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Theorem logfacbnd3 24137
Description: Show the stronger statement  log ( x ! )  =  x log x  -  x  +  O ( log x
) alluded to in logfacrlim 24138. (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 458 . . . . . . . . . . . 12  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  A  e.  RR+ )
21rprege0d 11348 . . . . . . . . . . 11  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( A  e.  RR  /\  0  <_  A ) )
3 flge0nn0 12053 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( |_ `  A
)  e.  NN0 )
42, 3syl 17 . . . . . . . . . 10  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( |_ `  A )  e. 
NN0 )
5 faccl 12468 . . . . . . . . . 10  |-  ( ( |_ `  A )  e.  NN0  ->  ( ! `
 ( |_ `  A ) )  e.  NN )
64, 5syl 17 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( ! `  ( |_ `  A ) )  e.  NN )
76nnrpd 11339 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( ! `  ( |_ `  A ) )  e.  RR+ )
8 relogcl 23511 . . . . . . . 8  |-  ( ( ! `  ( |_
`  A ) )  e.  RR+  ->  ( log `  ( ! `  ( |_ `  A ) ) )  e.  RR )
97, 8syl 17 . . . . . . 7  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( log `  ( ! `  ( |_ `  A ) ) )  e.  RR )
10 rpre 11308 . . . . . . . . 9  |-  ( A  e.  RR+  ->  A  e.  RR )
1110adantr 466 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  A  e.  RR )
12 relogcl 23511 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  ( log `  A )  e.  RR )
1312adantr 466 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( log `  A )  e.  RR )
14 peano2rem 9941 . . . . . . . . 9  |-  ( ( log `  A )  e.  RR  ->  (
( log `  A
)  -  1 )  e.  RR )
1513, 14syl 17 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( log `  A
)  -  1 )  e.  RR )
1611, 15remulcld 9671 . . . . . . 7  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( A  x.  ( ( log `  A )  - 
1 ) )  e.  RR )
179, 16resubcld 10047 . . . . . 6  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  e.  RR )
1817recnd 9669 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  e.  CC )
1918abscld 13485 . . . 4  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( abs `  ( ( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A )  -  1 ) ) ) )  e.  RR )
20 peano2rem 9941 . . . 4  |-  ( ( abs `  ( ( log `  ( ! `
 ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )  e.  RR  ->  ( ( abs `  ( ( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A )  -  1 ) ) ) )  -  1 )  e.  RR )
2119, 20syl 17 . . 3  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( abs `  (
( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )  - 
1 )  e.  RR )
22 ax-1cn 9597 . . . . 5  |-  1  e.  CC
23 subcl 9874 . . . . 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 666 . . . 4  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( ( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  -  1 )  e.  CC )
2524abscld 13485 . . 3  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( abs `  ( ( ( log `  ( ! `
 ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  -  1 ) )  e.  RR )
26 abs1 13348 . . . . 5  |-  ( abs `  1 )  =  1
2726oveq2i 6312 . . . 4  |-  ( ( abs `  ( ( log `  ( ! `
 ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )  -  ( abs `  1 ) )  =  ( ( abs `  ( ( log `  ( ! `
 ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )  - 
1 )
28 abs2dif 13383 . . . . 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 666 . . . 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 4455 . . 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 5877 . . . . . . . . . . . 12  |-  ( x  =  A  ->  ( |_ `  x )  =  ( |_ `  A
