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Theorem logfacrlim 20411
Description: Combine the estimates logfacubnd 20408 and logfaclbnd 20409, 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 8791 . . . 4  |-  1  e.  RR
21a1i 12 . . 3  |-  (  T. 
->  1  e.  RR )
3 ax-1cn 8749 . . . 4  |-  1  e.  CC
43a1i 12 . . 3  |-  (  T. 
->  1  e.  CC )
5 relogcl 19880 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
65adantl 454 . . . . . . . 8  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( log `  x )  e.  RR )
76recnd 8815 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( log `  x )  e.  CC )
83a1i 12 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR+ )  ->  1  e.  CC )
9 rpcnne0 10324 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( x  e.  CC  /\  x  =/=  0 ) )
109adantl 454 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( x  e.  CC  /\  x  =/=  0 ) )
11 divdir 9401 . . . . . . 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 1187 . . . . . 6  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( ( log `  x
)  +  1 )  /  x )  =  ( ( ( log `  x )  /  x
)  +  ( 1  /  x ) ) )
1312mpteq2dva 4066 . . . . 5  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( ( log `  x )  +  1 )  /  x ) )  =  ( x  e.  RR+  |->  ( ( ( log `  x
)  /  x )  +  ( 1  /  x ) ) ) )
14 simpr 449 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR+ )  ->  x  e.  RR+ )
156, 14rerpdivcld 10370 . . . . . 6  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( log `  x )  /  x )  e.  RR )
16 rpreccl 10330 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR+ )
1716adantl 454 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( 1  /  x )  e.  RR+ )
1817rpred 10343 . . . . . 6  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( 1  /  x )  e.  RR )
1910simpld 447 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  RR+ )  ->  x  e.  CC )
2019cxp1d 20001 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( x  ^ c  1 )  =  x )
2120oveq2d 5794 . . . . . . . 8  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( log `  x )  /  ( x  ^ c  1 ) )  =  ( ( log `  x )  /  x
) )
2221mpteq2dva 4066 . . . . . . 7  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( log `  x
)  /  ( x  ^ c  1 ) ) )  =  ( x  e.  RR+  |->  ( ( log `  x )  /  x ) ) )
23 1rp 10311 . . . . . . . 8  |-  1  e.  RR+
24 cxploglim 20220 . . . . . . . 8  |-  ( 1  e.  RR+  ->  ( x  e.  RR+  |->  ( ( log `  x )  /  ( x  ^ c  1 ) ) )  ~~> r  0 )
2523, 24mp1i 13 . . . . . . 7  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( log `  x
)  /  ( x  ^ c  1 ) ) )  ~~> r  0 )
2622, 25eqbrtrrd 4005 . . . . . 6  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( log `  x
)  /  x ) )  ~~> r  0 )
27 divrcnv 12259 . . . . . . 7  |-  ( 1  e.  CC  ->  (
x  e.  RR+  |->  ( 1  /  x ) )  ~~> r  0 )
283, 27mp1i 13 . . . . . 6  |-  (  T. 
