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Theorem chpchtlim 20740
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 1re 8927 . . . 4  |-  1  e.  RR
21a1i 10 . . 3  |-  (  T. 
->  1  e.  RR )
31a1i 10 . . . . 5  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  e.  RR )
4 2re 9905 . . . . . . . . . . 11  |-  2  e.  RR
5 elicopnf 10831 . . . . . . . . . . 11  |-  ( 2  e.  RR  ->  (
x  e.  ( 2 [,)  +oo )  <->  ( x  e.  RR  /\  2  <_  x ) ) )
64, 5ax-mp 8 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  <->  ( x  e.  RR  /\  2  <_  x ) )
76simplbi 446 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  RR )
87adantl 452 . . . . . . . 8  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  x  e.  RR )
9 0re 8928 . . . . . . . . . . . . 13  |-  0  e.  RR
109a1i 10 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) 
+oo )  ->  0  e.  RR )
114a1i 10 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) 
+oo )  ->  2  e.  RR )
12 2pos 9918 . . . . . . . . . . . . 13  |-  0  <  2
1312a1i 10 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) 
+oo )  ->  0  <  2 )
146simprbi 450 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) 
+oo )  ->  2  <_  x )
1510, 11, 7, 13, 14ltletrd 9066 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  0  <  x )
167, 15elrpd 10480 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  RR+ )
1716adantl 452 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  x  e.  RR+ )
1817rpge0d 10486 . . . . . . . 8  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  0  <_  x )
198, 18resqrcld 11996 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( sqr `  x )  e.  RR )
2017relogcld 20085 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( log `  x )  e.  RR )
2119, 20remulcld 8953 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( sqr `  x
)  x.  ( log `  x ) )  e.  RR )
2214adantl 452 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  2  <_  x )
23 chtrpcl 20525 . . . . . . 7  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
( theta `  x )  e.  RR+ )
248, 22, 23syl2anc 642 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( theta `  x )  e.  RR+ )
2521, 24rerpdivcld 10509 . . . . 5  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
)  e.  RR )
267ssriv 3260 . . . . . 6  |-  ( 2 [,)  +oo )  C_  RR
272recnd 8951 . . . . . 6  |-  (  T. 
->  1  e.  CC )
28 rlimconst 12114 . . . . . 6  |-  ( ( ( 2 [,)  +oo )  C_  RR  /\  1  e.  CC )  ->  (
x  e.  ( 2 [,)  +oo )  |->  1 )  ~~> r  1 )
2926, 27, 28sylancr 644 . . . . 5  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  1 )  ~~> r  1 )
30 ovex 5970 . . . . . . . . 9  |-  ( 2 [,)  +oo )  e.  _V
3130a1i 10 . . . . . . . 8  |-  (  T. 
->  ( 2 [,)  +oo )  e.  _V )
328, 24rerpdivcld 10509 . . . . . . . 8  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  /  ( theta `  x ) )  e.  RR )
33 ovex 5970 . . . . . . . . 9  |-  ( ( ( sqr `  x
)  x.  ( log `  x ) )  /  x )  e.  _V
3433a1i 10 . . . . . . . 8  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( ( sqr `  x
)  x.  ( log `  x ) )  /  x )  e.  _V )
35 eqidd 2359 . . . . . . . 8  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( x  /  ( theta `  x ) ) )  =  ( x  e.  ( 2 [,) 
+oo )  |->  ( x  /  ( theta `  x
) ) ) )
368recnd 8951 . . . . . . . . . . . 12  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  x  e.  CC )
37 cxpsqr 20161 . . . . . . . . . . . 12  |-  ( x  e.  CC  ->  (
x  ^ c  ( 1  /  2 ) )  =  ( sqr `  x ) )
3836, 37syl 15 . . . . . . . . . . 11  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  ^ c  ( 1  /  2 ) )  =  ( sqr `  x ) )
3938oveq2d 5961 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( log `  x
)  /  ( x  ^ c  ( 1  /  2 ) ) )  =  ( ( log `  x )  /  ( sqr `  x
) ) )
4020recnd 8951 . . . . . . . . . . 11  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( log `  x )  e.  CC )
4117rpsqrcld 11990 . . . . . . . . . . . 12  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( sqr `  x )  e.  RR+ )
4241rpcnne0d 10491 . . . . . . . . . . 11  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( sqr `  x
)  e.  CC  /\  ( sqr `  x )  =/=  0 ) )
43 divcan5 9552 . . . . . . . . . . 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 ) ) )
4440, 42, 42, 43syl3anc 1182 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( ( sqr `  x
)  x.  ( log `  x ) )  / 
( ( sqr `  x
)  x.  ( sqr `  x ) ) )  =  ( ( log `  x )  /  ( sqr `  x ) ) )
45 remsqsqr 11838 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  0  <_  x )  -> 
( ( sqr `  x
)  x.  ( sqr `  x ) )  =  x )
468, 18, 45syl2anc 642 . . . . . . . . . . 11  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( sqr `  x
)  x.  ( sqr `  x ) )  =  x )
4746oveq2d 5961 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( ( sqr `  x
)  x.  ( log `  x ) )  / 
( ( sqr `  x
)  x.  ( sqr `  x ) ) )  =  ( ( ( sqr `  x )  x.  ( log `  x
) )  /  x
) )
4839, 44, 473eqtr2d 2396 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( log `  x
)  /  ( x  ^ c  ( 1  /  2 ) ) )  =  ( ( ( sqr `  x
)  x.  ( log `  x ) )  /  x ) )
4948mpteq2dva 4187 . . . . . . . 8  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( log `  x
)  /  ( x  ^ c  ( 1  /  2 ) ) ) )  =  ( x  e.  ( 2 [,)  +oo )  |->  ( ( ( sqr `  x
)  x.  ( log `  x ) )  /  x ) ) )
5031, 32, 34, 35, 49offval2 6182 . . . . . . 7  |-  (  T. 
