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Theorem logfacrlim 20463
Description: Combine the estimates logfacubnd 20460 and logfaclbnd 20461, to get  log ( x ! )  =  x log x  +  O
( x ). Equation 9.2.9 of [Shapiro], p. 329. This is a weak form of the even stronger statement,  log ( x ! )  =  x log x  -  x  +  O ( log x
). (Contributed by Mario Carneiro, 16-Apr-2016.) (Revised by Mario Carneiro, 21-May-2016.)
Assertion
Ref Expression
logfacrlim  |-  ( x  e.  RR+  |->  ( ( log `  x )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) )  ~~> r  1

Proof of Theorem logfacrlim
StepHypRef Expression
1 1re 8837 . . . 4  |-  1  e.  RR
21a1i 10 . . 3  |-  (  T. 
->  1  e.  RR )
3 ax-1cn 8795 . . . 4  |-  1  e.  CC
43a1i 10 . . 3  |-  (  T. 
->  1  e.  CC )
5 relogcl 19932 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
65adantl 452 . . . . . . . 8  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( log `  x )  e.  RR )
76recnd 8861 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( log `  x )  e.  CC )
83a1i 10 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR+ )  ->  1  e.  CC )
9 rpcnne0 10371 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( x  e.  CC  /\  x  =/=  0 ) )
109adantl 452 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( x  e.  CC  /\  x  =/=  0 ) )
11 divdir 9447 . . . . . . 7  |-  ( ( ( log `  x
)  e.  CC  /\  1  e.  CC  /\  (
x  e.  CC  /\  x  =/=  0 ) )  ->  ( ( ( log `  x )  +  1 )  /  x )  =  ( ( ( log `  x
)  /  x )  +  ( 1  /  x ) ) )
127, 8, 10, 11syl3anc 1182 . . . . . 6  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( ( log `  x
)  +  1 )  /  x )  =  ( ( ( log `  x )  /  x
)  +  ( 1  /  x ) ) )
1312mpteq2dva 4106 . . . . 5  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( ( log `  x )  +  1 )  /  x ) )  =  ( x  e.  RR+  |->  ( ( ( log `  x
)  /  x )  +  ( 1  /  x ) ) ) )
14 simpr 447 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR+ )  ->  x  e.  RR+ )
156, 14rerpdivcld 10417 . . . . . 6  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( log `  x )  /  x )  e.  RR )
16 rpreccl 10377 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR+ )
1716adantl 452 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( 1  /  x )  e.  RR+ )
1817rpred 10390 . . . . . 6  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( 1  /  x )  e.  RR )
1910simpld 445 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  RR+ )  ->  x  e.  CC )
2019cxp1d 20053 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( x  ^ c  1 )  =  x )
2120oveq2d 5874 . . . . . . . 8  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( log `  x )  /  ( x  ^ c  1 ) )  =  ( ( log `  x )  /  x
) )
2221mpteq2dva 4106 . . . . . . 7  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( log `  x
)  /  ( x  ^ c  1 ) ) )  =  ( x  e.  RR+  |->  ( ( log `  x )  /  x ) ) )
23 1rp 10358 . . . . . . . 8  |-  1  e.  RR+
24 cxploglim 20272 . . . . . . . 8  |-  ( 1  e.  RR+  ->  ( x  e.  RR+  |->  ( ( log `  x )  /  ( x  ^ c  1 ) ) )  ~~> r  0 )
2523, 24mp1i 11 . . . . . . 7  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( log `  x
)  /  ( x  ^ c  1 ) ) )  ~~> r  0 )
2622, 25eqbrtrrd 4045 . . . . . 6  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( log `  x
)  /  x ) )  ~~> r  0 )
27 divrcnv 12311 . . . . . . 7  |-  ( 1  e.  CC  ->  (
x  e.  RR+  |->  ( 1  /  x ) )  ~~> r  0 )
283, 27mp1i 11 . . . . . 6  |-  (  T. 