->  ( x  e.  RR+  |->  ( 1  /  x
) )  ~~> r  0 )
2915, 18, 26, 28rlimadd 12067 . . . . 5  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( ( log `  x )  /  x
)  +  ( 1  /  x ) ) )  ~~> r  ( 0  +  0 ) )
3013, 29eqbrtrd 4003 . . . 4  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( ( log `  x )  +  1 )  /  x ) )  ~~> r  ( 0  +  0 ) )
31 00id 8941 . . . 4  |-  ( 0  +  0 )  =  0
3230, 31syl6breq 4022 . . 3  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( ( log `  x )  +  1 )  /  x ) )  ~~> r  0 )
33 peano2re 8939 . . . . . 6  |-  ( ( log `  x )  e.  RR  ->  (
( log `  x
)  +  1 )  e.  RR )
346, 33syl 17 . . . . 5  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( log `  x )  +  1 )  e.  RR )
3534, 14rerpdivcld 10370 . . . 4  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( ( log `  x
)  +  1 )  /  x )  e.  RR )
3635recnd 8815 . . 3  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( ( log `  x
)  +  1 )  /  x )  e.  CC )
37 rprege0 10321 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( x  e.  RR  /\  0  <_  x ) )
3837adantl 454 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( x  e.  RR  /\  0  <_  x ) )
39 flge0nn0 10900 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  0  <_  x )  -> 
( |_ `  x
)  e.  NN0 )
40 faccl 11250 . . . . . . . . 9  |-  ( ( |_ `  x )  e.  NN0  ->  ( ! `
 ( |_ `  x ) )  e.  NN )
4138, 39, 403syl 20 . . . . . . . 8  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ! `
 ( |_ `  x ) )  e.  NN )
4241nnrpd 10342 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ! `
 ( |_ `  x ) )  e.  RR+ )
43 relogcl 19880 . . . . . . 7  |-  ( ( ! `  ( |_
`  x ) )  e.  RR+  ->  ( log `  ( ! `  ( |_ `  x ) ) )  e.  RR )
4442, 43syl 17 . . . . . 6  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( log `  ( ! `  ( |_ `  x ) ) )  e.  RR )
4544, 14rerpdivcld 10370 . . . . 5  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x )  e.  RR )
4645recnd 8815 . . . 4  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x )  e.  CC )
477, 46subcld 9111 . . 3  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( log `  x )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) )  e.  CC )
48 logfacbnd3 20410 . . . . . 6  |-  ( ( x  e.  RR+  /\  1  <_  x )  ->  ( abs `  ( ( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x )  -  1 ) ) ) )  <_  ( ( log `  x )  +  1 ) )
4948adantl 454 . . . . 5  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) ) )  <_ 
( ( log `  x
)  +  1 ) )
5044recnd 8815 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( log `  ( ! `  ( |_ `  x ) ) )  e.  CC )
5150adantrr 700 . . . . . . . 8  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( log `  ( ! `  ( |_ `  x ) ) )  e.  CC )
529ad2antrl 711 . . . . . . . . . 10  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( x  e.  CC  /\  x  =/=  0 ) )
5352simpld 447 . . . . . . . . 9  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  ->  x  e.  CC )
547adantrr 700 . . . . . . . . . 10  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( log `  x
)  e.  CC )
55 subcl 9005 . . . . . . . . . 10  |-  ( ( ( log `  x
)  e.  CC  /\  1  e.  CC )  ->  ( ( log `  x
)  -  1 )  e.  CC )
5654, 3, 55sylancl 646 . . . . . . . . 9  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( log `  x
)  -  1 )  e.  CC )
5753, 56mulcld 8809 . . . . . . . 8  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( x  x.  (
( log `  x
)  -  1 ) )  e.  CC )
5851, 57subcld 9111 . . . . . . 7  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) )  e.  CC )
5958abscld 11869 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) ) )  e.  RR )
606adantrr 700 . . . . . . 7  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( log `  x
)  e.  RR )
6160, 33syl 17 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( log `  x
)  +  1 )  e.  RR )
62 rpregt0 10320 . . . . . . 7  |-  ( x  e.  RR+  ->  ( x  e.  RR  /\  0  <  x ) )
6362ad2antrl 711 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( x  e.  RR  /\  0  <  x ) )
