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Theorem logfacbnd3 20997
Description: Show the stronger statement  log ( x ! )  =  x log x  -  x  +  O ( log x
) alluded to in logfacrlim 20998. (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 444 . . . . . . . . . . . 12  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  A  e.  RR+ )
21rprege0d 10645 . . . . . . . . . . 11  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( A  e.  RR  /\  0  <_  A ) )
3 flge0nn0 11215 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( |_ `  A
)  e.  NN0 )
42, 3syl 16 . . . . . . . . . 10  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( |_ `  A )  e. 
NN0 )
5 faccl 11566 . . . . . . . . . 10  |-  ( ( |_ `  A )  e.  NN0  ->  ( ! `
 ( |_ `  A ) )  e.  NN )
64, 5syl 16 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( ! `  ( |_ `  A ) )  e.  NN )
76nnrpd 10637 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( ! `  ( |_ `  A ) )  e.  RR+ )
8 relogcl 20463 . . . . . . . 8  |-  ( ( ! `  ( |_
`  A ) )  e.  RR+  ->  ( log `  ( ! `  ( |_ `  A ) ) )  e.  RR )
97, 8syl 16 . . . . . . 7  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( log `  ( ! `  ( |_ `  A ) ) )  e.  RR )
10 rpre 10608 . . . . . . . . 9  |-  ( A  e.  RR+  ->  A  e.  RR )
1110adantr 452 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  A  e.  RR )
12 relogcl 20463 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  ( log `  A )  e.  RR )
1312adantr 452 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( log `  A )  e.  RR )
14 peano2rem 9357 . . . . . . . . 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 9106 . . . . . . 7  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( A  x.  ( ( log `  A )  - 
1 ) )  e.  RR )
179, 16resubcld 9455 . . . . . 6  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  e.  RR )
1817recnd 9104 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  e.  CC )
1918abscld 12228 . . . 4  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( abs `  ( ( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A )  -  1 ) ) ) )  e.  RR )
20 peano2rem 9357 . . . 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 9038 . . . . 5  |-  1  e.  CC
23 subcl 9295 . . . . 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 644 . . . 4  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( ( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  -  1 )  e.  CC )
2524abscld 12228 . . 3  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( abs `  ( ( ( log `  ( ! `
 ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  -  1 ) )  e.  RR )
26 abs1 12092 . . . . 5  |-  ( abs `  1 )  =  1
2726oveq2i 6084 . . . 4  |-  ( ( abs `  ( ( log `  ( ! `
 ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )  -  ( abs `  1 ) )  =  ( ( abs `  ( ( log `  ( ! `
 ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )  - 
1 )
28 abs2dif 12126 . . . . 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 644 . . . 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 4238 . . 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 5720 . . . . . . . . . . . 12  |-  ( x  =  A  ->  ( |_ `  x )  =  ( |_ `  A
) )
3231oveq2d 6089 . . . . . . . . . . 11  |-  ( x  =  A  ->  (
1 ... ( |_ `  x ) )  =  ( 1 ... ( |_ `  A ) ) )
3332sumeq1d 12485 . . . . . . . . . 10  |-  ( x  =  A  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( log `  n
)  =  sum_ n  e.  ( 1 ... ( |_ `  A ) ) ( log `  n
) )
34 id 20 . . . . . . . . . . 11  |-  ( x  =  A  ->  x  =  A )
35 fveq2 5720 . . . . . . . . . . . 12  |-  ( x  =  A  ->  ( log `  x )  =  ( log `  A