->  ( x  e.  RR+  |->  ( 1  /  x
) )  ~~> r  0 )
2915, 18, 26, 28rlimadd 12116 . . . . 5  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( ( log `  x )  /  x
)  +  ( 1  /  x ) ) )  ~~> r  ( 0  +  0 ) )
3013, 29eqbrtrd 4043 . . . 4  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( ( log `  x )  +  1 )  /  x ) )  ~~> r  ( 0  +  0 ) )
31 00id 8987 . . . 4  |-  ( 0  +  0 )  =  0
3230, 31syl6breq 4062 . . 3  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( ( log `  x )  +  1 )  /  x ) )  ~~> r  0 )
33 peano2re 8985 . . . . . 6  |-  ( ( log `  x )  e.  RR  ->  (
( log `  x
)  +  1 )  e.  RR )
346, 33syl 15 . . . . 5  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( log `  x )  +  1 )  e.  RR )
3534, 14rerpdivcld 10417 . . . 4  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( ( log `  x
)  +  1 )  /  x )  e.  RR )
3635recnd 8861 . . 3  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( ( log `  x
)  +  1 )  /  x )  e.  CC )
37 rprege0 10368 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( x  e.  RR  /\  0  <_  x ) )
3837adantl 452 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( x  e.  RR  /\  0  <_  x ) )
39 flge0nn0 10948 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  0  <_  x )  -> 
( |_ `  x
)  e.  NN0 )
40 faccl 11298 . . . . . . . . 9  |-  ( ( |_ `  x )  e.  NN0  ->  ( ! `
 ( |_ `  x ) )  e.  NN )
4138, 39, 403syl 18 . . . . . . . 8  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ! `
 ( |_ `  x ) )  e.  NN )
4241nnrpd 10389 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ! `
 ( |_ `  x ) )  e.  RR+ )
43 relogcl 19932 . . . . . . 7  |-  ( ( ! `  ( |_
`  x ) )  e.  RR+  ->  ( log `  ( ! `  ( |_ `  x ) ) )  e.  RR )
4442, 43syl 15 . . . . . 6  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( log `  ( ! `  ( |_ `  x ) ) )  e.  RR )
4544, 14rerpdivcld 10417 . . . . 5  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x )  e.  RR )
4645recnd 8861 . . . 4  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x )  e.  CC )
477, 46subcld 9157 . . 3  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( log `  x )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) )  e.  CC )
48 logfacbnd3 20462 . . . . . 6  |-  ( ( x  e.  RR+  /\  1  <_  x )  ->  ( abs `  ( ( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x )  -  1 ) ) ) )  <_  ( ( log `  x )  +  1 ) )
4948adantl 452 . . . . 5  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) ) )  <_ 
( ( log `  x
)  +  1 ) )
5044recnd 8861 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( log `  ( ! `  ( |_ `  x ) ) )  e.  CC )
5150adantrr 697 . . . . . . . 8  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( log `  ( ! `  ( |_ `  x ) ) )  e.  CC )
529ad2antrl 708 . . . . . . . . . 10  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( x  e.  CC  /\  x  =/=  0 ) )
5352simpld 445 . . . . . . . . 9  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  ->  x  e.  CC )
547adantrr 697 . . . . . . . . . 10  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( log `  x
)  e.  CC )
55 subcl 9051 . . . . . . . . . 10  |-  ( ( ( log `  x
)  e.  CC  /\  1  e.  CC )  ->  ( ( log `  x
)  -  1 )  e.  CC )
5654, 3, 55sylancl 643 . . . . . . . . 9  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( log `  x
)  -  1 )  e.  CC )
5753, 56mulcld 8855 . . . . . . . 8  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( x  x.  (
( log `  x
)  -  1 ) )  e.  CC )
5851, 57subcld 9157 . . . . . . 7  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) )  e.  CC )
5958abscld 11918 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) ) )  e.  RR )
606adantrr 697 . . . . . . 7  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( log `  x
)  e.  RR )
6160, 33syl 15 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( log `  x
)  +  1 )  e.  RR )
62 rpregt0 10367 . . . . . . 7  |-  ( x  e.  RR+  ->  ( x  e.  RR  /\  0  <  x ) )
