MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  chpchtlim Structured version   Unicode version

Theorem chpchtlim 21178
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 9095 . . . 4  |-  1  e.  RR
21a1i 11 . . 3  |-  (  T. 
->  1  e.  RR )
31a1i 11 . . . . 5  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  e.  RR )
4 2re 10074 . . . . . . . . . . 11  |-  2  e.  RR
5 elicopnf 11005 . . . . . . . . . . 11  |-  ( 2  e.  RR  ->  (
x  e.  ( 2 [,)  +oo )  <->  ( x  e.  RR  /\  2  <_  x ) ) )
64, 5ax-mp 5 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  <->  ( x  e.  RR  /\  2  <_  x ) )
76simplbi 448 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  RR )
87adantl 454 . . . . . . . 8  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  x  e.  RR )
9 0re 9096 . . . . . . . . . . . . 13  |-  0  e.  RR
109a1i 11 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) 
+oo )  ->  0  e.  RR )
114a1i 11 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) 
+oo )  ->  2  e.  RR )
12 2pos 10087 . . . . . . . . . . . . 13  |-  0  <  2
1312a1i 11 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) 
+oo )  ->  0  <  2 )
146simprbi 452 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) 
+oo )  ->  2  <_  x )
1510, 11, 7, 13, 14ltletrd 9235 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  0  <  x )
167, 15elrpd 10651 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  RR+ )
1716adantl 454 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  x  e.  RR+ )
1817rpge0d 10657 . . . . . . . 8  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  0  <_  x )
198, 18resqrcld 12225 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( sqr `  x )  e.  RR )
2017relogcld 20523 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( log `  x )  e.  RR )
2119, 20remulcld 9121 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( sqr `  x
)  x.  ( log `  x ) )  e.  RR )
2214adantl 454 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  2  <_  x )
23 chtrpcl 20963 . . . . . . 7  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
( theta `  x )  e.  RR+ )
248, 22, 23syl2anc 644 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( theta `  x )  e.  RR+ )
2521, 24rerpdivcld 10680 . . . . 5  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
)  e.  RR )
267ssriv 3354 . . . . . 6  |-  ( 2 [,)  +oo )  C_  RR
272recnd 9119 . . . . . 6  |-  (  T. 
->  1  e.  CC )
28 rlimconst 12343 . . . . . 6  |-  ( ( ( 2 [,)  +oo )  C_  RR  /\  1  e.  CC )  ->  (
x  e.  ( 2 [,)  +oo )  |->  1 )  ~~> r  1 )
2926, 27, 28sylancr 646 . . . . 5  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  1 )  ~~> r  1 )
30 ovex 6109 . . . . . . . . 9  |-  ( 2 [,)  +oo )  e.  _V
3130a1i 11 . . . . . . . 8  |-  (  T. 