) )
3231oveq2d 6317 . . . . . . . . . . 11  |-  ( x  =  A  ->  (
1 ... ( |_ `  x ) )  =  ( 1 ... ( |_ `  A ) ) )
3332sumeq1d 13754 . . . . . . . . . 10  |-  ( x  =  A  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( log `  n
)  =  sum_ n  e.  ( 1 ... ( |_ `  A ) ) ( log `  n
) )
34 id 23 . . . . . . . . . . 11  |-  ( x  =  A  ->  x  =  A )
35 fveq2 5877 . . . . . . . . . . . 12  |-  ( x  =  A  ->  ( log `  x )  =  ( log `  A
) )
3635oveq1d 6316 . . . . . . . . . . 11  |-  ( x  =  A  ->  (
( log `  x
)  -  1 )  =  ( ( log `  A )  -  1 ) )
3734, 36oveq12d 6319 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  x.  ( ( log `  x )  -  1 ) )  =  ( A  x.  ( ( log `  A
)  -  1 ) ) )
3833, 37oveq12d 6319 . . . . . . . . 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 2422 . . . . . . . . 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 6329 . . . . . . . . 9  |-  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( log `  n )  -  (
x  x.  ( ( log `  x )  -  1 ) ) )  e.  _V
4138, 39, 40fvmpt3i 5965 . . . . . . . 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 466 . . . . . . 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 23536 . . . . . . . . 9  |-  ( ( |_ `  A )  e.  NN0  ->  ( log `  ( ! `  ( |_ `  A ) ) )  =  sum_ n  e.  ( 1 ... ( |_ `  A ) ) ( log `  n
) )
444, 43syl 17 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( log `  ( ! `  ( |_ `  A ) ) )  =  sum_ n  e.  ( 1 ... ( |_ `  A
) ) ( log `  n ) )
4544oveq1d 6316 . . . . . . 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 2466 . . . . . 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 11306 . . . . . . 7  |-  1  e.  RR+
48 fveq2 5877 . . . . . . . . . . . . . 14  |-  ( x  =  1  ->  ( |_ `  x )  =  ( |_ `  1
) )
49 1z 10967 . . . . . . . . . . . . . . 15  |-  1  e.  ZZ
50 flid 12043 . . . . . . . . . . . . . . 15  |-  ( 1  e.  ZZ  ->  ( |_ `  1 )  =  1 )
5149, 50ax-mp 5 . . . . . . . . . . . . . 14  |-  ( |_
`  1 )  =  1
5248, 51syl6eq 2479 . . . . . . . . . . . . 13  |-  ( x  =  1  ->  ( |_ `  x )  =  1 )
5352oveq2d 6317 . . . . . . . . . . . 12  |-  ( x  =  1  ->  (
1 ... ( |_ `  x ) )  =  ( 1 ... 1
) )
5453sumeq1d 13754 . . . . . . . . . . 11  |-  ( x  =  1  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( log `  n
)  =  sum_ n  e.  ( 1 ... 1
) ( log `  n
) )
55 0cn 9635 . . . . . . . . . . . 12  |-  0  e.  CC
56 fveq2 5877 . . . . . . . . . . . . . 14  |-  ( n  =  1  ->  ( log `  n )  =  ( log `  1
) )
57 log1 23521 . . . . . . . . . . . . . 14  |-  ( log `  1 )  =  0
5856, 57syl6eq 2479 . . . . . . . . . . . . 13  |-  ( n  =  1  ->  ( log `  n )  =  0 )
5958fsum1 13795 . . . . . . . . . . . 12  |-  ( ( 1  e.  ZZ  /\  0  e.  CC )  -> 
sum_ n  e.  (
1 ... 1 ) ( log `  n )  =  0 )
6049, 55, 59mp2an 676 . . . . . . . . . . 11  |-  sum_ n  e.  ( 1 ... 1
) ( log `  n
)  =  0
6154, 60syl6eq 2479 . . . . . . . . . 10  |-  ( x  =  1  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( log `  n
)  =  0 )
62 id 23 . . . . . . . . . . . 12  |-  ( x  =  1  ->  x  =  1 )
63 fveq2 5877 . . . . . . . . . . . . . 14  |-  ( x  =  1  ->  ( log `  x )  =  ( log `  1
) )
6463, 57syl6eq 2479 . . . . . . . . . . . . 13  |-  ( x  =  1  ->  ( log `  x )  =  0 )
6564oveq1d 6316 . . . . . . . . . . . 12  |-  ( x  =  1  ->  (
( log `  x
)  -  1 )  =  ( 0  -  1 ) )
6662, 65oveq12d 6319 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
x  x.  ( ( log `  x )  -  1 ) )  =  ( 1  x.  ( 0  -  1 ) ) )
6755, 22subcli 9950 . . . . . . . . . . . 12  |-  ( 0  -  1 )  e.  CC
6867mulid2i 9646 . . . . . . . . . . 11  |-  ( 1  x.  ( 0  -  1 ) )  =  ( 0  -  1 )
