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Theorem logfacbnd3 22537
Description: Show the stronger statement  log ( x ! )  =  x log x  -  x  +  O ( log x
) alluded to in logfacrlim 22538. (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 457 . . . . . . . . . . . 12  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  A  e.  RR+ )
21rprege0d 11026 . . . . . . . . . . 11  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( A  e.  RR  /\  0  <_  A ) )
3 flge0nn0 11658 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( |_ `  A
)  e.  NN0 )
42, 3syl 16 . . . . . . . . . 10  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( |_ `  A )  e. 
NN0 )
5 faccl 12053 . . . . . . . . . 10  |-  ( ( |_ `  A )  e.  NN0  ->  ( ! `
 ( |_ `  A ) )  e.  NN )
64, 5syl 16 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( ! `  ( |_ `  A ) )  e.  NN )
76nnrpd 11018 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( ! `  ( |_ `  A ) )  e.  RR+ )
8 relogcl 22002 . . . . . . . 8  |-  ( ( ! `  ( |_
`  A ) )  e.  RR+  ->  ( log `  ( ! `  ( |_ `  A ) ) )  e.  RR )
97, 8syl 16 . . . . . . 7  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( log `  ( ! `  ( |_ `  A ) ) )  e.  RR )
10 rpre 10989 . . . . . . . . 9  |-  ( A  e.  RR+  ->  A  e.  RR )
1110adantr 465 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  A  e.  RR )
12 relogcl 22002 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  ( log `  A )  e.  RR )
1312adantr 465 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( log `  A )  e.  RR )
14 peano2rem 9667 . . . . . . . . 9  |-  ( ( log `  A )  e.  RR  ->  (
( log `  A
)  -  1 )  e.  RR )
1513, 14syl 16 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( log `  A
)  -  1 )  e.  RR )
1611, 15remulcld 9406 . . . . . . 7  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( A  x.  ( ( log `  A )  - 
1 ) )  e.  RR )
179, 16resubcld 9768 . . . . . 6  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  e.  RR )
1817recnd 9404 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  e.  CC )
1918abscld 12914 . . . 4  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( abs `  ( ( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A )  -  1 ) ) ) )  e.  RR )
20 peano2rem 9667 . . . 4  |-  ( ( abs `  ( ( log `  ( ! `
 ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )  e.  RR  ->  ( ( abs `  ( ( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A )  -  1 ) ) ) )  -  1 )  e.  RR )
2119, 20syl 16 . . 3  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( abs `  (
( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )  - 
1 )  e.  RR )
22 ax-1cn 9332 . . . . 5  |-  1  e.  CC
23 subcl 9601 . . . . 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 662 . . . 4  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( ( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  -  1 )  e.  CC )
2524abscld 12914 . . 3  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( abs `  ( ( ( log `  ( ! `
 ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  -  1 ) )  e.  RR )
26 abs1 12778 . . . . 5  |-  ( abs `  1 )  =  1
2726oveq2i 6097 . . . 4  |-  ( ( abs `  ( ( log `  ( ! `
 ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )  -  ( abs `  1 ) )  =  ( ( abs `  ( ( log `  ( ! `
 ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )  - 
1 )
28 abs2dif 12812 . . . . 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 662 . . . 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 4321 . . 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 5686 . . . . . . . . . . . 12  |-  ( x  =  A  ->  ( |_ `  x )  =  ( |_ `  A