->  ( 2 [,)  +oo )  e.  _V )
328, 24rerpdivcld 10680 . . . . . . . 8  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  /  ( theta `  x ) )  e.  RR )
33 ovex 6109 . . . . . . . . 9  |-  ( ( ( sqr `  x
)  x.  ( log `  x ) )  /  x )  e.  _V
3433a1i 11 . . . . . . . 8  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( ( sqr `  x
)  x.  ( log `  x ) )  /  x )  e.  _V )
35 eqidd 2439 . . . . . . . 8  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( x  /  ( theta `  x ) ) )  =  ( x  e.  ( 2 [,) 
+oo )  |->  ( x  /  ( theta `  x
) ) ) )
368recnd 9119 . . . . . . . . . . . 12  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  x  e.  CC )
37 cxpsqr 20599 . . . . . . . . . . . 12  |-  ( x  e.  CC  ->  (
x  ^ c  ( 1  /  2 ) )  =  ( sqr `  x ) )
3836, 37syl 16 . . . . . . . . . . 11  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  ^ c  ( 1  /  2 ) )  =  ( sqr `  x ) )
3938oveq2d 6100 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( log `  x
)  /  ( x  ^ c  ( 1  /  2 ) ) )  =  ( ( log `  x )  /  ( sqr `  x
) ) )
4020recnd 9119 . . . . . . . . . . 11  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( log `  x )  e.  CC )
4117rpsqrcld 12219 . . . . . . . . . . . 12  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( sqr `  x )  e.  RR+ )
4241rpcnne0d 10662 . . . . . . . . . . 11  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( sqr `  x
)  e.  CC  /\  ( sqr `  x )  =/=  0 ) )
43 divcan5 9721 . . . . . . . . . . 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 1185 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( ( sqr `  x
)  x.  ( log `  x ) )  / 
( ( sqr `  x
)  x.  ( sqr `  x ) ) )  =  ( ( log `  x )  /  ( sqr `  x ) ) )
45 remsqsqr 12067 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  0  <_  x )  -> 
( ( sqr `  x
)  x.  ( sqr `  x ) )  =  x )
468, 18, 45syl2anc 644 . . . . . . . . . . 11  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( sqr `  x
)  x.  ( sqr `  x ) )  =  x )
4746oveq2d 6100 . . . . . . . . . 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 2476 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( log `  x
)  /  ( x  ^ c  ( 1  /  2 ) ) )  =  ( ( ( sqr `  x
)  x.  ( log `  x ) )  /  x ) )
4948mpteq2dva 4298 . . . . . . . 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 6325 . . . . . . 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 10658 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  x  =/=  0 )
5224rpcnne0d 10662 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  e.  CC  /\  ( theta `  x )  =/=  0
) )
5321recnd 9119 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( sqr `  x
)  x.  ( log `  x ) )  e.  CC )
54 dmdcan 9729 . . . . . . . . 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 1191 . . . . . . . 8  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( x  /  ( theta `  x ) )  x.  ( ( ( sqr `  x )  x.  ( log `  x
) )  /  x
) )  =  ( ( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) )
5655mpteq2dva 4298 . . . . . . 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 2470 . . . . . 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 21177 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( x  /  ( theta `  x
) ) )  e.  O ( 1 )
5916ssriv 3354 . . . . . . . . 9  |-  ( 2 [,)  +oo )  C_  RR+
6059a1i 11 . . . . . . . 8  |-  (  T. 
->  ( 2 [,)  +oo )  C_  RR+ )
61 1rp 10621 . . . . . . . . . . 11  |-  1  e.  RR+
62 rphalfcl 10641 . . . . . . . . . . 11  |-  ( 1  e.  RR+  ->  ( 1  /  2 )  e.  RR+ )
6361, 62ax-mp 5 . . . . . . . . . 10  |-  ( 1  /  2 )  e.  RR+
64 cxploglim 20821 . . . . . . . . . 10  |-  ( ( 1  /  2 )  e.  RR+  ->  ( x  e.  RR+  |->  ( ( log `  x )  /  ( x  ^ c  ( 1  / 
2 ) ) ) )  ~~> r  0 )