6966, 68syl6eq 2479 . . . . . . . . . 10  |-  ( x  =  1  ->  (
x  x.  ( ( log `  x )  -  1 ) )  =  ( 0  -  1 ) )
7061, 69oveq12d 6319 . . . . . . . . 9  |-  ( x  =  1  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( log `  n )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) )  =  ( 0  -  ( 0  -  1 ) ) )
71 nncan 9903 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  1  e.  CC )  ->  ( 0  -  (
0  -  1 ) )  =  1 )
7255, 22, 71mp2an 676 . . . . . . . . 9  |-  ( 0  -  ( 0  -  1 ) )  =  1
7370, 72syl6eq 2479 . . . . . . . 8  |-  ( x  =  1  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( log `  n )  -  ( x  x.  ( ( log `  x
)  -  1 ) ) )  =  1 )
7473, 39, 40fvmpt3i 5965 . . . . . . 7  |-  ( 1  e.  RR+  ->  ( ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( log `  n )  -  (
x  x.  ( ( log `  x )  -  1 ) ) ) ) `  1
)  =  1 )
7547, 74mp1i 13 . . . . . 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 6319 . . . . 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 5881 . . . 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 11712 . . . . . 6  |-  ( 0 (,) +oo )  = 
RR+
7978eqcomi 2435 . . . . 5  |-  RR+  =  ( 0 (,) +oo )
80 nnuz 11194 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
8149a1i 11 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  1  e.  ZZ )
82 1re 9642 . . . . . 6  |-  1  e.  RR
8382a1i 11 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  1  e.  RR )
84 pnfxr 11412 . . . . . 6  |- +oo  e.  RR*
8584a1i 11 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  -> +oo  e.  RR* )
86 1nn0 10885 . . . . . . 7  |-  1  e.  NN0
8782, 86nn0addge1i 10918 . . . . . 6  |-  1  <_  ( 1  +  1 )
8887a1i 11 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  1  <_  ( 1  +  1 ) )
89 0red 9644 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  0  e.  RR )
90 rpre 11308 . . . . . . 7  |-  ( x  e.  RR+  ->  x  e.  RR )
9190adantl 467 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  x  e.  RR+ )  ->  x  e.  RR )
92 relogcl 23511 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
9392adantl 467 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  x  e.  RR+ )  ->  ( log `  x
)  e.  RR )
94 peano2rem 9941 . . . . . . 7  |-  ( ( log `  x )  e.  RR  ->  (
( log `  x
)  -  1 )  e.  RR )
9593, 94syl 17 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  x  e.  RR+ )  ->  ( ( log `  x
)  -  1 )  e.  RR )
9691, 95remulcld 9671 . . . . 5  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  x  e.  RR+ )  ->  ( x  x.  (
( log `  x
)  -  1 ) )  e.  RR )
97 nnrp 11311 . . . . . 6  |-  ( x  e.  NN  ->  x  e.  RR+ )
9897, 93sylan2 476 . . . . 5  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  x  e.  NN )  ->  ( log `  x
)  e.  RR )
99 advlog 23585 . . . . . 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 5877 . . . . 5  |-  ( x  =  n  ->  ( log `  x )  =  ( log `  n
) )
102 simp32 1042 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  n  e.  RR+ )  /\  ( 1  <_  x  /\  x  <_  n  /\  n  <_ +oo ) )  ->  x  <_  n )
103 logleb 23538 . . . . . . 7  |-  ( ( x  e.  RR+  /\  n  e.  RR+ )  ->  (
x  <_  n  <->  ( log `  x )  <_  ( log `  n ) ) )
1041033ad2ant2 1027 . . . . . 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 213 . . . . 5  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  n  e.  RR+ )  /\  ( 1  <_  x  /\  x  <_  n  /\  n  <_ +oo ) )  -> 
( log `  x
)  <_  ( log `  n ) )
106 simprr 764 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  1  <_  x )
)  ->  1  <_  x )