->  ( ( x  e.  ( 2 [,)  +oo )  |->  ( x  / 
( theta `  x )
) )  o F  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 ) ) ) )
5117rpne0d 10487 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  x  =/=  0 )
5224rpcnne0d 10491 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  e.  CC  /\  ( theta `  x )  =/=  0
) )
5321recnd 8951 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( sqr `  x
)  x.  ( log `  x ) )  e.  CC )
54 dmdcan 9560 . . . . . . . . 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 )
) )
5536, 51, 52, 53, 54syl211anc 1188 . . . . . . . 8  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( x  /  ( theta `  x ) )  x.  ( ( ( sqr `  x )  x.  ( log `  x
) )  /  x
) )  =  ( ( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) )
5655mpteq2dva 4187 . . . . . . 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 ) ) ) )
5750, 56eqtrd 2390 . . . . . 6  |-  (  T. 
->  ( ( x  e.  ( 2 [,)  +oo )  |->  ( x  / 
( theta `  x )
) )  o F  x.  ( x  e.  ( 2 [,)  +oo )  |->  ( ( log `  x )  /  (
x  ^ c  ( 1  /  2 ) ) ) ) )  =  ( x  e.  ( 2 [,)  +oo )  |->  ( ( ( sqr `  x )  x.  ( log `  x
) )  /  ( theta `  x ) ) ) )
58 chto1lb 20739 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( x  /  ( theta `  x
) ) )  e.  O ( 1 )
5916ssriv 3260 . . . . . . . . 9  |-  ( 2 [,)  +oo )  C_  RR+
6059a1i 10 . . . . . . . 8  |-  (  T. 
->  ( 2 [,)  +oo )  C_  RR+ )
61 1rp 10450 . . . . . . . . . . 11  |-  1  e.  RR+
62 rphalfcl 10470 . . . . . . . . . . 11  |-  ( 1  e.  RR+  ->  ( 1  /  2 )  e.  RR+ )
6361, 62ax-mp 8 . . . . . . . . . 10  |-  ( 1  /  2 )  e.  RR+
64 cxploglim 20383 . . . . . . . . . 10  |-  ( ( 1  /  2 )  e.  RR+  ->  ( x  e.  RR+  |->  ( ( log `  x )  /  ( x  ^ c  ( 1  / 
2 ) ) ) )  ~~> r  0 )
6563, 64ax-mp 8 . . . . . . . . 9  |-  ( x  e.  RR+  |->  ( ( log `  x )  /  ( x  ^ c  ( 1  / 
2 ) ) ) )  ~~> r  0
6665a1i 10 . . . . . . . 8  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( log `  x
)  /  ( x  ^ c  ( 1  /  2 ) ) ) )  ~~> r  0 )
6760, 66rlimres2 12131 . . . . . . 7  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( log `  x
)  /  ( x  ^ c  ( 1  /  2 ) ) ) )  ~~> r  0 )
68 o1rlimmul 12188 . . . . . . 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
) ) )  o F  x.  ( x  e.  ( 2 [,) 
+oo )  |->  ( ( log `  x )  /  ( x  ^ c  ( 1  / 
2 ) ) ) ) )  ~~> r  0 )
6958, 67, 68sylancr 644 . . . . . 6  |-  (  T. 
