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Theorem logfacrlim 20996
Description: Combine the estimates logfacubnd 20993 and logfaclbnd 20994, 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 9079 . . . 4  |-  1  e.  RR
21a1i 11 . . 3  |-  (  T. 
->  1  e.  RR )
3 ax-1cn 9037 . . . 4  |-  1  e.  CC
43a1i 11 . . 3  |-  (  T. 
->  1  e.  CC )
5 relogcl 20461 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
65adantl 453 . . . . . . . 8  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( log `  x )  e.  RR )
76recnd 9103 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( log `  x )  e.  CC )
83a1i 11 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR+ )  ->  1  e.  CC )
9 rpcnne0 10618 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( x  e.  CC  /\  x  =/=  0 ) )
109adantl 453 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( x  e.  CC  /\  x  =/=  0 ) )
11 divdir 9690 . . . . . . 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 4287 . . . . 5  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( ( log `  x )  +  1 )  /  x ) )  =  ( x  e.  RR+  |->  ( ( ( log `  x
)  /  x )  +  ( 1  /  x ) ) ) )
14 simpr 448 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR+ )  ->  x  e.  RR+ )
156, 14rerpdivcld 10664 . . . . . 6  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( log `  x )  /  x )  e.  RR )
16 rpreccl 10624 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR+ )
1716adantl 453 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( 1  /  x )  e.  RR+ )
1817rpred 10637 . . . . . 6  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( 1  /  x )  e.  RR )
1910simpld 446 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  RR+ )  ->  x  e.  CC )
2019cxp1d 20585 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( x  ^ c  1 )  =  x )
2120oveq2d 6088 . . . . . . . 8  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( log `  x )  /  ( x  ^ c  1 ) )  =  ( ( log `  x )  /  x
) )
2221mpteq2dva 4287 . . . . . . 7  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( log `  x
)  /  ( x  ^ c  1 ) ) )  =  ( x  e.  RR+  |->  ( ( log `  x )  /  x ) ) )
23 1rp 10605 . . . . . . . 8  |-  1  e.  RR+
24 cxploglim 20804 . . . . . . . 8  |-  ( 1  e.  RR+  ->  ( x  e.  RR+  |->  ( ( log `  x )  /  ( x  ^ c  1 ) ) )  ~~> r  0 )
2523, 24mp1i 12 . . . . . . 7  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( log `  x
)  /  ( x  ^ c  1 ) ) )  ~~> r  0 )
2622, 25eqbrtrrd 4226 . . . . . 6  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( log `  x
)  /  x ) )  ~~> r  0 )
27 divrcnv 12620 . . . . . . 7  |-  ( 1  e.  CC  ->  (
x  e.  RR+  |->  ( 1  /  x ) )  ~~> r  0 )
283, 27mp1i 12 . . . . . 6  |-  (  T. 
