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

Theorem logfacbnd3 20462
Description: Show the stronger statement  log ( x ! )  =  x log x  -  x  +  O ( log x
) alluded to in logfacrlim 20463. (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 10397 . . . . . . . . . . 11  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( A  e.  RR  /\  0  <_  A ) )
3 flge0nn0 10948 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( |_ `  A
)  e.  NN0 )
42, 3syl 15 . . . . . . . . . 10  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( |_ `  A )  e. 
NN0 )
5 faccl 11298 . . . . . . . . . 10  |-  ( ( |_ `  A )  e.  NN0  ->  ( ! `
 ( |_ `  A ) )  e.  NN )
64, 5syl 15 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( ! `  ( |_ `  A ) )  e.  NN )
76nnrpd 10389 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( ! `  ( |_ `  A ) )  e.  RR+ )
8 relogcl 19932 . . . . . . . 8  |-  ( ( ! `  ( |_
`  A ) )  e.  RR+  ->  ( log `  ( ! `  ( |_ `  A ) ) )  e.  RR )
97, 8syl 15 . . . . . . 7  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( log `  ( ! `  ( |_ `  A ) ) )  e.  RR )
10 rpre 10360 . . . . . . . . 9  |-  ( A  e.  RR+  ->  A  e.  RR )
1110adantr 451 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  A  e.  RR )
12 relogcl 19932 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  ( log `  A )  e.  RR )
1312adantr 451 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( log `  A )  e.  RR )
14 peano2rem 9113 . . . . . . . . 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 8863 . . . . . . 7  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( A  x.  ( ( log `  A )  - 
1 ) )  e.  RR )
179, 16resubcld 9211 . . . . . 6  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  e.  RR )
1817recnd 8861 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  e.  CC )
1918abscld 11918 . . . 4  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( abs `  ( ( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A )  -  1 ) ) ) )  e.  RR )
20 peano2rem 9113 . . . 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 8795 . . . . 5  |-  1  e.  CC
23 subcl 9051 . . . . 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 11918 . . 3  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( abs `  ( ( ( log `  ( ! `
 ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  -  1 ) )  e.  RR )
26 abs1 11782 . . . . 5  |-  ( abs `  1 )  =  1
2726oveq2i 5869 . . . 4  |-  ( ( abs `  ( ( log `  ( ! `
 ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )  -  ( abs `  1 ) )  =  ( ( abs `  ( ( log `  ( ! `
 ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )  - 
1 )
28 abs2dif 11816 . . . . 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 4057 . . 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 5525 . . . . . . . . . . . 12  |-  ( x  =  A  ->  ( |_ `  x )  =  ( |_ `  A
) )
3231oveq2d 5874 . . . . . . . . . . 11  |-  ( x  =  A  ->  (
1 ... ( |_ `  x ) )  =  ( 1 ... ( |_ `  A ) ) )
3332sumeq1d 12174 . . . . . . . . . 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 5525 . . . . . . . . . . . 12  |-  ( x  =  A  ->  ( log `  x )  =  ( log `  A
) )
3635oveq1d 5873 . . . . . . . . . . 11  |-  ( x  =  A  ->  (
( log `  x
)  -  1 )  =  ( ( log `  A )  -  1 ) )
3734, 36oveq12d 5876 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  x.  ( ( log `  x )  -  1 ) )  =  ( A  x.  ( ( log `  A
)  -  1 ) ) )
3833, 37oveq12d 5876 . . . . . . . . 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 2283 . . . . . . . . 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 5883 . . . . . . . . 9  |-  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( log `  n )  -  (
x  x.  ( ( log `  x )  -  1 ) ) )  e.  _V
4138, 39, 40fvmpt3i 5605 . . . . . . . 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 19954 . . . . . . . . 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 5873 . . . . . . 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 2318 . . . . . 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 10358 . . . . . . 7  |-  1  e.  RR+
48 fveq2 5525 . . . . . . . . . . . . . 14  |-  ( x  =  1  ->  ( |_ `  x )  =  ( |_ `  1
) )
49 1z 10053 . . . . . . . . . . . . . . 15  |-  1  e.  ZZ
50 flid 10939 . . . . . . . . . . . . . . 15  |-  ( 1  e.  ZZ  ->  ( |_ `  1 )  =  1 )
5149, 50ax-mp 8 . . . . . . . . . . . . . 14  |-  ( |_
`  1 )  =  1
5248, 51syl6eq 2331 . . . . . . . . . . . . 13  |-  ( x  =  1  ->  ( |_ `  x )  =  1 )
5352oveq2d 5874 . . . . . . . . . . . 12  |-  ( x  =  1  ->  (
1 ... ( |_ `  x ) )  =  ( 1 ... 1
) )
5453sumeq1d 12174 . . . . . . . . . . 11  |-  ( x  =  1  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( log `  n
)  =  sum_ n  e.  ( 1 ... 1
) ( log `  n
) )
55 0cn 8831 . . . . . . . . . . . 12  |-  0  e.  CC
56 fveq2 5525 . . . . . . . . . . . . . 14  |-  ( n  =  1  ->  ( log `  n )  =  ( log `  1