->  ( ( x  e.  ( 2 [,)  +oo )  |->  ( x  / 
( theta `  x )
) )  o F  x.  ( x  e.  ( 2 [,)  +oo )  |->  ( ( log `  x )  /  (
x  ^ c  ( 1  /  2 ) ) ) ) )  ~~> r  0 )
7057, 69eqbrtrrd 4126 . . . . 5  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( ( sqr `  x )  x.  ( log `  x ) )  /  ( theta `  x
) ) )  ~~> r  0 )
713, 25, 29, 70rlimadd 12212 . . . 4  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( 1  +  ( ( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) ) )  ~~> r  ( 1  +  0 ) )
72 ax-1cn 8885 . . . . 5  |-  1  e.  CC
7372addid1i 9089 . . . 4  |-  ( 1  +  0 )  =  1
7471, 73syl6breq 4143 . . 3  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( 1  +  ( ( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) ) )  ~~> r  1 )
75 readdcl 8910 . . . 4  |-  ( ( 1  e.  RR  /\  ( ( ( sqr `  x )  x.  ( log `  x ) )  /  ( theta `  x
) )  e.  RR )  ->  ( 1  +  ( ( ( sqr `  x )  x.  ( log `  x ) )  /  ( theta `  x
) ) )  e.  RR )
761, 25, 75sylancr 644 . . 3  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
1  +  ( ( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) )  e.  RR )
77 chpcl 20474 . . . . 5  |-  ( x  e.  RR  ->  (ψ `  x )  e.  RR )
788, 77syl 15 . . . 4  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (ψ `  x )  e.  RR )
7978, 24rerpdivcld 10509 . . 3  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
(ψ `  x )  /  ( theta `  x
) )  e.  RR )
80 chtcl 20459 . . . . . . . 8  |-  ( x  e.  RR  ->  ( theta `  x )  e.  RR )
818, 80syl 15 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( theta `  x )  e.  RR )
82 readdcl 8910 . . . . . . 7  |-  ( ( ( theta `  x )  e.  RR  /\  ( ( sqr `  x )  x.  ( log `  x
) )  e.  RR )  ->  ( ( theta `  x )  +  ( ( sqr `  x
)  x.  ( log `  x ) ) )  e.  RR )
8381, 21, 82syl2anc 642 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  +  ( ( sqr `  x )  x.  ( log `  x ) ) )  e.  RR )
844a1i 10 . . . . . . . 8  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  2  e.  RR )
85 1lt2 9978 . . . . . . . . . 10  |-  1  <  2
861, 4, 85ltleii 9031 . . . . . . . . 9  |-  1  <_  2
8786a1i 10 . . . . . . . 8  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  <_  2 )
883, 84, 8, 87, 22letrd 9063 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  <_  x )
89 chpub 20571 . . . . . . 7  |-  ( ( x  e.  RR  /\  1  <_  x )  -> 
(ψ `  x )  <_  ( ( theta `  x
)  +  ( ( sqr `  x )  x.  ( log `  x
) ) ) )
908, 88, 89syl2anc 642 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (ψ `  x )  <_  (
( theta `  x )  +  ( ( sqr `  x )  x.  ( log `  x ) ) ) )
9178, 83, 24, 90lediv1dd 10536 . . . . 5  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
(ψ `  x )  /  ( theta `  x
) )  <_  (
( ( theta `  x
)  +  ( ( sqr `  x )  x.  ( log `  x
) ) )  / 
( theta `  x )
) )
9224rpcnd 10484 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( theta `  x )  e.  CC )
93 divdir 9537 . . . . . . 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
) ) ) )
9492, 53, 52, 93syl3anc 1182 . . . . . 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 )
) ) )
95 divid 9541 . . . . . . . 8  |-  ( ( ( theta `  x )  e.  CC  /\  ( theta `  x )  =/=  0
)  ->  ( ( theta `  x )  / 
( theta `  x )
)  =  1 )
9652, 95syl 15 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  ( theta `  x
) )  =  1 )
9796oveq1d 5960 . . . . . 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 )
) ) )
9894, 97eqtrd 2390 . . . . 5  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( ( theta `  x
)  +  ( ( sqr `  x )  x.  ( log `  x
) ) )  / 
( theta `  x )
)  =  ( 1  +  ( ( ( sqr `  x )  x.  ( log `  x
) )  /  ( theta `  x ) ) ) )
9991, 98breqtrd 4128 . . . 4  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
(ψ `  x )  /  ( theta `  x
) )  <_  (
1  +  ( ( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) ) )
10099adantrr 697 . . 3  |-  ( (  T.  /\  ( x  e.  ( 2 [,) 
+oo )  /\  1  <_  x ) )  -> 
( (ψ `  x
)  /  ( theta `  x ) )  <_ 
