MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  logfacbnd3 Unicode version

Theorem logfacbnd3 20478
Description: Show the stronger statement  log ( x ! )  =  x log x  -  x  +  O ( log x
) alluded to in logfacrlim 20479. (Contributed by Mario Carneiro, 20-May-2016.)
Assertion
Ref Expression
logfacbnd3  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( abs `  ( ( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A )  -  1 ) ) ) )  <_  ( ( log `  A )  +  1 ) )

Proof of Theorem logfacbnd3
Dummy variables  x  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 443 . . . . . . . . . . . 12  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  A  e.  RR+ )
21rprege0d 10413 . . . . . . . . . . 11  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( A  e.  RR  /\  0  <_  A ) )
3 flge0nn0 10964 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( |_ `  A
)  e.  NN0 )
42, 3syl 15 . . . . . . . . . 10  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( |_ `  A )  e. 
NN0 )
5 faccl 11314 . . . . . . . . . 10  |-  ( ( |_ `  A )  e.  NN0  ->  ( ! `
 ( |_ `  A ) )  e.  NN )
64, 5syl 15 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( ! `  ( |_ `  A ) )  e.  NN )
76nnrpd 10405 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( ! `  ( |_ `  A ) )  e.  RR+ )
8 relogcl 19948 . . . . . . . 8  |-  ( ( ! `  ( |_
`  A ) )  e.  RR+  ->  ( log `  ( ! `  ( |_ `  A ) ) )  e.  RR )
97, 8syl 15 . . . . . . 7  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( log `  ( ! `  ( |_ `  A ) ) )  e.  RR )
10 rpre 10376 . . . . . . . . 9  |-  ( A  e.  RR+  ->  A  e.  RR )
1110adantr 451 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  A  e.  RR )
12 relogcl 19948 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  ( log `  A )  e.  RR )
1312adantr 451 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( log `  A )  e.  RR )
14 peano2rem 9129 . . . . . . . . 9  |-  ( ( log `  A )  e.  RR  ->  (
( log `  A
)  -  1 )  e.  RR )
1513, 14syl 15 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( log `  A
)  -  1 )  e.  RR )
1611, 15remulcld 8879 . . . . . . 7  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( A  x.  ( ( log `  A )  - 
1 ) )  e.  RR )
179, 16resubcld 9227 . . . . . 6  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  e.  RR )
1817recnd 8877 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  e.  CC )
1918abscld 11934 . . . 4  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( abs `  ( ( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A )  -  1 ) ) ) )  e.  RR )
20 peano2rem 9129 . . . 4  |-  ( ( abs `  ( ( log `  ( ! `
 ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )  e.  RR  ->  ( ( abs `  ( ( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A )  -  1 ) ) ) )  -  1 )  e.  RR )
2119, 20syl 15 . . 3  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( abs `  (
( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )  - 
1 )  e.  RR )
22 ax-1cn 8811 . . . . 5  |-  1  e.  CC
23 subcl 9067 . . . . 5  |-  ( ( ( ( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  e.  CC  /\  1  e.  CC )  ->  ( ( ( log `  ( ! `
 ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  -  1 )  e.  CC )
2418, 22, 23sylancl 643 . . . 4  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( ( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  -  1 )  e.  CC )
2524abscld 11934 . . 3  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( abs `  ( ( ( log `  ( ! `
 ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  -  1 ) )  e.  RR )
26 abs1 11798 . . . . 5  |-  ( abs `  1 )  =  1
2726oveq2i 5885 . . . 4  |-  ( ( abs `  ( ( log `  ( ! `
 ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )  -  ( abs `  1 ) )  =  ( ( abs `  ( ( log `  ( ! `
 ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )  - 
1 )
28 abs2dif 11832 . . . . 5  |-  ( ( ( ( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  e.  CC  /\  1  e.  CC )  ->  ( ( abs `  ( ( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )  -  ( abs `  1 ) )  <_  ( abs `  ( ( ( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A )  -  1 ) ) )  - 
1 ) ) )
2918, 22, 28sylancl 643 . . . 4  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( abs `  (
( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )  -  ( abs `  1 ) )  <_  ( abs `  ( ( ( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A )  -  1 ) ) )  - 
1 ) ) )
3027, 29syl5eqbrr 4073 . . 3  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( abs `  (
( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )  - 
1 )  <_  ( abs `  ( ( ( log `  ( ! `
 ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  -  1 ) ) )
31 fveq2 5541 . . . . . . . . . . . 12  |-  ( x  =  A  ->  ( |_ `  x )  =  ( |_ `  A
) )
3231oveq2d 5890 . . . . . . . . . . 11  |-  ( x  =  A  ->  (
1 ... ( |_ `  x ) )  =  ( 1 ... ( |_ `  A ) ) )
3332sumeq1d 12190 . . . . . . . . . 10  |-  ( x  =  A  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( log `  n
)  =  sum_ n  e.  ( 1 ... ( |_ `  A ) ) ( log `  n
) )
34 id 19 . . . . . . . . . . 11  |-  ( x  =  A  ->  x  =  A )
35 fveq2 5541 . . . . . . . . . . . 12  |-  ( x  =  A  ->  ( log `  x )  =  ( log `  A
) )
3635oveq1d 5889 . . . . . . . . . . 11  |-  ( x  =  A  ->  (
( log `  x
)  -  1 )  =  ( ( log `  A )  -  1 ) )
3734, 36oveq12d 5892 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  x.  ( ( log `  x )  -  1 ) )  =  ( A  x.  ( ( log `  A
)  -  1 ) ) )
3833, 37oveq12d 5892 . . . . . . . . 9  |-  ( x  =  A  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( log `  n )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) )  =  (
sum_ n  e.  (
1 ... ( |_ `  A ) ) ( log `  n )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )
39 eqid 2296 . . . . . . . . 9  |-  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( log `  n )  -  (
x  x.  ( ( log `  x )  -  1 ) ) ) )  =  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( log `  n )  -  (
x  x.  ( ( log `  x )  -  1 ) ) ) )
40 ovex 5899 . . . . . . . . 9  |-  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( log `  n )  -  (
x  x.  ( ( log `  x )  -  1 ) ) )  e.  _V
4138, 39, 40fvmpt3i 5621 . . . . . . . 8  |-  ( A  e.  RR+  ->  ( ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( log `  n )  -  (
x  x.  ( ( log `  x )  -  1 ) ) ) ) `  A
)  =  ( sum_ n  e.  ( 1 ... ( |_ `  A
) ) ( log `  n )  -  ( A  x.  ( ( log `  A )  - 
1 ) ) ) )
4241adantr 451 . . . . . . 7  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( log `  n
)  -  ( x  x.  ( ( log `  x )  -  1 ) ) ) ) `
 A )  =  ( sum_ n  e.  ( 1 ... ( |_
`  A ) ) ( log `  n
)  -  ( A  x.  ( ( log `  A )  -  1 ) ) ) )
43 logfac 19970 . . . . . . . . 9  |-  ( ( |_ `  A )  e.  NN0  ->  ( log `  ( ! `  ( |_ `  A ) ) )  =  sum_ n  e.  ( 1 ... ( |_ `  A ) ) ( log `  n
) )
444, 43syl 15 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( log `  ( ! `  ( |_ `  A ) ) )  =  sum_ n  e.  ( 1 ... ( |_ `  A
) ) ( log `  n ) )
4544oveq1d 5889 . . . . . . 7  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  =  (
sum_ n  e.  (
1 ... ( |_ `  A ) ) ( log `  n )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )
4642, 45eqtr4d 2331 . . . . . 6  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( log `  n
)  -  ( x  x.  ( ( log `  x )  -  1 ) ) ) ) `
 A )  =  ( ( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )
47 1rp 10374 . . . . . . 7  |-  1  e.  RR+
48 fveq2 5541 . . . . . . . . . . . . . 14  |-  ( x  =  1  ->  ( |_ `  x )  =  ( |_ `  1
) )
49 1z 10069 . . . . . . . . . . . . . . 15  |-  1  e.  ZZ
50 flid 10955 . . . . . . . . . . . . . . 15  |-  ( 1  e.  ZZ  ->  ( |_ `  1 )  =  1 )
5149, 50ax-mp 8 . . . . . . . . . . . . . 14  |-  ( |_
`  1 )  =  1
5248, 51syl6eq 2344 . . . . . . . . . . . . 13  |-  ( x  =  1  ->  ( |_ `  x )  =  1 )
5352oveq2d 5890 . . . . . . . . . . . 12  |-  ( x  =  1  ->  (
1 ... ( |_ `  x ) )  =  ( 1 ... 1
) )
5453sumeq1d 12190 . . . . . . . . . . 11  |-  ( x  =  1  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( log `  n
)  =  sum_ n  e.  ( 1 ... 1
) ( log `  n
) )
55 0cn 8847 . . . . . . . . . . . 12  |-  0  e.  CC
56 fveq2 5541 . . . . . . . . . . . . . 14  |-  ( n  =  1  ->  ( log `  n )  =  ( log `  1
) )
57 log1 19955 . . . . . . . . . . . . . 14  |-  ( log `  1 )  =  0
5856, 57syl6eq 2344 . . . . . . . . . . . . 13  |-  ( n  =  1  ->  ( log `  n )  =  0 )
5958fsum1 12230 . . . . . . . . . . . 12  |-  ( ( 1  e.  ZZ  /\  0  e.  CC )  -> 
sum_ n  e.  (
1 ... 1 ) ( log `  n )  =  0 )