64 lediv1 9575 . . . . . 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 1187 . . . . 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 203 . . . 4  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( abs `  (
( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) ) )  /  x )  <_  (
( ( log `  x
)  +  1 )  /  x ) )
6752simprd 451 . . . . . . . . 9  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  ->  x  =/=  0 )
6856, 53, 67divcan3d 9495 . . . . . . . 8  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( x  x.  ( ( log `  x
)  -  1 ) )  /  x )  =  ( ( log `  x )  -  1 ) )
6968oveq1d 5793 . . . . . . 7  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( ( x  x.  ( ( log `  x )  -  1 ) )  /  x
)  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) )  =  ( ( ( log `  x )  -  1 )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) )
70 divsubdir 9410 . . . . . . . 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 1187 . . . . . . 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 700 . . . . . . . 8  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( log `  ( ! `  ( |_ `  x ) ) )  /  x )  e.  CC )
733a1i 12 . . . . . . . 8  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
1  e.  CC )
7454, 72, 73sub32d 9143 . . . . . . 7  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( ( log `  x )  -  (
( log `  ( ! `  ( |_ `  x ) ) )  /  x ) )  -  1 )  =  ( ( ( log `  x )  -  1 )  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) ) )
7569, 71, 743eqtr4rd 2299 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( ( log `  x )  -  (
( log `  ( ! `  ( |_ `  x ) ) )  /  x ) )  -  1 )  =  ( ( ( x  x.  ( ( log `  x )  -  1 ) )  -  ( log `  ( ! `  ( |_ `  x ) ) ) )  /  x ) )
7675fveq2d 5448 . . . . 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 9111 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( x  x.  ( ( log `  x
)  -  1 ) )  -  ( log `  ( ! `  ( |_ `  x ) ) ) )  e.  CC )
7877, 53, 67absdivd 11888 . . . . 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 11886 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( x  x.  (
( log `  x
)  -  1 ) )  -  ( log `  ( ! `  ( |_ `  x ) ) ) ) )  =  ( abs `  (
( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) ) ) )
8037ad2antrl 711 . . . . . . 7  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( x  e.  RR  /\  0  <_  x )
)
81 absid 11732 . . . . . . 7  |-  ( ( x  e.  RR  /\  0  <_  x )  -> 
( abs `  x
)  =  x )
8280, 81syl 17 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  x
)  =  x )
8379, 82oveq12d 5796 . . . . 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 2292 . . . 4  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( ( log `  x
)  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) )  -  1 ) )  =  ( ( abs `  ( ( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) ) )  /  x ) )
8536adantrr 700 . . . . . . 7  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( ( log `  x )  +  1 )  /  x )  e.  CC )
8685subid1d 9100 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( ( ( log `  x )  +  1 )  /  x )  -  0 )  =  ( ( ( log `  x
)  +  1 )  /  x ) )
8786fveq2d 5448 . . . . 5  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( ( ( log `  x )  +  1 )  /  x )  -  0 ) )  =  ( abs `  (
( ( log `  x
)  +  1 )  /  x ) ) )
88 log1 19887 . . . . . . . . 9  |-  ( log `  1 )  =  0
89 simprr 736 . . . . . . . . . 10  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
1  <_  x )
9014adantrr 700 . . . . . . . . . . 11  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  ->  x  e.  RR+ )
91 logleb 19905 . . . . . . . . . . 11  |-  ( ( 1  e.  RR+  /\  x  e.  RR+ )  ->  (
1  <_  x  <->  ( log `  1 )  <_  ( log `  x ) ) )
9223, 90, 91sylancr 647 . . . . . . . . . 10  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( 1  <_  x  <->  ( log `  1 )  <_  ( log `  x
) ) )
9389, 92mpbid 203 . . . . . . . . 9  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( log `  1
)  <_  ( log `  x ) )
9488, 93syl5eqbrr 4017 . . . . . . . 8  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
