   class class class wbr 4297    e. cmpt 4355   ` cfv 5423  (class class class)co 6096    oFcof 6323   CCcc 9285   RRcr 9286   0cc0 9287   1c1 9288    + caddc 9290    x. cmul 9292   +oocpnf 9420    < clt 9423    <_ cle 9424    / cdiv 9998   2c2 10376   RR+crp 10996   [,)cico 11307   sqrcsqr 12727    ~~> r crli 12968   O(1)co1 12969   logclog 22011    ^c ccxp 22012   thetaccht 22433  ψcchp 22435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365  ax-addf 9366  ax-mulf 9367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-fi 7666  df-sup 7696  df-oi 7729  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-ioo 11309  df-ioc 11310  df-ico 11311  df-icc 11312  df-fz 11443  df-fzo 11554  df-fl 11647  df-mod 11714  df-seq 11812  df-exp 11871  df-fac 12057  df-bc 12084  df-hash 12109  df-shft 12561  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-limsup 12954  df-clim 12971  df-rlim 12972  df-o1 12973  df-lo1 12974  df-sum 13169  df-ef 13358  df-e 13359  df-sin 13360  df-cos 13361  df-pi 13363  df-dvds 13541  df-gcd 13696  df-prm 13769  df-pc 13909  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-starv 14258  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-unif 14266  df-hom 14267  df-cco 14268  df-rest 14366  df-topn 14367  df-0g 14385  df-gsum 14386  df-topgen 14387  df-pt 14388  df-prds 14391  df-xrs 14445  df-qtop 14450  df-imas 14451  df-xps 14453  df-mre 14529  df-mrc 14530  df-acs 14532  df-mnd 15420  df-submnd 15470  df-mulg 15553  df-cntz 15840  df-cmn 16284  df-psmet 17814  df-xmet 17815  df-met 17816  df-bl 17817  df-mopn 17818  df-fbas 17819  df-fg 17820  df-cnfld 17824  df-top 18508  df-bases 18510  df-topon 18511  df-topsp 18512  df-cld 18628  df-ntr 18629  df-cls 18630  df-nei 18707  df-lp 18745  df-perf 18746  df-cn 18836  df-cnp 18837  df-haus 18924  df-tx 19140  df-hmeo 19333  df-fil 19424  df-fm 19516  df-flim 19517  df-flf 19518  df-xms 19900  df-ms 19901  df-tms 19902  df-cncf 20459  df-limc 21346  df-dv 21347  df-log 22013  df-cxp 22014  df-cht 22439  df-vma 22440  df-chp 22441  df-ppi 22442
This theorem is referenced by:  chpo1ub  22734  pnt2  22867
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