) )
57 log1 19939 . . . . . . . . . . . . . 14  |-  ( log `  1 )  =  0
5856, 57syl6eq 2331 . . . . . . . . . . . . 13  |-  ( n  =  1  ->  ( log `  n )  =  0 )
5958fsum1 12214 . . . . . . . . . . . 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 2331 . . . . . . . . . 10  |-  ( x  =  1  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( log `  n
)  =  0 )
62 id 19 . . . . . . . . . . . 12  |-  ( x  =  1  ->  x  =  1 )
63 fveq2 5525 . . . . . . . . . . . . . 14  |-  ( x  =  1  ->  ( log `  x )  =  ( log `  1
) )
6463, 57syl6eq 2331 . . . . . . . . . . . . 13  |-  ( x  =  1  ->  ( log `  x )  =  0 )
6564oveq1d 5873 . . . . . . . . . . . 12  |-  ( x  =  1  ->  (
( log `  x
)  -  1 )  =  ( 0  -  1 ) )
6662, 65oveq12d 5876 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
x  x.  ( ( log `  x )  -  1 ) )  =  ( 1  x.  ( 0  -  1 ) ) )
6755, 22subcli 9122 . . . . . . . . . . . 12  |-  ( 0  -  1 )  e.  CC
6867mulid2i 8840 . . . . . . . . . . 11  |-  ( 1  x.  ( 0  -  1 ) )  =  ( 0  -  1 )
6966, 68syl6eq 2331 . . . . . . . . . 10  |-  ( x  =  1  ->  (
x  x.  ( ( log `  x )  -  1 ) )  =  ( 0  -  1 ) )
7061, 69oveq12d 5876 . . . . . . . . 9  |-  ( x  =  1  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( log `  n )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) )  =  ( 0  -  ( 0  -  1 ) ) )
71 nncan 9076 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  1  e.  CC )  ->  ( 0  -  (
0  -  1 ) )  =  1 )
7255, 22, 71mp2an 653 . . . . . . . . 9  |-  ( 0  -  ( 0  -  1 ) )  =  1
7370, 72syl6eq 2331 . . . . . . . 8  |-  ( x  =  1  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( log `  n )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) )  =  1 )
7473, 39, 40fvmpt3i 5605 . . . . . . 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 5876 . . . . 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 5529 . . . 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 10727 . . . . . 6  |-  ( 0 (,)  +oo )  =  RR+
7978eqcomi 2287 . . . . 5  |-  RR+  =  ( 0 (,)  +oo )
80 nnuz 10263 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
8149a1i 10 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  1  e.  ZZ )
82 1re 8837 . . . . . 6  |-  1  e.  RR
8382a1i 10 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  1  e.  RR )
84 pnfxr 10455 . . . . . 6  |-  +oo  e.  RR*
8584a1i 10 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  +oo  e.  RR* )
86 1nn0 9981 . . . . . . 7  |-  1  e.  NN0
8782, 86nn0addge1i 10012 . . . . . 6  |-  1  <_  ( 1  +  1 )
8887a1i 10 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  1  <_  ( 1  +  1 ) )
89 0re 8838 . . . . . 6  |-  0  e.  RR
9089a1i 10 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  0  e.  RR )
91 rpre 10360 . . . . . . 7  |-  ( x  e.  RR+  ->  x  e.  RR )
9291adantl 452 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  x  e.  RR+ )  ->  x  e.  RR )
93 relogcl 19932 . . . . . . . 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 9113 . . . . . . 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 8863 . . . . 5  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  x  e.  RR+ )  ->  ( x  x.  (
( log `  x
)  -  1 ) )  e.  RR )
98 nnrp 10363 . . . . . 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 20001 . . . . . 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 5525 . . . . 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 19957 . . . . . . 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 19957 . . . . . . . 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 4057 . . . . 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 9396 . . . . . 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 8881 . . . . . 6  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  A  e.  RR* )
118 pnfge 10469 . . . . . 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 19381 . . . 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 4045 . . 3  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( abs `  ( ( ( log `  ( ! `
 ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  -  1 ) )  <_  ( log `  A ) )
12221, 25, 13, 30, 121letrd 8973 . 2  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( abs `  (
( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )  - 
1 )  <_  ( log `  A ) )
12319, 83, 13lesubaddd 9369 . 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 1623    e. wcel 1684   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    +oocpnf 8864   RR*cxr 8866    <_ cle 8868    - cmin 9037   NNcn 9746   NN0cn0 9965   ZZcz 10024   RR+crp 10354   (,)cioo 10656   ...cfz 10782   |_cfl 10924   !cfa 11288   abscabs 11719   sum_csu 12158    _D cdv 19213   logclog 19912
This theorem is referenced by:  logfacrlim  20463
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
  Copyright terms: Public domain W3C validator