) )
3231oveq2d 6102 . . . . . . . . . . 11  |-  ( x  =  A  ->  (
1 ... ( |_ `  x ) )  =  ( 1 ... ( |_ `  A ) ) )
3332sumeq1d 13170 . . . . . . . . . 10  |-  ( x  =  A  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( log `  n
)  =  sum_ n  e.  ( 1 ... ( |_ `  A ) ) ( log `  n
) )
34 id 22 . . . . . . . . . . 11  |-  ( x  =  A  ->  x  =  A )
35 fveq2 5686 . . . . . . . . . . . 12  |-  ( x  =  A  ->  ( log `  x )  =  ( log `  A
) )
3635oveq1d 6101 . . . . . . . . . . 11  |-  ( x  =  A  ->  (
( log `  x
)  -  1 )  =  ( ( log `  A )  -  1 ) )
3734, 36oveq12d 6104 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  x.  ( ( log `  x )  -  1 ) )  =  ( A  x.  ( ( log `  A
)  -  1 ) ) )
3833, 37oveq12d 6104 . . . . . . . . 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 2438 . . . . . . . . 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 6111 . . . . . . . . 9  |-  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( log `  n )  -  (
x  x.  ( ( log `  x )  -  1 ) ) )  e.  _V
4138, 39, 40fvmpt3i 5773 . . . . . . . 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 465 . . . . . . 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 22024 . . . . . . . . 9  |-  ( ( |_ `  A )  e.  NN0  ->  ( log `  ( ! `  ( |_ `  A ) ) )  =  sum_ n  e.  ( 1 ... ( |_ `  A ) ) ( log `  n
) )
444, 43syl 16 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( log `  ( ! `  ( |_ `  A ) ) )  =  sum_ n  e.  ( 1 ... ( |_ `  A
) ) ( log `  n ) )
4544oveq1d 6101 . . . . . . 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 2473 . . . . . 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 10987 . . . . . . 7  |-  1  e.  RR+
48 fveq2 5686 . . . . . . . . . . . . . 14  |-  ( x  =  1  ->  ( |_ `  x )  =  ( |_ `  1
) )
49 1z 10668 . . . . . . . . . . . . . . 15  |-  1  e.  ZZ
50 flid 11649 . . . . . . . . . . . . . . 15  |-  ( 1  e.  ZZ  ->  ( |_ `  1 )  =  1 )
5149, 50ax-mp 5 . . . . . . . . . . . . . 14  |-  ( |_
`  1 )  =  1
5248, 51syl6eq 2486 . . . . . . . . . . . . 13  |-  ( x  =  1  ->  ( |_ `  x )  =  1 )
5352oveq2d 6102 . . . . . . . . . . . 12  |-  ( x  =  1  ->  (
1 ... ( |_ `  x ) )  =  ( 1 ... 1
) )
5453sumeq1d 13170 . . . . . . . . . . 11  |-  ( x  =  1  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( log `  n
)  =  sum_ n  e.  ( 1 ... 1
) ( log `  n
) )
55 0cn 9370 . . . . . . . . . . . 12  |-  0  e.  CC
56 fveq2 5686 . . . . . . . . . . . . . 14  |-  ( n  =  1  ->  ( log `  n )  =  ( log `  1
) )
57 log1 22009 . . . . . . . . . . . . . 14  |-  ( log `  1 )  =  0
5856, 57syl6eq 2486 . . . . . . . . . . . . 13  |-  ( n  =  1  ->  ( log `  n )  =  0 )
5958fsum1 13210 . . . . . . . . . . . 12  |-  ( ( 1  e.  ZZ  /\  0  e.  CC )  -> 
sum_ n  e.  (
1 ... 1 ) ( log `  n )  =  0 )
6049, 55, 59mp2an 672 . . . . . . . . . . 11  |-  sum_ n  e.  ( 1 ... 1
) ( log `  n
)  =  0
6154, 60syl6eq 2486 . . . . . . . . . 10  |-  ( x  =  1  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( log `  n
)  =  0 )
62 id 22 . . . . . . . . . . . 12  |-  ( x  =  1  ->  x  =  1 )
63 fveq2 5686 . . . . . . . . . . . . . 14  |-  ( x  =  1  ->  ( log `  x )  =  ( log `  1
) )
6463, 57syl6eq 2486 . . . . . . . . . . . . 13  |-  ( x  =  1  ->  ( log `  x )  =  0 )
6564oveq1d 6101 . . . . . . . . . . . 12  |-  ( x  =  1  ->  (
( log `  x
)  -  1 )  =  ( 0  -  1 ) )
6662, 65oveq12d 6104 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
x  x.  ( ( log `  x )  -  1 ) )  =  ( 1  x.  ( 0  -  1 ) ) )
