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Theorem logfacrlim 20457
Description: Combine the estimates logfacubnd 20454 and logfaclbnd 20455, 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 8832 . . . 4  |-  1  e.  RR
21a1i 12 . . 3  |-  (  T. 
->  1  e.  RR )
3 ax-1cn 8790 . . . 4  |-  1  e.  CC
43a1i 12 . . 3  |-  (  T. 
->  1  e.  CC )
5 relogcl 19926 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
65adantl 454 . . . . . . . 8  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( log `  x )  e.  RR )
76recnd 8856 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( log `  x )  e.  CC )
83a1i 12 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR+ )  ->  1  e.  CC )
9 rpcnne0 10366 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( x  e.  CC  /\  x  =/=  0 ) )
109adantl 454 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( x  e.  CC  /\  x  =/=  0 ) )
11 divdir 9442 . . . . . . 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 1184 . . . . . 6  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( ( log `  x
)  +  1 )  /  x )  =  ( ( ( log `  x )  /  x
)  +  ( 1  /  x ) ) )
1312mpteq2dva 4107 . . . . 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 10412 . . . . . 6  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( log `  x )  /  x )  e.  RR )
16 rpreccl 10372 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR+ )
1716adantl 454 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( 1  /  x )  e.  RR+ )
1817rpred 10385 . . . . . 6  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( 1  /  x )  e.  RR )
1910simpld 447 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  RR+ )  ->  x  e.  CC )
2019cxp1d 20047 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( x  ^ c  1 )  =  x )
2120oveq2d 5835 . . . . . . . 8  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( log `  x )  /  ( x  ^ c  1 ) )  =  ( ( log `  x )  /  x
) )
2221mpteq2dva 4107 . . . . . . 7  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( log `  x
)  /  ( x  ^ c  1 ) ) )  =  ( x  e.  RR+  |->  ( ( log `  x )  /  x ) ) )
23 1rp 10353 . . . . . . . 8  |-  1  e.  RR+
24 cxploglim 20266 . . . . . . . 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 4046 . . . . . 6  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( log `  x
)  /  x ) )  ~~> r  0 )
27 divrcnv 12305 . . . . . . 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 12110 . . . . 5  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( ( log `  x )  /  x
)  +  ( 1  /  x ) ) )  ~~> r  ( 0  +  0 ) )
3013, 29eqbrtrd 4044 . . . 4  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( ( log `  x )  +  1 )  /  x ) )  ~~> r  ( 0  +  0 ) )
31 00id 8982 . . . 4  |-  ( 0  +  0 )  =  0
3230, 31syl6breq 4063 . . 3  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( ( log `  x )  +  1 )  /  x ) )  ~~> r  0 )
33 peano2re 8980 . . . . . 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 10412 . . . 4  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( ( log `  x
)  +  1 )  /  x )  e.  RR )
3635recnd 8856 . . 3  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( ( log `  x
)  +  1 )  /  x )  e.  CC )
37 rprege0 10363 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( x  e.  RR  /\  0  <_  x ) )
3837adantl 454 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( x  e.  RR  /\  0  <_  x ) )
39 flge0nn0 10942 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  0  <_  x )  -> 
( |_ `  x
)  e.  NN0 )
40 faccl 11292 . . . . . . . . 9  |-  ( ( |_ `  x )  e.  NN0  ->  ( ! `
 ( |_ `  x ) )  e.  NN )
4138, 39, 403syl 20 . . . . . . . 8  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ! `
 ( |_ `  x ) )  e.  NN )
4241nnrpd 10384 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ! `
 ( |_ `  x ) )  e.  RR+ )
43 relogcl 19926 . . . . . . 7  |-  ( ( ! `  ( |_
`  x ) )  e.  RR+  ->  ( log `  ( ! `  ( |_ `  x ) ) )  e.  RR )
4442, 43syl 17 . . . . . 6  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( log `  ( ! `  ( |_ `  x ) ) )  e.  RR )
4544, 14rerpdivcld 10412 . . . . 5  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x )  e.  RR )
4645recnd 8856 . . . 4  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x )  e.  CC )