) )
3635oveq1d 6088 . . . . . . . . . . 11  |-  ( x  =  A  ->  (
( log `  x
)  -  1 )  =  ( ( log `  A )  -  1 ) )
3734, 36oveq12d 6091 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  x.  ( ( log `  x )  -  1 ) )  =  ( A  x.  ( ( log `  A
)  -  1 ) ) )
3833, 37oveq12d 6091 . . . . . . . . 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 2435 . . . . . . . . 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 6098 . . . . . . . . 9  |-  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( log `  n )  -  (
x  x.  ( ( log `  x )  -  1 ) ) )  e.  _V
4138, 39, 40fvmpt3i 5801 . . . . . . . 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 452 . . . . . . 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 20485 . . . . . . . . 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 6088 . . . . . . 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 2470 . . . . . 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 10606 . . . . . . 7  |-  1  e.  RR+
48 fveq2 5720 . . . . . . . . . . . . . 14  |-  ( x  =  1  ->  ( |_ `  x )  =  ( |_ `  1
) )
49 1z 10301 . . . . . . . . . . . . . . 15  |-  1  e.  ZZ
50 flid 11206 . . . . . . . . . . . . . . 15  |-  ( 1  e.  ZZ  ->  ( |_ `  1 )  =  1 )
5149, 50ax-mp 8 . . . . . . . . . . . . . 14  |-  ( |_
`  1 )  =  1
5248, 51syl6eq 2483 . . . . . . . . . . . . 13  |-  ( x  =  1  ->  ( |_ `  x )  =  1 )
5352oveq2d 6089 . . . . . . . . . . . 12  |-  ( x  =  1  ->  (
1 ... ( |_ `  x ) )  =  ( 1 ... 1
) )
5453sumeq1d 12485 . . . . . . . . . . 11  |-  ( x  =  1  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( log `  n
)  =  sum_ n  e.  ( 1 ... 1
) ( log `  n
) )
55 0cn 9074 . . . . . . . . . . . 12  |-  0  e.  CC
56 fveq2 5720 . . . . . . . . . . . . . 14  |-  ( n  =  1  ->  ( log `  n )  =  ( log `  1
) )
57 log1 20470 . . . . . . . . . . . . . 14  |-  ( log `  1 )  =  0
5856, 57syl6eq 2483 . . . . . . . . . . . . 13  |-  ( n  =  1  ->  ( log `  n )  =  0 )
5958fsum1 12525 . . . . . . . . . . . 12  |-  ( ( 1  e.  ZZ  /\  0  e.  CC )  -> 
sum_ n  e.  (
1 ... 1 ) ( log `  n )  =  0 )
6049, 55, 59mp2an 654 . . . . . . . . . . 11  |-  sum_ n  e.  ( 1 ... 1
) ( log `  n
)  =  0
6154, 60syl6eq 2483 . . . . . . . . . 10  |-  ( x  =  1  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( log `  n
)  =  0 )
62 id 20 . . . . . . . . . . . 12  |-  ( x  =  1  ->  x  =  1 )
63 fveq2 5720 . . . . . . . . . . . . . 14  |-  ( x  =  1  ->  ( log `  x )  =  ( log `  1
) )
6463, 57syl6eq 2483 . . . . . . . . . . . . 13  |-  ( x  =  1  ->  ( log `  x )  =  0 )
6564oveq1d 6088 . . . . . . . . . . . 12  |-  ( x  =  1  ->  (
( log `  x
)  -  1 )  =  ( 0  -  1 ) )
6662, 65oveq12d 6091 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
x  x.  ( ( log `  x )  -  1 ) )  =  ( 1  x.  ( 0  -  1 ) ) )
6755, 22subcli 9366 . . . . . . . . . . . 12  |-  ( 0  -  1 )  e.  CC
6867mulid2i 9083 . . . . . . . . . . 11  |-  ( 1  x.  ( 0  -  1 ) )  =  ( 0  -  1 )
6966, 68syl6eq 2483 . . . . . . . . . 10  |-  ( x  =  1  ->  (
x  x.  ( ( log `  x )  -  1 ) )  =  ( 0  -  1 ) )
7061, 69oveq12d 6091 . . . . . . . . 9  |-  ( x  =  1  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( log `  n )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) )  =  ( 0  -  ( 0  -  1 ) ) )
71 nncan 9320 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  1  e.  CC )  ->  ( 0  -  (
0  -  1 ) )  =  1 )
7255, 22, 71mp2an 654 . . . . . . . . 9  |-  ( 0  -  ( 0  -  1 ) )  =  1
7370, 72syl6eq 2483 . . . . . . . 8  |-  ( x  =  1  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( log `  n )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) )  =  1 )