6755, 22subcli 9676 . . . . . . . . . . . 12  |-  ( 0  -  1 )  e.  CC
6867mulid2i 9381 . . . . . . . . . . 11  |-  ( 1  x.  ( 0  -  1 ) )  =  ( 0  -  1 )
6966, 68syl6eq 2486 . . . . . . . . . 10  |-  ( x  =  1  ->  (
x  x.  ( ( log `  x )  -  1 ) )  =  ( 0  -  1 ) )
7061, 69oveq12d 6104 . . . . . . . . 9  |-  ( x  =  1  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( log `  n )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) )  =  ( 0  -  ( 0  -  1 ) ) )
71 nncan 9630 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  1  e.  CC )  ->  ( 0  -  (
0  -  1 ) )  =  1 )
7255, 22, 71mp2an 672 . . . . . . . . 9  |-  ( 0  -  ( 0  -  1 ) )  =  1
7370, 72syl6eq 2486 . . . . . . . 8  |-  ( x  =  1  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( log `  n )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) )  =  1 )
7473, 39, 40fvmpt3i 5773 . . . . . . 7  |-  ( 1  e.  RR+  ->  ( ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( log `  n )  -  (
x  x.  ( ( log `  x )  -  1 ) ) ) ) `  1
)  =  1 )
7547, 74mp1i 12 . . . . . 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 6104 . . . . 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 5690 . . . 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 11365 . . . . . 6  |-  ( 0 (,) +oo )  = 
RR+
7978eqcomi 2442 . . . . 5  |-  RR+  =  ( 0 (,) +oo )
80 nnuz 10888 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
8149a1i 11 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  1  e.  ZZ )
82 1re 9377 . . . . . 6  |-  1  e.  RR
8382a1i 11 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  1  e.  RR )
84 pnfxr 11084 . . . . . 6  |- +oo  e.  RR*
8584a1i 11 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  -> +oo  e.  RR* )
86 1nn0 10587 . . . . . . 7  |-  1  e.  NN0
8782, 86nn0addge1i 10620 . . . . . 6  |-  1  <_  ( 1  +  1 )
8887a1i 11 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  1  <_  ( 1  +  1 ) )
89 0red 9379 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  0  e.  RR )
90 rpre 10989 . . . . . . 7  |-  ( x  e.  RR+  ->  x  e.  RR )
9190adantl 466 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  x  e.  RR+ )  ->  x  e.  RR )
92 relogcl 22002 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
9392adantl 466 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  x  e.  RR+ )  ->  ( log `  x
)  e.  RR )
94 peano2rem 9667 . . . . . . 7  |-  ( ( log `  x )  e.  RR  ->  (
( log `  x
)  -  1 )  e.  RR )
9593, 94syl 16 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  x  e.  RR+ )  ->  ( ( log `  x
)  -  1 )  e.  RR )
9691, 95remulcld 9406 . . . . 5  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  x  e.  RR+ )  ->  ( x  x.  (
( log `  x
)  -  1 ) )  e.  RR )
97 nnrp 10992 . . . . . 6  |-  ( x  e.  NN  ->  x  e.  RR+ )
9897, 93sylan2 474 . . . . 5  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  x  e.  NN )  ->  ( log `  x
)  e.  RR )
99 advlog 22074 . . . . . 6  |-  ( RR 
_D  ( x  e.  RR+  |->  ( x  x.  ( ( log `  x
)  -  1 ) ) ) )  =  ( x  e.  RR+  |->  ( log `  x ) )
10099a1i 11 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( RR  _D  ( x  e.  RR+  |->  ( x  x.  ( ( log `  x
)  -  1 ) ) ) )  =  ( x  e.  RR+  |->  ( log `  x ) ) )
101 fveq2 5686 . . . . 5  |-  ( x  =  n  ->  ( log `  x )  =  ( log `  n
) )
102 simp32 1025 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  n  e.  RR+ )  /\  ( 1  <_  x  /\  x  <_  n  /\  n  <_ +oo ) )  ->  x  <_  n )
103 logleb 22027 . . . . . . 7  |-  ( ( x  e.  RR+  /\  n  e.  RR+ )  ->  (
x  <_  n  <->  ( log `  x )  <_  ( log `  n ) ) )