6362ad2antrl 708 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( x  e.  RR  /\  0  <  x ) )
64 lediv1 9621 . . . . . 6  |-  ( ( ( abs `  (
( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) ) )  e.  RR  /\  ( ( log `  x )  +  1 )  e.  RR  /\  ( x  e.  RR  /\  0  <  x ) )  -> 
( ( abs `  (
( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) ) )  <_ 
( ( log `  x
)  +  1 )  <-> 
( ( abs `  (
( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) ) )  /  x )  <_  (
( ( log `  x
)  +  1 )  /  x ) ) )
6559, 61, 63, 64syl3anc 1182 . . . . 5  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( abs `  (
( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) ) )  <_ 
( ( log `  x
)  +  1 )  <-> 
( ( abs `  (
( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) ) )  /  x )  <_  (
( ( log `  x
)  +  1 )  /  x ) ) )
6649, 65mpbid 201 . . . 4  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( abs `  (
( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) ) )  /  x )  <_  (
( ( log `  x
)  +  1 )  /  x ) )
6752simprd 449 . . . . . . . . 9  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  ->  x  =/=  0 )
6856, 53, 67divcan3d 9541 . . . . . . . 8  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( x  x.  ( ( log `  x
)  -  1 ) )  /  x )  =  ( ( log `  x )  -  1 ) )
6968oveq1d 5873 . . . . . . 7  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( ( x  x.  ( ( log `  x )  -  1 ) )  /  x
)  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) )  =  ( ( ( log `  x )  -  1 )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) )
70 divsubdir 9456 . . . . . . . 8  |-  ( ( ( x  x.  (
( log `  x
)  -  1 ) )  e.  CC  /\  ( log `  ( ! `
 ( |_ `  x ) ) )  e.  CC  /\  (
x  e.  CC  /\  x  =/=  0 ) )  ->  ( ( ( x  x.  ( ( log `  x )  -  1 ) )  -  ( log `  ( ! `  ( |_ `  x ) ) ) )  /  x )  =  ( ( ( x  x.  ( ( log `  x )  -  1 ) )  /  x )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) )
7157, 51, 52, 70syl3anc 1182 . . . . . . 7  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( ( x  x.  ( ( log `  x )  -  1 ) )  -  ( log `  ( ! `  ( |_ `  x ) ) ) )  /  x )  =  ( ( ( x  x.  ( ( log `  x
)  -  1 ) )  /  x )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) )
7246adantrr 697 . . . . . . . 8  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( log `  ( ! `  ( |_ `  x ) ) )  /  x )  e.  CC )
733a1i 10 . . . . . . . 8  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
1  e.  CC )
7454, 72, 73sub32d 9189 . . . . . . 7  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( ( log `  x )  -  (
( log `  ( ! `  ( |_ `  x ) ) )  /  x ) )  -  1 )  =  ( ( ( log `  x )  -  1 )  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) ) )
7569, 71, 743eqtr4rd 2326 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( ( log `  x )  -  (
( log `  ( ! `  ( |_ `  x ) ) )  /  x ) )  -  1 )  =  ( ( ( x  x.  ( ( log `  x )  -  1 ) )  -  ( log `  ( ! `  ( |_ `  x ) ) ) )  /  x ) )
7675fveq2d 5529 . . . . 5  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( ( log `  x
)  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) )  -  1 ) )  =  ( abs `  (
( ( x  x.  ( ( log `  x
)  -  1 ) )  -  ( log `  ( ! `  ( |_ `  x ) ) ) )  /  x
) ) )
7757, 51subcld 9157 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( x  x.  ( ( log `  x
)  -  1 ) )  -  ( log `  ( ! `  ( |_ `  x ) ) ) )  e.  CC )
7877, 53, 67absdivd 11937 . . . . 5  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( ( x  x.  ( ( log `  x
)  -  1 ) )  -  ( log `  ( ! `  ( |_ `  x ) ) ) )  /  x
) )  =  ( ( abs `  (
( x  x.  (
( log `  x
)  -  1 ) )  -  ( log `  ( ! `  ( |_ `  x ) ) ) ) )  / 
( abs `  x
) ) )
7957, 51abssubd 11935 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( x  x.  (
( log `  x
)  -  1 ) )  -  ( log `  ( ! `  ( |_ `  x ) ) ) ) )  =  ( abs `  (
( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) ) ) )
8037ad2antrl 708 . . . . . . 7  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( x  e.  RR  /\  0  <_  x )