7473, 39, 40fvmpt3i 5801 . . . . . . 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 6091 . . . . 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 5724 . . . 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 10978 . . . . . 6  |-  ( 0 (,)  +oo )  =  RR+
7978eqcomi 2439 . . . . 5  |-  RR+  =  ( 0 (,)  +oo )
80 nnuz 10511 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
8149a1i 11 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  1  e.  ZZ )
82 1re 9080 . . . . . 6  |-  1  e.  RR
8382a1i 11 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  1  e.  RR )
84 pnfxr 10703 . . . . . 6  |-  +oo  e.  RR*
8584a1i 11 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  +oo  e.  RR* )
86 1nn0 10227 . . . . . . 7  |-  1  e.  NN0
8782, 86nn0addge1i 10258 . . . . . 6  |-  1  <_  ( 1  +  1 )
8887a1i 11 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  1  <_  ( 1  +  1 ) )
89 0re 9081 . . . . . 6  |-  0  e.  RR
9089a1i 11 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  0  e.  RR )
91 rpre 10608 . . . . . . 7  |-  ( x  e.  RR+  ->  x  e.  RR )
9291adantl 453 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  x  e.  RR+ )  ->  x  e.  RR )
93 relogcl 20463 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
9493adantl 453 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  x  e.  RR+ )  ->  ( log `  x
)  e.  RR )
95 peano2rem 9357 . . . . . . 7  |-  ( ( log `  x )  e.  RR  ->  (
( log `  x
)  -  1 )  e.  RR )
9694, 95syl 16 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  x  e.  RR+ )  ->  ( ( log `  x
)  -  1 )  e.  RR )
9792, 96remulcld 9106 . . . . 5  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  x  e.  RR+ )  ->  ( x  x.  (
( log `  x
)  -  1 ) )  e.  RR )
98 nnrp 10611 . . . . . 6  |-  ( x  e.  NN  ->  x  e.  RR+ )
9998, 94sylan2 461 . . . . 5  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  x  e.  NN )  ->  ( log `  x
)  e.  RR )
100 advlog 20535 . . . . . 6  |-  ( RR 
_D  ( x  e.  RR+  |->  ( x  x.  ( ( log `  x
)  -  1 ) ) ) )  =  ( x  e.  RR+  |->  ( log `  x ) )
101100a1i 11 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( RR  _D  ( x  e.  RR+  |->  ( x  x.  ( ( log `  x
)  -  1 ) ) ) )  =  ( x  e.  RR+  |->  ( log `  x ) ) )
102 fveq2 5720 . . . . 5  |-  ( x  =  n  ->  ( log `  x )  =  ( log `  n
) )
103 simp32 994 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  n  e.  RR+ )  /\  ( 1  <_  x  /\  x  <_  n  /\  n  <_  +oo ) )  ->  x  <_  n )
104 logleb 20488 . . . . . . 7  |-  ( ( x  e.  RR+  /\  n  e.  RR+ )  ->  (
x  <_  n  <->  ( log `  x )  <_  ( log `  n ) ) )
1051043ad2ant2 979 . . . . . 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 202 . . . . 5  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  n  e.  RR+ )  /\  ( 1  <_  x  /\  x  <_  n  /\  n  <_  +oo ) )  -> 
( log `  x
)  <_  ( log `  n ) )
107 simprr 734 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  1  <_  x )
)  ->  1  <_  x )
108 simprl 733 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  1  <_  x )
)  ->  x  e.  RR+ )
109 logleb 20488 . . . . . . . 8  |-  ( ( 1  e.  RR+  /\  x  e.  RR+ )  ->  (
1  <_  x  <->  ( log `  1 )  <_  ( log `  x ) ) )
11047, 108, 109sylancr 645 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  1  <_  x )
)  ->  ( 1  <_  x  <->  ( log `  1 )  <_  ( log `  x ) ) )
111107, 110mpbid 202 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  1  <_  x )
)  ->  ( log `  1 )  <_  ( log `  x ) )
11257, 111syl5eqbrr 4238 . . . . 5  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  1  <_  x )