->  ( x  e.  RR+  |->  ( 1  /  x
) )  ~~> r  0 )
2915, 18, 26, 28rlimadd 12424 . . . . 5  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( ( log `  x )  /  x
)  +  ( 1  /  x ) ) )  ~~> r  ( 0  +  0 ) )
3013, 29eqbrtrd 4224 . . . 4  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( ( log `  x )  +  1 )  /  x ) )  ~~> r  ( 0  +  0 ) )
31 00id 9230 . . . 4  |-  ( 0  +  0 )  =  0
3230, 31syl6breq 4243 . . 3  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( ( log `  x )  +  1 )  /  x ) )  ~~> r  0 )
33 peano2re 9228 . . . . . 6  |-  ( ( log `  x )  e.  RR  ->  (
( log `  x
)  +  1 )  e.  RR )
346, 33syl 16 . . . . 5  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( log `  x )  +  1 )  e.  RR )
3534, 14rerpdivcld 10664 . . . 4  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( ( log `  x
)  +  1 )  /  x )  e.  RR )
3635recnd 9103 . . 3  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( ( log `  x
)  +  1 )  /  x )  e.  CC )
37 rprege0 10615 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( x  e.  RR  /\  0  <_  x ) )
3837adantl 453 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( x  e.  RR  /\  0  <_  x ) )
39 flge0nn0 11213 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  0  <_  x )  -> 
( |_ `  x
)  e.  NN0 )
40 faccl 11564 . . . . . . . . 9  |-  ( ( |_ `  x )  e.  NN0  ->  ( ! `
 ( |_ `  x ) )  e.  NN )
4138, 39, 403syl 19 . . . . . . . 8  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ! `
 ( |_ `  x ) )  e.  NN )
4241nnrpd 10636 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ! `
 ( |_ `  x ) )  e.  RR+ )
43 relogcl 20461 . . . . . . 7  |-  ( ( ! `  ( |_
`  x ) )  e.  RR+  ->  ( log `  ( ! `  ( |_ `  x ) ) )  e.  RR )
4442, 43syl 16 . . . . . 6  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( log `  ( ! `  ( |_ `  x ) ) )  e.  RR )
4544, 14rerpdivcld 10664 . . . . 5  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x )  e.  RR )
4645recnd 9103 . . . 4  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x )  e.  CC )
477, 46subcld 9400 . . 3  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( log `  x )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) )  e.  CC )
48 logfacbnd3 20995 . . . . . 6  |-  ( ( x  e.  RR+  /\  1  <_  x )  ->  ( abs `  ( ( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x )  -  1 ) ) ) )  <_  ( ( log `  x )  +  1 ) )
4948adantl 453 . . . . 5  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) ) )  <_ 
( ( log `  x
)  +  1 ) )
5044recnd 9103 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( log `  ( ! `  ( |_ `  x ) ) )  e.  CC )
5150adantrr 698 . . . . . . . 8  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( log `  ( ! `  ( |_ `  x ) ) )  e.  CC )
529ad2antrl 709 . . . . . . . . . 10  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( x  e.  CC  /\  x  =/=  0 ) )
5352simpld 446 . . . . . . . . 9  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  ->  x  e.  CC )
547adantrr 698 . . . . . . . . . 10  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( log `  x
)  e.  CC )
55 subcl 9294 . . . . . . . . . 10  |-  ( ( ( log `  x
)  e.  CC  /\  1  e.  CC )  ->  ( ( log `  x
)  -  1 )  e.  CC )
5654, 3, 55sylancl 644 . . . . . . . . 9  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( log `  x
)  -  1 )  e.  CC )
5753, 56mulcld 9097 . . . . . . . 8  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( x  x.  (
( log `  x
)  -  1 ) )  e.  CC )
5851, 57subcld 9400 . . . . . . 7  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) )  e.  CC )
5958abscld 12226 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) ) )  e.  RR )
606adantrr 698 . . . . . . 7  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( log `  x
)  e.  RR )
6160, 33syl 16 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( log `  x
)  +  1 )  e.  RR )
62 rpregt0 10614 . . . . . . 7  |-  ( x  e.  RR+  ->  ( x  e.  RR  /\  0  <  x ) )
6362ad2antrl 709 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( x  e.  RR  /\  0  <  x ) )