6049, 55, 59mp2an 653 . . . . . . . . . . 11  |-  sum_ n  e.  ( 1 ... 1
) ( log `  n
)  =  0
6154, 60syl6eq 2344 . . . . . . . . . 10  |-  ( x  =  1  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( log `  n
)  =  0 )
62 id 19 . . . . . . . . . . . 12  |-  ( x  =  1  ->  x  =  1 )
63 fveq2 5541 . . . . . . . . . . . . . 14  |-  ( x  =  1  ->  ( log `  x )  =  ( log `  1
) )
6463, 57syl6eq 2344 . . . . . . . . . . . . 13  |-  ( x  =  1  ->  ( log `  x )  =  0 )
6564oveq1d 5889 . . . . . . . . . . . 12  |-  ( x  =  1  ->  (
( log `  x
)  -  1 )  =  ( 0  -  1 ) )
6662, 65oveq12d 5892 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
x  x.  ( ( log `  x )  -  1 ) )  =  ( 1  x.  ( 0  -  1 ) ) )
6755, 22subcli 9138 . . . . . . . . . . . 12  |-  ( 0  -  1 )  e.  CC
6867mulid2i 8856 . . . . . . . . . . 11  |-  ( 1  x.  ( 0  -  1 ) )  =  ( 0  -  1 )
6966, 68syl6eq 2344 . . . . . . . . . 10  |-  ( x  =  1  ->  (
x  x.  ( ( log `  x )  -  1 ) )  =  ( 0  -  1 ) )
7061, 69oveq12d 5892 . . . . . . . . 9  |-  ( x  =  1  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( log `  n )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) )  =  ( 0  -  ( 0  -  1 ) ) )
71 nncan 9092 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  1  e.  CC )  ->  ( 0  -  (
0  -  1 ) )  =  1 )
7255, 22, 71mp2an 653 . . . . . . . . 9  |-  ( 0  -  ( 0  -  1 ) )  =  1
7370, 72syl6eq 2344 . . . . . . . 8  |-  ( x  =  1  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( log `  n )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) )  =  1 )
7473, 39, 40fvmpt3i 5621 . . . . . . 7  |-  ( 1  e.  RR+  ->  ( ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( log `  n )  -  (
x  x.  ( ( log `  x )  -  1 ) ) ) ) `  1
)  =  1 )
7547, 74mp1i 11 . . . . . 6  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( log `  n
)  -  ( x  x.  ( ( log `  x )  -  1 ) ) ) ) `
 1 )  =  1 )
7646, 75oveq12d 5892 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( log `  n
)  -  ( x  x.  ( ( log `  x )  -  1 ) ) ) ) `
 A )  -  ( ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( log `  n
)  -  ( x  x.  ( ( log `  x )  -  1 ) ) ) ) `
 1 ) )  =  ( ( ( log `  ( ! `
 ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  -  1 ) )
7776fveq2d 5545 . . . 4  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( abs `  ( ( ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( log `  n )  -  (
x  x.  ( ( log `  x )  -  1 ) ) ) ) `  A
)  -  ( ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( log `  n )  -  (
x  x.  ( ( log `  x )  -  1 ) ) ) ) `  1
) ) )  =  ( abs `  (
( ( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  -  1 ) ) )
78 ioorp 10743 . . . . . 6  |-  ( 0 (,)  +oo )  =  RR+
7978eqcomi 2300 . . . . 5  |-  RR+  =  ( 0 (,)  +oo )
80 nnuz 10279 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
8149a1i 10 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  1  e.  ZZ )
82 1re 8853 . . . . . 6  |-  1  e.  RR
8382a1i 10 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  1  e.  RR )
84 pnfxr 10471 . . . . . 6  |-  +oo  e.  RR*
8584a1i 10 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  +oo  e.  RR* )
86 1nn0 9997 . . . . . . 7  |-  1  e.  NN0
8782, 86nn0addge1i 10028 . . . . . 6  |-  1  <_  ( 1  +  1 )
8887a1i 10 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  1  <_  ( 1  +  1 ) )
89 0re 8854 . . . . . 6  |-  0  e.  RR
9089a1i 10 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  0  e.  RR )
91 rpre 10376 . . . . . . 7  |-  ( x  e.  RR+  ->  x  e.  RR )
9291adantl 452 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  x  e.  RR+ )  ->  x  e.  RR )
93 relogcl 19948 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
9493adantl 452 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  x  e.  RR+ )  ->  ( log `  x
)  e.  RR )
95 peano2rem 9129 . . . . . . 7  |-  ( ( log `  x )  e.  RR  ->  (
( log `  x
)  -  1 )  e.  RR )
9694, 95syl 15 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  x  e.  RR+ )  ->  ( ( log `  x
)  -  1 )  e.  RR )
9792, 96remulcld 8879 . . . . 5  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  x  e.  RR+ )  ->  ( x  x.  (
( log `  x
)  -  1 ) )  e.  RR )
98 nnrp 10379 . . . . . 6  |-  ( x  e.  NN  ->  x  e.  RR+ )