477, 46subcld 9152 . . 3  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( log `  x )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) )  e.  CC )
48 logfacbnd3 20456 . . . . . 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 8856 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( log `  ( ! `  ( |_ `  x ) ) )  e.  CC )
5150adantrr 699 . . . . . . . 8  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( log `  ( ! `  ( |_ `  x ) ) )  e.  CC )
529ad2antrl 710 . . . . . . . . . 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 699 . . . . . . . . . 10  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( log `  x
)  e.  CC )
55 subcl 9046 . . . . . . . . . 10  |-  ( ( ( log `  x
)  e.  CC  /\  1  e.  CC )  ->  ( ( log `  x
)  -  1 )  e.  CC )
5654, 3, 55sylancl 645 . . . . . . . . 9  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( log `  x
)  -  1 )  e.  CC )
5753, 56mulcld 8850 . . . . . . . 8  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( x  x.  (
( log `  x
)  -  1 ) )  e.  CC )
5851, 57subcld 9152 . . . . . . 7  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) )  e.  CC )
5958abscld 11912 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) ) )  e.  RR )
606adantrr 699 . . . . . . 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 10362 . . . . . . 7  |-  ( x  e.  RR+  ->  ( x  e.  RR  /\  0  <  x ) )
6362ad2antrl 710 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( x  e.  RR  /\  0  <  x ) )
64 lediv1 9616 . . . . . 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 1184 . . . . 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 9536 . . . . . . . 8  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( x  x.  ( ( log `  x
)  -  1 ) )  /  x )  =  ( ( log `  x )  -  1 ) )
6968oveq1d 5834 . . . . . . 7  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( ( x  x.  ( ( log `  x )  -  1 ) )  /  x
)  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) )  =  ( ( ( log `  x )  -  1 )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) )
70 divsubdir 9451 . . . . . . . 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 1184 . . . . . . 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 699 . . . . . . . 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 9184 . . . . . . 7  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( ( log `  x )  -  (
( log `  ( ! `  ( |_ `  x ) ) )  /  x ) )  -  1 )  =  ( ( ( log `  x )  -  1 )  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) ) )
7569, 71, 743eqtr4rd 2327 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( ( log `  x )  -  (
( log `  ( ! `  ( |_ `  x ) ) )  /  x ) )  -  1 )  =  ( ( ( x  x.  ( ( log `  x )  -  1 ) )  -  ( log `  ( ! `  ( |_ `  x ) ) ) )  /  x ) )
7675fveq2d 5489 . . . . 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 9152 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( x  x.  ( ( log `  x
)  -  1 ) )  -  ( log `  ( ! `  ( |_ `  x ) ) ) )  e.  CC )
7877, 53, 67absdivd 11931 . . . . 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 11929 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( x  x.  (
( log `  x
)  -  1 ) )  -  ( log `  ( ! `  ( |_ `  x ) ) ) ) )  =  ( abs `  (
( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) ) ) )
8037ad2antrl 710 . . . . . . 7  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( x  e.  RR  /\  0  <_  x )
)
81 absid 11775 . . . . . . 7  |-  ( ( x  e.  RR  /\  0  <_  x )  -> 
( abs `  x
)  =  x )
8280, 81syl 17 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  x
)  =  x )
8379, 82oveq12d 5837 . . . . 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 2320 . . . 4  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( ( log `  x
)  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) )  -  1 ) )  =  ( ( abs `  ( ( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) ) )  /  x ) )
8536adantrr 699 . . . . . . 7  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( ( log `  x )  +  1 )  /  x )  e.  CC )
8685subid1d 9141 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( ( ( log `  x )  +  1 )  /  x )  -  0 )  =  ( ( ( log `  x
)  +  1 )  /  x ) )