64 lediv1 9864 . . . . . 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 202 . . . 4  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( abs `  (
( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) ) )  /  x )  <_  (
( ( log `  x
)  +  1 )  /  x ) )
6752simprd 450 . . . . . . . . 9  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  ->  x  =/=  0 )
6856, 53, 67divcan3d 9784 . . . . . . . 8  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( x  x.  ( ( log `  x
)  -  1 ) )  /  x )  =  ( ( log `  x )  -  1 ) )
6968oveq1d 6087 . . . . . . 7  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( ( x  x.  ( ( log `  x )  -  1 ) )  /  x
)  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) )  =  ( ( ( log `  x )  -  1 )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) )
70 divsubdir 9699 . . . . . . . 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 698 . . . . . . . 8  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( log `  ( ! `  ( |_ `  x ) ) )  /  x )  e.  CC )
733a1i 11 . . . . . . . 8  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
1  e.  CC )
7454, 72, 73sub32d 9432 . . . . . . 7  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( ( log `  x )  -  (
( log `  ( ! `  ( |_ `  x ) ) )  /  x ) )  -  1 )  =  ( ( ( log `  x )  -  1 )  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) ) )
7569, 71, 743eqtr4rd 2478 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( ( log `  x )  -  (
( log `  ( ! `  ( |_ `  x ) ) )  /  x ) )  -  1 )  =  ( ( ( x  x.  ( ( log `  x )  -  1 ) )  -  ( log `  ( ! `  ( |_ `  x ) ) ) )  /  x ) )
7675fveq2d 5723 . . . . 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 9400 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( x  x.  ( ( log `  x
)  -  1 ) )  -  ( log `  ( ! `  ( |_ `  x ) ) ) )  e.  CC )
7877, 53, 67absdivd 12245 . . . . 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 12243 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( x  x.  (
( log `  x
)  -  1 ) )  -  ( log `  ( ! `  ( |_ `  x ) ) ) ) )  =  ( abs `  (
( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) ) ) )
8037ad2antrl 709 . . . . . . 7  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( x  e.  RR  /\  0  <_  x )
)
81 absid 12089 . . . . . . 7  |-  ( ( x  e.  RR  /\  0  <_  x )  -> 
( abs `  x
)  =  x )
8280, 81syl 16 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  x
)  =  x )
8379, 82oveq12d 6090 . . . . 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 2471 . . . 4  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( ( log `  x
)  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) )  -  1 ) )  =  ( ( abs `  ( ( log `  ( ! `  ( |_ `  x ) ) )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) ) )  /  x ) )
8536adantrr 698 . . . . . . 7  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( ( log `  x )  +  1 )  /  x )  e.  CC )
8685subid1d 9389 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( ( ( log `  x )  +  1 )  /  x )  -  0 )  =  ( ( ( log `  x
)  +  1 )  /  x ) )
8786fveq2d 5723 . . . . 5  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( ( ( log `  x )  +  1 )  /  x )  -  0 ) )  =  ( abs `  (
( ( log `  x
)  +  1 )  /  x ) ) )
88 log1 20468 . . . . . . . . 9  |-  ( log `  1 )  =  0
89 simprr 734 . . . . . . . . . 10  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
1  <_  x )
9014adantrr 698 . . . . . . . . . . 11  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  ->  x  e.  RR+ )
91 logleb 20486 . . . . . . . . . . 11  |-  ( ( 1  e.  RR+  /\  x  e.  RR+ )  ->  (
1  <_  x  <->  ( log `  1 )  <_  ( log `  x ) ) )
9223, 90, 91sylancr 645 . . . . . . . . . 10  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( 1  <_  x  <->  ( log `  1 )  <_  ( log `  x
) ) )
9389, 92mpbid 202 . . . . . . . . 9  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( log `  1
)  <_  ( log `  x ) )
9488, 93syl5eqbrr 4238 . . . . . . . 8  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