0  <_  ( log `  x ) )
9560, 94ge0p1rpd 10369 . . . . . . 7  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( log `  x
)  +  1 )  e.  RR+ )
9695, 90rpdivcld 10360 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( ( log `  x )  +  1 )  /  x )  e.  RR+ )
97 rprege0 10321 . . . . . 6  |-  ( ( ( ( log `  x
)  +  1 )  /  x )  e.  RR+  ->  ( ( ( ( log `  x
)  +  1 )  /  x )  e.  RR  /\  0  <_ 
( ( ( log `  x )  +  1 )  /  x ) ) )
98 absid 11732 . . . . . 6  |-  ( ( ( ( ( log `  x )  +  1 )  /  x )  e.  RR  /\  0  <_  ( ( ( log `  x )  +  1 )  /  x ) )  ->  ( abs `  ( ( ( log `  x )  +  1 )  /  x ) )  =  ( ( ( log `  x
)  +  1 )  /  x ) )
9996, 97, 983syl 20 . . . . 5  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( ( log `  x
)  +  1 )  /  x ) )  =  ( ( ( log `  x )  +  1 )  /  x ) )
10087, 99eqtrd 2288 . . . 4  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( ( ( log `  x )  +  1 )  /  x )  -  0 ) )  =  ( ( ( log `  x )  +  1 )  /  x ) )
10166, 84, 1003brtr4d 4013 . . 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 12073 . 2  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( log `  x
)  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) ) )  ~~> r  1 )
103102trud 1320 1  |-  ( x  e.  RR+  |->  ( ( log `  x )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) )  ~~> r  1
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    T. wtru 1312    = wceq 1619    e. wcel 1621    =/= wne 2419   class class class wbr 3983    e. cmpt 4037   ` cfv 4659  (class class class)co 5778   CCcc 8689   RRcr 8690   0cc0 8691   1c1 8692    + caddc 8694    x. cmul 8696    < clt 8821    <_ cle 8822    - cmin 8991    / cdiv 9377   NNcn 9700   NN0cn0 9918   RR+crp 10307   |_cfl 10876   !cfa 11240   abscabs 11670    ~~> r crli 11910   logclog 19860    ^ c ccxp 19861
This theorem is referenced by:  vmadivsum  20579
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-inf2 7296  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768  ax-pre-sup 8769  ax-addf 8770  ax-mulf 8771
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-of 5998  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-2o 6434  df-oadd 6437  df-er 6614  df-map 6728  df-pm 6729  df-ixp 6772  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-fi 7119  df-sup 7148  df-oi 7179  df-card 7526  df-cda 7748  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378  df-n 9701  df-2 9758  df-3 9759  df-4 9760  df-5 9761  df-6 9762  df-7 9763  df-8 9764  df-9 9765  df-10 9766  df-n0 9919  df-z 9978  df-dec 10078  df-uz 10184  df-q 10270  df-rp 10308  df-xneg 10405  df-xadd 10406  df-xmul 10407  df-ioo 10612  df-ioc 10613  df-ico 10614  df-icc 10615  df-fz 10735  df-fzo 10823  df-fl 10877  df-mod 10926  df-seq 10999  df-exp 11057  df-fac 11241  df-bc 11268  df-hash 11290  df-shft 11513  df-cj 11535  df-re 11536  df-im 11537  df-sqr 11671  df-abs 11672  df-limsup 11896  df-clim 11913  df-rlim 11914  df-sum 12110  df-ef 12297  df-sin 12299  df-cos 12300  df-pi 12302  df-struct 13098  df-ndx 13099  df-slot 13100  df-base 13101  df-sets 13102  df-ress 13103  df-plusg 13169  df-mulr 13170  df-starv 13171  df-sca 13172  df-vsca 13173  df-tset 13175  df-ple 13176  df-ds 13178  df-hom 13180  df-cco 13181  df-rest 13275  df-topn 13276  df-topgen 13292  df-pt 13293  df-prds 13296  df-xrs 13351  df-0g 13352  df-gsum 13353  df-qtop 13358  df-imas 13359  df-xps 13361  df-mre 13436  df-mrc 13437  df-acs 13439  df-mnd 14315  df-submnd 14364  df-mulg 14440  df-cntz 14741  df-cmn 15039  df-xmet 16321  df-met 16322  df-bl 16323  df-mopn 16324  df-cnfld 16326  df-top 16584  df-bases 16586  df-topon 16587  df-topsp 16588  df-cld 16704  df-ntr 16705  df-cls 16706  df-nei 16783  df-lp 16816  df-perf 16817  df-cn 16905  df-cnp 16906  df-haus 16991  df-cmp 17062  df-tx 17205  df-hmeo 17394  df-fbas 17468  df-fg 17469  df-fil 17489  df-fm 17581  df-flim 17582  df-flf 17583  df-xms 17833  df-ms 17834  df-tms 17835  df-cncf 18330  df-limc 19164  df-dv 19165  df-log 19862  df-cxp 19863
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