9998, 94sylan2 460 . . . . 5  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  x  e.  NN )  ->  ( log `  x
)  e.  RR )
100 advlog 20017 . . . . . 6  |-  ( RR 
_D  ( x  e.  RR+  |->  ( x  x.  ( ( log `  x
)  -  1 ) ) ) )  =  ( x  e.  RR+  |->  ( log `  x ) )
101100a1i 10 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( RR  _D  ( x  e.  RR+  |->  ( x  x.  ( ( log `  x
)  -  1 ) ) ) )  =  ( x  e.  RR+  |->  ( log `  x ) ) )
102 fveq2 5541 . . . . 5  |-  ( x  =  n  ->  ( log `  x )  =  ( log `  n
) )
103 simp32 992 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  n  e.  RR+ )  /\  ( 1  <_  x  /\  x  <_  n  /\  n  <_  +oo ) )  ->  x  <_  n )
104 logleb 19973 . . . . . . 7  |-  ( ( x  e.  RR+  /\  n  e.  RR+ )  ->  (
x  <_  n  <->  ( log `  x )  <_  ( log `  n ) ) )
1051043ad2ant2 977 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  n  e.  RR+ )  /\  ( 1  <_  x  /\  x  <_  n  /\  n  <_  +oo ) )  -> 
( x  <_  n  <->  ( log `  x )  <_  ( log `  n
) ) )
106103, 105mpbid 201 . . . . 5  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  n  e.  RR+ )  /\  ( 1  <_  x  /\  x  <_  n  /\  n  <_  +oo ) )  -> 
( log `  x
)  <_  ( log `  n ) )
107 simprr 733 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  1  <_  x )
)  ->  1  <_  x )
108 simprl 732 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  1  <_  x )
)  ->  x  e.  RR+ )
109 logleb 19973 . . . . . . . 8  |-  ( ( 1  e.  RR+  /\  x  e.  RR+ )  ->  (
1  <_  x  <->  ( log `  1 )  <_  ( log `  x ) ) )
11047, 108, 109sylancr 644 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  1  <_  x )
)  ->  ( 1  <_  x  <->  ( log `  1 )  <_  ( log `  x ) ) )
111107, 110mpbid 201 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  1  <_  x )
)  ->  ( log `  1 )  <_  ( log `  x ) )
11257, 111syl5eqbrr 4073 . . . . 5  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  1  <_  x )
)  ->  0  <_  ( log `  x ) )
11347a1i 10 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  1  e.  RR+ )
114 1le1 9412 . . . . . 6  |-  1  <_  1
115114a1i 10 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  1  <_  1 )
116 simpr 447 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  1  <_  A )
11711rexrd 8897 . . . . . 6  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  A  e.  RR* )
118 pnfge 10485 . . . . . 6  |-  ( A  e.  RR*  ->  A  <_  +oo )
119117, 118syl 15 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  A  <_  +oo )
12079, 80, 81, 83, 85, 88, 90, 97, 94, 99, 101, 102, 106, 39, 112, 113, 1, 115, 116, 119, 35dvfsum2 19397 . . . 4  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( abs `  ( ( ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( log `  n )  -  (
x  x.  ( ( log `  x )  -  1 ) ) ) ) `  A
)  -  ( ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( log `  n )  -  (
x  x.  ( ( log `  x )  -  1 ) ) ) ) `  1
) ) )  <_ 
( log `  A
) )
12177, 120eqbrtrrd 4061 . . 3  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( abs `  ( ( ( log `  ( ! `
 ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  -  1 ) )  <_  ( log `  A ) )
12221, 25, 13, 30, 121letrd 8989 . 2  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( abs `  (
( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )  - 
1 )  <_  ( log `  A ) )
12319, 83, 13lesubaddd 9385 . 2  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( ( abs `  (
( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )  - 
1 )  <_  ( log `  A )  <->  ( abs `  ( ( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )  <_ 
( ( log `  A
)  +  1 ) ) )
124122, 123mpbid 201 1  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( abs `  ( ( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A )  -  1 ) ) ) )  <_  ( ( log `  A )  +  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    +oocpnf 8880   RR*cxr 8882    <_ cle 8884    - cmin 9053   NNcn 9762   NN0cn0 9981   ZZcz 10040   RR+crp 10370   (,)cioo 10672   ...cfz 10798   |_cfl 10940   !cfa 11304   abscabs 11735   sum_csu 12174    _D cdv 19229   logclog 19928
This theorem is referenced by:  logfacrlim  20479
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-pi 12370  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-cmp 17130  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930
  Copyright terms: Public domain W3C validator