0  <_  ( log `  x ) )
9560, 94ge0p1rpd 10663 . . . . . . 7  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( log `  x
)  +  1 )  e.  RR+ )
9695, 90rpdivcld 10654 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( ( log `  x )  +  1 )  /  x )  e.  RR+ )
97 rprege0 10615 . . . . . 6  |-  ( ( ( ( log `  x
)  +  1 )  /  x )  e.  RR+  ->  ( ( ( ( log `  x
)  +  1 )  /  x )  e.  RR  /\  0  <_ 
( ( ( log `  x )  +  1 )  /  x ) ) )
98 absid 12089 . . . . . 6  |-  ( ( ( ( ( log `  x )  +  1 )  /  x )  e.  RR  /\  0  <_  ( ( ( log `  x )  +  1 )  /  x ) )  ->  ( abs `  ( ( ( log `  x )  +  1 )  /  x ) )  =  ( ( ( log `  x
)  +  1 )  /  x ) )
9996, 97, 983syl 19 . . . . 5  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( ( log `  x
)  +  1 )  /  x ) )  =  ( ( ( log `  x )  +  1 )  /  x ) )
10087, 99eqtrd 2467 . . . 4  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  (
( ( ( log `  x )  +  1 )  /  x )  -  0 ) )  =  ( ( ( log `  x )  +  1 )  /  x ) )
10166, 84, 1003brtr4d 4234 . . 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 12430 . 2  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( log `  x
)  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) ) )  ~~> r  1 )
103102trud 1332 1  |-  ( x  e.  RR+  |->  ( ( log `  x )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) )  ~~> r  1
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    T. wtru 1325    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204    e. cmpt 4258   ` cfv 5445  (class class class)co 6072   CCcc 8977   RRcr 8978   0cc0 8979   1c1 8980    + caddc 8982    x. cmul 8984    < clt 9109    <_ cle 9110    - cmin 9280    / cdiv 9666   NNcn 9989   NN0cn0 10210   RR+crp 10601   |_cfl 11189   !cfa 11554   abscabs 12027    ~~> r crli 12267   logclog 20440    ^ c ccxp 20441
This theorem is referenced by:  vmadivsum  21164
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057  ax-addf 9058  ax-mulf 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-of 6296  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-2o 6716  df-oadd 6719  df-er 6896  df-map 7011  df-pm 7012  df-ixp 7055  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-fi 7407  df-sup 7437  df-oi 7468  df-card 7815  df-cda 8037  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-5 10050  df-6 10051  df-7 10052  df-8 10053  df-9 10054  df-10 10055  df-n0 10211  df-z 10272  df-dec 10372  df-uz 10478  df-q 10564  df-rp 10602  df-xneg 10699  df-xadd 10700  df-xmul 10701  df-ioo 10909  df-ioc 10910  df-ico 10911  df-icc 10912  df-fz 11033  df-fzo 11124  df-fl 11190  df-mod 11239  df-seq 11312  df-exp 11371  df-fac 11555  df-bc 11582  df-hash 11607  df-shft 11870  df-cj 11892  df-re 11893  df-im 11894  df-sqr 12028  df-abs 12029  df-limsup 12253  df-clim 12270  df-rlim 12271  df-sum 12468  df-ef 12658  df-sin 12660  df-cos 12661  df-pi 12663  df-struct 13459  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-ress 13464  df-plusg 13530  df-mulr 13531  df-starv 13532  df-sca 13533  df-vsca 13534  df-tset 13536  df-ple 13537  df-ds 13539  df-unif 13540  df-hom 13541  df-cco 13542  df-rest 13638  df-topn 13639  df-topgen 13655  df-pt 13656  df-prds 13659  df-xrs 13714  df-0g 13715  df-gsum 13716  df-qtop 13721  df-imas 13722  df-xps 13724  df-mre 13799  df-mrc 13800  df-acs 13802  df-mnd 14678  df-submnd 14727  df-mulg 14803  df-cntz 15104  df-cmn 15402  df-psmet 16682  df-xmet 16683  df-met 16684  df-bl 16685  df-mopn 16686  df-fbas 16687  df-fg 16688  df-cnfld 16692  df-top 16951  df-bases 16953  df-topon 16954  df-topsp 16955  df-cld 17071  df-ntr 17072  df-cls 17073  df-nei 17150  df-lp 17188  df-perf 17189  df-cn 17279  df-cnp 17280  df-haus 17367  df-cmp 17438  df-tx 17582  df-hmeo 17775  df-fil 17866  df-fm 17958  df-flim 17959  df-flf 17960  df-xms 18338  df-ms 18339  df-tms 18340  df-cncf 18896  df-limc 19741  df-dv 19742  df-log 20442  df-cxp 20443
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