6563, 64ax-mp 5 . . . . . . . . 9  |-  ( x  e.  RR+  |->  ( ( log `  x )  /  ( x  ^ c  ( 1  / 
2 ) ) ) )  ~~> r  0
6665a1i 11 . . . . . . . 8  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( log `  x
)  /  ( x  ^ c  ( 1  /  2 ) ) ) )  ~~> r  0 )
6760, 66rlimres2 12360 . . . . . . 7  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( log `  x
)  /  ( x  ^ c  ( 1  /  2 ) ) ) )  ~~> r  0 )
68 o1rlimmul 12417 . . . . . . 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 646 . . . . . 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 4237 . . . . 5  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( ( sqr `  x )  x.  ( log `  x ) )  /  ( theta `  x
) ) )  ~~> r  0 )
713, 25, 29, 70rlimadd 12441 . . . 4  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( 1  +  ( ( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) ) )  ~~> r  ( 1  +  0 ) )
72 ax-1cn 9053 . . . . 5  |-  1  e.  CC
7372addid1i 9258 . . . 4  |-  ( 1  +  0 )  =  1
7471, 73syl6breq 4254 . . 3  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( 1  +  ( ( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) ) )  ~~> r  1 )
75 readdcl 9078 . . . 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 646 . . 3  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
1  +  ( ( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) )  e.  RR )
77 chpcl 20912 . . . . 5  |-  ( x  e.  RR  ->  (ψ `  x )  e.  RR )
788, 77syl 16 . . . 4  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (ψ `  x )  e.  RR )
7978, 24rerpdivcld 10680 . . 3  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
(ψ `  x )  /  ( theta `  x
) )  e.  RR )
80 chtcl 20897 . . . . . . . 8  |-  ( x  e.  RR  ->  ( theta `  x )  e.  RR )
818, 80syl 16 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( theta `  x )  e.  RR )
82 readdcl 9078 . . . . . . 7  |-  ( ( ( theta `  x )  e.  RR  /\  ( ( sqr `  x )  x.  ( log `  x
) )  e.  RR )  ->  ( ( theta `  x )  +  ( ( sqr `  x
)  x.  ( log `  x ) ) )  e.  RR )
8381, 21, 82syl2anc 644 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  +  ( ( sqr `  x )  x.  ( log `  x ) ) )  e.  RR )
844a1i 11 . . . . . . . 8  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  2  e.  RR )
85 1lt2 10147 . . . . . . . . . 10  |-  1  <  2
861, 4, 85ltleii 9201 . . . . . . . . 9  |-  1  <_  2
8786a1i 11 . . . . . . . 8  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  <_  2 )
883, 84, 8, 87, 22letrd 9232 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  <_  x )
89 chpub 21009 . . . . . . 7  |-  ( ( x  e.  RR  /\  1  <_  x )  -> 
(ψ `  x )  <_  ( ( theta `  x
)  +  ( ( sqr `  x )  x.  ( log `  x
) ) ) )
908, 88, 89syl2anc 644 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (ψ `  x )  <_  (
( theta `  x )  +  ( ( sqr `  x )  x.  ( log `  x ) ) ) )
9178, 83, 24, 90lediv1dd 10707 . . . . 5  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
(ψ `  x )  /  ( theta `  x
) )  <_  (
( ( theta `  x
)  +  ( ( sqr `  x )  x.  ( log `  x
) ) )  / 
( theta `  x )
) )
9224rpcnd 10655 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( theta `  x )  e.  CC )
93 divdir 9706 . . . . . . 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 1185 . . . . . 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 9710 . . . . . . . 8  |-  ( ( ( theta `  x )  e.  CC  /\  ( theta `  x )  =/=  0
)  ->  ( ( theta `  x )  / 
( theta `  x )
)  =  1 )
9652, 95syl 16 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  ( theta `  x
) )  =  1 )
9796oveq1d 6099 . . . . . 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 2470 . . . . 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 4239 . . . 4  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
(ψ `  x )  /  ( theta `  x
) )  <_  (