8786fveq2d 5489 . . . . 5  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( ( ( log `  x )  +  1 )  /  x )  -  0 ) )  =  ( abs `  (
( ( log `  x
)  +  1 )  /  x ) ) )
88 log1 19933 . . . . . . . . 9  |-  ( log `  1 )  =  0
89 simprr 735 . . . . . . . . . 10  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
1  <_  x )
9014adantrr 699 . . . . . . . . . . 11  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  ->  x  e.  RR+ )
91 logleb 19951 . . . . . . . . . . 11  |-  ( ( 1  e.  RR+  /\  x  e.  RR+ )  ->  (
1  <_  x  <->  ( log `  1 )  <_  ( log `  x ) ) )
9223, 90, 91sylancr 646 . . . . . . . . . 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 4058 . . . . . . . 8  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
0  <_  ( log `  x ) )
9560, 94ge0p1rpd 10411 . . . . . . 7  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( log `  x
)  +  1 )  e.  RR+ )
9695, 90rpdivcld 10402 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( ( log `  x )  +  1 )  /  x )  e.  RR+ )
97 rprege0 10363 . . . . . 6  |-  ( ( ( ( log `  x
)  +  1 )  /  x )  e.  RR+  ->  ( ( ( ( log `  x
)  +  1 )  /  x )  e.  RR  /\  0  <_ 
( ( ( log `  x )  +  1 )  /  x ) ) )
98 absid 11775 . . . . . 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 2316 . . . 4  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( ( ( log `  x )  +  1 )  /  x )  -  0 ) )  =  ( ( ( log `  x )  +  1 )  /  x ) )
10166, 84, 1003brtr4d 4054 . . 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 12116 . 2  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( log `  x
)  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) ) )  ~~> r  1 )
103102trud 1316 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 1309    = wceq 1624    e. wcel 1685    =/= wne 2447   class class class wbr 4024    e. cmpt 4078   ` cfv 5221  (class class class)co 5819   CCcc 8730   RRcr 8731   0cc0 8732   1c1 8733    + caddc 8735    x. cmul 8737    < clt 8862    <_ cle 8863    - cmin 9032    / cdiv 9418   NNcn 9741   NN0cn0 9960   RR+crp 10349   |_cfl 10918   !cfa 11282   abscabs 11713    ~~> r crli 11953   logclog 19906    ^ c ccxp 19907
This theorem is referenced by:  vmadivsum  20625
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810  ax-addf 8811  ax-mulf 8812
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-of 6039  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-2o 6475  df-oadd 6478  df-er 6655  df-map 6769  df-pm 6770  df-ixp 6813  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-fi 7160  df-sup 7189  df-oi 7220  df-card 7567  df-cda 7789  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-4 9801  df-5 9802  df-6 9803  df-7 9804  df-8 9805  df-9 9806  df-10 9807  df-n0 9961  df-z 10020  df-dec 10120  df-uz 10226  df-q 10312  df-rp 10350  df-xneg 10447  df-xadd 10448  df-xmul 10449  df-ioo 10654  df-ioc 10655  df-ico 10656  df-icc 10657  df-fz 10777  df-fzo 10865  df-fl 10919  df-mod 10968  df-seq 11041  df-exp 11099  df-fac 11283  df-bc 11310  df-hash 11332  df-shft 11556  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-abs 11715  df-limsup 11939  df-clim 11956  df-rlim 11957  df-sum 12153  df-ef 12343  df-sin 12345  df-cos 12346  df-pi 12348  df-struct 13144  df-ndx 13145  df-slot 13146  df-base 13147  df-sets 13148  df-ress 13149  df-plusg 13215  df-mulr 13216  df-starv 13217  df-sca 13218  df-vsca 13219  df-tset 13221  df-ple 13222  df-ds 13224  df-hom 13226  df-cco 13227  df-rest 13321  df-topn 13322  df-topgen 13338  df-pt 13339  df-prds 13342  df-xrs 13397  df-0g 13398  df-gsum 13399  df-qtop 13404  df-imas 13405  df-xps 13407  df-mre 13482  df-mrc 13483  df-acs 13485  df-mnd 14361  df-submnd 14410  df-mulg 14486  df-cntz 14787  df-cmn 15085  df-xmet 16367  df-met 16368  df-bl 16369  df-mopn 16370  df-cnfld 16372  df-top 16630  df-bases 16632  df-topon 16633  df-topsp 16634  df-cld 16750  df-ntr 16751  df-cls 16752  df-nei 16829  df-lp 16862  df-perf 16863  df-cn 16951  df-cnp 16952  df-haus 17037  df-cmp 17108  df-tx 17251  df-hmeo 17440  df-fbas 17514  df-fg 17515  df-fil 17535  df-fm 17627  df-flim 17628  df-flf 17629  df-xms 17879  df-ms 17880  df-tms 17881  df-cncf 18376  df-limc 19210  df-dv 19211  df-log 19908  df-cxp 19909
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