1041033ad2ant2 1010 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  n  e.  RR+ )  /\  ( 1  <_  x  /\  x  <_  n  /\  n  <_ +oo ) )  -> 
( x  <_  n  <->  ( log `  x )  <_  ( log `  n
) ) )
105102, 104mpbid 210 . . . . 5  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  n  e.  RR+ )  /\  ( 1  <_  x  /\  x  <_  n  /\  n  <_ +oo ) )  -> 
( log `  x
)  <_  ( log `  n ) )
106 simprr 756 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  1  <_  x )
)  ->  1  <_  x )
107 simprl 755 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  1  <_  x )
)  ->  x  e.  RR+ )
108 logleb 22027 . . . . . . . 8  |-  ( ( 1  e.  RR+  /\  x  e.  RR+ )  ->  (
1  <_  x  <->  ( log `  1 )  <_  ( log `  x ) ) )
10947, 107, 108sylancr 663 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  1  <_  x )
)  ->  ( 1  <_  x  <->  ( log `  1 )  <_  ( log `  x ) ) )
110106, 109mpbid 210 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  1  <_  x )
)  ->  ( log `  1 )  <_  ( log `  x ) )
11157, 110syl5eqbrr 4321 . . . . 5  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  1  <_  x )
)  ->  0  <_  ( log `  x ) )
11247a1i 11 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  1  e.  RR+ )
113 1le1 9956 . . . . . 6  |-  1  <_  1
114113a1i 11 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  1  <_  1 )
115 simpr 461 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  1  <_  A )
11611rexrd 9425 . . . . . 6  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  A  e.  RR* )
117 pnfge 11102 . . . . . 6  |-  ( A  e.  RR*  ->  A  <_ +oo )
118116, 117syl 16 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  A  <_ +oo )
11979, 80, 81, 83, 85, 88, 89, 96, 93, 98, 100, 101, 105, 39, 111, 112, 1, 114, 115, 118, 35dvfsum2 21481 . . . 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
) )
12077, 119eqbrtrrd 4309 . . 3  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( abs `  ( ( ( log `  ( ! `
 ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  -  1 ) )  <_  ( log `  A ) )
12121, 25, 13, 30, 120letrd 9520 . 2  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( abs `  (
( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )  - 
1 )  <_  ( log `  A ) )
12219, 83, 13lesubaddd 9928 . 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 ) ) )
123121, 122mpbid 210 1  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( abs `  ( ( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A )  -  1 ) ) ) )  <_  ( ( log `  A )  +  1 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4287    e. cmpt 4345   ` cfv 5413  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279   +oocpnf 9407   RR*cxr 9409    <_ cle 9411    - cmin 9587   NNcn 10314   NN0cn0 10571   ZZcz 10638   RR+crp 10983   (,)cioo 11292   ...cfz 11429   |_cfl 11632   !cfa 12043   abscabs 12715   sum_csu 13155    _D cdv 21313   logclog 21981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-sum 13156  df-ef 13345  df-sin 13347  df-cos 13348  df-pi 13350  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-fbas 17789  df-fg 17790  df-cnfld 17794  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cld 18598  df-ntr 18599  df-cls 18600  df-nei 18677  df-lp 18715  df-perf 18716  df-cn 18806  df-cnp 18807  df-haus 18894  df-cmp 18965  df-tx 19110  df-hmeo 19303  df-fil 19394  df-fm 19486  df-flim 19487  df-flf 19488  df-xms 19870  df-ms 19871  df-tms 19872  df-cncf 20429  df-limc 21316  df-dv 21317  df-log 21983
This theorem is referenced by:  logfacrlim  22538
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