1  +  ( ( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) ) )
10099adantrr 699 . . 3  |-  ( (  T.  /\  ( x  e.  ( 2 [,) 
+oo )  /\  1  <_  x ) )  -> 
( (ψ `  x
)  /  ( theta `  x ) )  <_ 
( 1  +  ( ( ( sqr `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) ) )
10192mulid2d 9111 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
1  x.  ( theta `  x ) )  =  ( theta `  x )
)
102 chtlepsi 20995 . . . . . . 7  |-  ( x  e.  RR  ->  ( theta `  x )  <_ 
(ψ `  x )
)
1038, 102syl 16 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( theta `  x )  <_ 
(ψ `  x )
)
104101, 103eqbrtrd 4235 . . . . 5  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
1  x.  ( theta `  x ) )  <_ 
(ψ `  x )
)
1053, 78, 24lemuldivd 10698 . . . . 5  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( 1  x.  ( theta `  x ) )  <_  (ψ `  x
)  <->  1  <_  (
(ψ `  x )  /  ( theta `  x
) ) ) )
106104, 105mpbid 203 . . . 4  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  <_  ( (ψ `  x
)  /  ( theta `  x ) ) )
107106adantrr 699 . . 3  |-  ( (  T.  /\  ( x  e.  ( 2 [,) 
+oo )  /\  1  <_  x ) )  -> 
1  <_  ( (ψ `  x )  /  ( theta `  x ) ) )
1082, 2, 74, 76, 79, 100, 107rlimsqz2 12449 . 2  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( (ψ `  x
)  /  ( theta `  x ) ) )  ~~> r  1 )
109108trud 1333 1  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( (ψ `  x )  /  ( theta `  x ) ) )  ~~> r  1
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    T. wtru 1326    = wceq 1653    e. wcel 1726    =/= wne 2601   _Vcvv 2958    C_ wss 3322   class class class wbr 4215    e. cmpt 4269   ` cfv 5457  (class class class)co 6084    o Fcof 6306   CCcc 8993   RRcr 8994   0cc0 8995   1c1 8996    + caddc 8998    x. cmul 9000    +oocpnf 9122    < clt 9125    <_ cle 9126    / cdiv 9682   2c2 10054   RR+crp 10617   [,)cico 10923   sqrcsqr 12043    ~~> r crli 12284   O ( 1 )co1 12285   logclog 20457    ^ c ccxp 20458   thetaccht 20878  ψcchp 20880
This theorem is referenced by:  chpo1ub  21179  pnt2  21312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074  ax-mulf 9075
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-fi 7419  df-sup 7449  df-oi 7482  df-card 7831  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-ioo 10925  df-ioc 10926  df-ico 10927  df-icc 10928  df-fz 11049  df-fzo 11141  df-fl 11207  df-mod 11256  df-seq 11329  df-exp 11388  df-fac 11572  df-bc 11599  df-hash 11624  df-shft 11887  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-limsup 12270  df-clim 12287  df-rlim 12288  df-o1 12289  df-lo1 12290  df-sum 12485  df-ef 12675  df-e 12676  df-sin 12677  df-cos 12678  df-pi 12680  df-dvds 12858  df-gcd 13012  df-prm 13085  df-pc 13216  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-starv 13549  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-unif 13557  df-hom 13558  df-cco 13559  df-rest 13655  df-topn 13656  df-topgen 13672  df-pt 13673  df-prds 13676  df-xrs 13731  df-0g 13732  df-gsum 13733  df-qtop 13738  df-imas 13739  df-xps 13741  df-mre 13816  df-mrc 13817  df-acs 13819  df-mnd 14695  df-submnd 14744  df-mulg 14820  df-cntz 15121  df-cmn 15419  df-psmet 16699  df-xmet 16700  df-met 16701  df-bl 16702  df-mopn 16703  df-fbas 16704  df-fg 16705  df-cnfld 16709  df-top 16968  df-bases 16970  df-topon 16971  df-topsp 16972  df-cld 17088  df-ntr 17089  df-cls 17090  df-nei 17167  df-lp 17205  df-perf 17206  df-cn 17296  df-cnp 17297  df-haus 17384  df-tx 17599  df-hmeo 17792  df-fil 17883  df-fm 17975  df-flim 17976  df-flf 17977  df-xms 18355  df-ms 18356  df-tms 18357  df-cncf 18913  df-limc 19758  df-dv 19759  df-log 20459  df-cxp 20460  df-cht 20884  df-vma 20885  df-chp 20886  df-ppi 20887
  Copyright terms: Public domain W3C validator