)  ->  0  <_  ( log `  x ) )
11347a1i 11 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  1  e.  RR+ )
114 1le1 9640 . . . . . 6  |-  1  <_  1
115114a1i 11 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  1  <_  1 )
116 simpr 448 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  1  <_  A )
11711rexrd 9124 . . . . . 6  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  A  e.  RR* )
118 pnfge 10717 . . . . . 6  |-  ( A  e.  RR*  ->  A  <_  +oo )
119117, 118syl 16 . . . . 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 19908 . . . 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 4226 . . 3  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( abs `  ( ( ( log `  ( ! `
 ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  -  1 ) )  <_  ( log `  A ) )
12221, 25, 13, 30, 121letrd 9217 . 2  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( abs `  (
( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )  - 
1 )  <_  ( log `  A ) )
12319, 83, 13lesubaddd 9613 . 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 202 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4204    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   CCcc 8978   RRcr 8979   0cc0 8980   1c1 8981    + caddc 8983    x. cmul 8985    +oocpnf 9107   RR*cxr 9109    <_ cle 9111    - cmin 9281   NNcn 9990   NN0cn0 10211   ZZcz 10272   RR+crp 10602   (,)cioo 10906   ...cfz 11033   |_cfl 11191   !cfa 11556   abscabs 12029   sum_csu 12469    _D cdv 19740   logclog 20442
This theorem is referenced by:  logfacrlim  20998
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 4693  ax-inf2 7586  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058  ax-addf 9059  ax-mulf 9060
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 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7469  df-card 7816  df-cda 8038  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-4 10050  df-5 10051  df-6 10052  df-7 10053  df-8 10054  df-9 10055  df-10 10056  df-n0 10212  df-z 10273  df-dec 10373  df-uz 10479  df-q 10565  df-rp 10603  df-xneg 10700  df-xadd 10701  df-xmul 10702  df-ioo 10910  df-ioc 10911  df-ico 10912  df-icc 10913  df-fz 11034  df-fzo 11126  df-fl 11192  df-mod 11241  df-seq 11314  df-exp 11373  df-fac 11557  df-bc 11584  df-hash 11609  df-shft 11872  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031  df-limsup 12255  df-clim 12272  df-rlim 12273  df-sum 12470  df-ef 12660  df-sin 12662  df-cos 12663  df-pi 12665  df-struct 13461  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-ress 13466  df-plusg 13532  df-mulr 13533  df-starv 13534  df-sca 13535  df-vsca 13536  df-tset 13538  df-ple 13539  df-ds 13541  df-unif 13542  df-hom 13543  df-cco 13544  df-rest 13640  df-topn 13641  df-topgen 13657  df-pt 13658  df-prds 13661  df-xrs 13716  df-0g 13717  df-gsum 13718  df-qtop 13723  df-imas 13724  df-xps 13726  df-mre 13801  df-mrc 13802  df-acs 13804  df-mnd 14680  df-submnd 14729  df-mulg 14805  df-cntz 15106  df-cmn 15404  df-psmet 16684  df-xmet 16685  df-met 16686  df-bl 16687  df-mopn 16688  df-fbas 16689  df-fg 16690  df-cnfld 16694  df-top 16953  df-bases 16955  df-topon 16956  df-topsp 16957  df-cld 17073  df-ntr 17074  df-cls 17075  df-nei 17152  df-lp 17190  df-perf 17191  df-cn 17281  df-cnp 17282  df-haus 17369  df-cmp 17440  df-tx 17584  df-hmeo 17777  df-fil 17868  df-fm 17960  df-flim 17961  df-flf 17962  df-xms 18340  df-ms 18341  df-tms 18342  df-cncf 18898  df-limc 19743  df-dv 19744  df-log 20444
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