)
81 absid 11781 . . . . . . 7  |-  ( ( x  e.  RR  /\  0  <_  x )  -> 
( abs `  x
)  =  x )
8280, 81syl 15 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  x
)  =  x )
8379, 82oveq12d 5876 . . . . 5  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( abs `  (
( x  x.  (
( log `  x
)  -  1 ) )  -  ( log `  ( ! `  ( |_ `  x ) ) ) ) )  / 
( abs `  x
) )  =  ( ( abs `  (
( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) ) )  /  x ) )
8476, 78, 833eqtrd 2319 . . . 4  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( ( log `  x
)  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) )  -  1 ) )  =  ( ( abs `  ( ( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) ) )  /  x ) )
8536adantrr 697 . . . . . . 7  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( ( log `  x )  +  1 )  /  x )  e.  CC )
8685subid1d 9146 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( ( ( log `  x )  +  1 )  /  x )  -  0 )  =  ( ( ( log `  x
)  +  1 )  /  x ) )
8786fveq2d 5529 . . . . 5  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( ( ( log `  x )  +  1 )  /  x )  -  0 ) )  =  ( abs `  (
( ( log `  x
)  +  1 )  /  x ) ) )
88 log1 19939 . . . . . . . . 9  |-  ( log `  1 )  =  0
89 simprr 733 . . . . . . . . . 10  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
1  <_  x )
9014adantrr 697 . . . . . . . . . . 11  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  ->  x  e.  RR+ )
91 logleb 19957 . . . . . . . . . . 11  |-  ( ( 1  e.  RR+  /\  x  e.  RR+ )  ->  (
1  <_  x  <->  ( log `  1 )  <_  ( log `  x ) ) )
9223, 90, 91sylancr 644 . . . . . . . . . 10  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( 1  <_  x  <->  ( log `  1 )  <_  ( log `  x
) ) )
9389, 92mpbid 201 . . . . . . . . 9  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( log `  1
)  <_  ( log `  x ) )
9488, 93syl5eqbrr 4057 . . . . . . . 8  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
0  <_  ( log `  x ) )
9560, 94ge0p1rpd 10416 . . . . . . 7  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( log `  x
)  +  1 )  e.  RR+ )
9695, 90rpdivcld 10407 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( ( log `  x )  +  1 )  /  x )  e.  RR+ )
97 rprege0 10368 . . . . . 6  |-  ( ( ( ( log `  x
)  +  1 )  /  x )  e.  RR+  ->  ( ( ( ( log `  x
)  +  1 )  /  x )  e.  RR  /\  0  <_ 
( ( ( log `  x )  +  1 )  /  x ) ) )
98 absid 11781 . . . . . 6  |-  ( ( ( ( ( log `  x )  +  1 )  /  x )  e.  RR  /\  0  <_  ( ( ( log `  x )  +  1 )  /  x ) )  ->  ( abs `  ( ( ( log `  x )  +  1 )  /  x ) )  =  ( ( ( log `  x
)  +  1 )  /  x ) )
9996, 97, 983syl 18 . . . . 5  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( ( log `  x
)  +  1 )  /  x ) )  =  ( ( ( log `  x )  +  1 )  /  x ) )
10087, 99eqtrd 2315 . . . 4  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( ( ( log `  x )  +  1 )  /  x )  -  0 ) )  =  ( ( ( log `  x )  +  1 )  /  x ) )
10166, 84, 1003brtr4d 4053 . . 3  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( ( log `  x
)  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) )  -  1 ) )  <_  ( abs `  (
( ( ( log `  x )  +  1 )  /  x )  -  0 ) ) )
1022, 4, 32, 36, 47, 101rlimsqzlem 12122 . 2  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( log `  x
)  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) ) )  ~~> r  1 )
103102trud 1314 1  |-  ( x  e.  RR+  |->  ( ( log `  x )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) )  ~~> r  1
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    T. wtru 1307    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   NN0cn0 9965   RR+crp 10354   |_cfl 10924   !cfa 11288   abscabs 11719    ~~> r crli 11959   logclog 19912    ^ c ccxp 19913
This theorem is referenced by:  vmadivsum  20631
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-cmp 17114  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914  df-cxp 19915
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