( 1  +  ( ( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) ) )
10192mulid2d 8943 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
1  x.  ( theta `  x ) )  =  ( theta `  x )
)
102 chtlepsi 20557 . . . . . . 7  |-  ( x  e.  RR  ->  ( theta `  x )  <_ 
(ψ `  x )
)
1038, 102syl 15 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( theta `  x )  <_ 
(ψ `  x )
)
104101, 103eqbrtrd 4124 . . . . 5  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
1  x.  ( theta `  x ) )  <_ 
(ψ `  x )
)
1053, 78, 24lemuldivd 10527 . . . . 5  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( 1  x.  ( theta `  x ) )  <_  (ψ `  x
)  <->  1  <_  (
(ψ `  x )  /  ( theta `  x
) ) ) )
106104, 105mpbid 201 . . . 4  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  <_  ( (ψ `  x
)  /  ( theta `  x ) ) )
107106adantrr 697 . . 3  |-  ( (  T.  /\  ( x  e.  ( 2 [,) 
+oo )  /\  1  <_  x ) )  -> 
1  <_  ( (ψ `  x )  /  ( theta `  x ) ) )
1082, 2, 74, 76, 79, 100, 107rlimsqz2 12220 . 2  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( (ψ `  x
)  /  ( theta `  x ) ) )  ~~> r  1 )
109108trud 1323 1  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( (ψ `  x )  /  ( theta `  x ) ) )  ~~> r  1
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    T. wtru 1316    = wceq 1642    e. wcel 1710    =/= wne 2521   _Vcvv 2864    C_ wss 3228   class class class wbr 4104    e. cmpt 4158   ` cfv 5337  (class class class)co 5945    o Fcof 6163   CCcc 8825   RRcr 8826   0cc0 8827   1c1 8828    + caddc 8830    x. cmul 8832    +oocpnf 8954    < clt 8957    <_ cle 8958    / cdiv 9513   2c2 9885   RR+crp 10446   [,)cico 10750   sqrcsqr 11814    ~~> r crli 12055   O ( 1 )co1 12056   logclog 20019    ^ c ccxp 20020   thetaccht 20440  ψcchp 20442
This theorem is referenced by:  chpo1ub  20741  pnt2  20874
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905  ax-addf 8906  ax-mulf 8907
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-iin 3989  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-2o 6567  df-oadd 6570  df-er 6747  df-map 6862  df-pm 6863  df-ixp 6906  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-fi 7255  df-sup 7284  df-oi 7315  df-card 7662  df-cda 7884  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-7 9899  df-8 9900  df-9 9901  df-10 9902  df-n0 10058  df-z 10117  df-dec 10217  df-uz 10323  df-q 10409  df-rp 10447  df-xneg 10544  df-xadd 10545  df-xmul 10546  df-ioo 10752  df-ioc 10753  df-ico 10754  df-icc 10755  df-fz 10875  df-fzo 10963  df-fl 11017  df-mod 11066  df-seq 11139  df-exp 11198  df-fac 11382  df-bc 11409  df-hash 11431  df-shft 11658  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-limsup 12041  df-clim 12058  df-rlim 12059  df-o1 12060  df-lo1 12061  df-sum 12256  df-ef 12446  df-e 12447  df-sin 12448  df-cos 12449  df-pi 12451  df-dvds 12629  df-gcd 12783  df-prm 12856  df-pc 12987  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-mulr 13319  df-starv 13320  df-sca 13321  df-vsca 13322  df-tset 13324  df-ple 13325  df-ds 13327  df-unif 13328  df-hom 13329  df-cco 13330  df-rest 13426  df-topn 13427  df-topgen 13443  df-pt 13444  df-prds 13447  df-xrs 13502  df-0g 13503  df-gsum 13504  df-qtop 13509  df-imas 13510  df-xps 13512  df-mre 13587  df-mrc 13588  df-acs 13590  df-mnd 14466  df-submnd 14515  df-mulg 14591  df-cntz 14892  df-cmn 15190  df-xmet 16475  df-met 16476  df-bl 16477  df-mopn 16478  df-fbas 16479  df-fg 16480  df-cnfld 16483  df-top 16742  df-bases 16744  df-topon 16745  df-topsp 16746  df-cld 16862  df-ntr 16863  df-cls 16864  df-nei 16941  df-lp 16974  df-perf 16975  df-cn 17063  df-cnp 17064  df-haus 17149  df-tx 17363  df-hmeo 17552  df-fil 17643  df-fm 17735  df-flim 17736  df-flf 17737  df-xms 17987  df-ms 17988  df-tms 17989  df-cncf 18485  df-limc 19320  df-dv 19321  df-log 20021  df-cxp 20022  df-cht 20446  df-vma 20447  df-chp 20448  df-ppi 20449
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