107 simprl 762 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  1  <_  x )
)  ->  x  e.  RR+ )
108 logleb 23538 . . . . . . . 8  |-  ( ( 1  e.  RR+  /\  x  e.  RR+ )  ->  (
1  <_  x  <->  ( log `  1 )  <_  ( log `  x ) ) )
10947, 107, 108sylancr 667 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  1  <_  x )
)  ->  ( 1  <_  x  <->  ( log `  1 )  <_  ( log `  x ) ) )
110106, 109mpbid 213 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  1  <_  A )  /\  ( x  e.  RR+  /\  1  <_  x )
)  ->  ( log `  1 )  <_  ( log `  x ) )
11157, 110syl5eqbrr 4455 . . . . 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 10240 . . . . . 6  |-  1  <_  1
114113a1i 11 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  1  <_  1 )
115 simpr 462 . . . . 5  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  1  <_  A )
11611rexrd 9690 . . . . . 6  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  A  e.  RR* )
117 pnfge 11432 . . . . . 6  |-  ( A  e.  RR*  ->  A  <_ +oo )
118116, 117syl 17 . . . . 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 22972 . . . 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 4443 . . 3  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  ( abs `  ( ( ( log `  ( ! `
 ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) )  -  1 ) )  <_  ( log `  A ) )
12121, 25, 13, 30, 120letrd 9792 . 2  |-  ( ( A  e.  RR+  /\  1  <_  A )  ->  (
( abs `  (
( log `  ( ! `  ( |_ `  A ) ) )  -  ( A  x.  ( ( log `  A
)  -  1 ) ) ) )  - 
1 )  <_  ( log `  A ) )
12219, 83, 13lesubaddd 10210 . 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 213 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   class class class wbr 4420    |-> cmpt 4479   ` cfv 5597  (class class class)co 6301   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544   +oocpnf 9672   RR*cxr 9674    <_ cle 9676    - cmin 9860   NNcn 10609   NN0cn0 10869   ZZcz 10937   RR+crp 11302   (,)cioo 11635   ...cfz 11784   |_cfl 12025   !cfa 12458   abscabs 13285   sum_csu 13739    _D cdv 22804   logclog 23490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-inf2 8148  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-se 4809  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-isom 5606  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-of 6541  df-om 6703  df-1st 6803  df-2nd 6804  df-supp 6922  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-2o 7187  df-oadd 7190  df-er 7367  df-map 7478  df-pm 7479  df-ixp 7527  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-fsupp 7886  df-fi 7927  df-sup 7958  df-inf 7959  df-oi 8027  df-card 8374  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12027  df-mod 12096  df-seq 12213  df-exp 12272  df-fac 12459  df-bc 12487  df-hash 12515  df-shft 13118  df-cj 13150  df-re 13151  df-im 13152  df-sqrt 13286  df-abs 13287  df-limsup 13513  df-clim 13539  df-rlim 13540  df-sum 13740  df-ef 14108  df-sin 14110  df-cos 14111  df-pi 14113  df-struct 15110  df-ndx 15111  df-slot 15112  df-base 15113  df-sets 15114  df-ress 15115  df-plusg 15190  df-mulr 15191  df-starv 15192  df-sca 15193  df-vsca 15194  df-ip 15195  df-tset 15196  df-ple 15197  df-ds 15199  df-unif 15200  df-hom 15201  df-cco 15202  df-rest 15308  df-topn 15309  df-0g 15327  df-gsum 15328  df-topgen 15329  df-pt 15330  df-prds 15333  df-xrs 15387  df-qtop 15393  df-imas 15394  df-xps 15397  df-mre 15479  df-mrc 15480  df-acs 15482  df-mgm 16475  df-sgrp 16514  df-mnd 16524  df-submnd 16570  df-mulg 16663  df-cntz 16958  df-cmn 17419  df-psmet 18949  df-xmet 18950  df-met 18951  df-bl 18952  df-mopn 18953  df-fbas 18954  df-fg 18955  df-cnfld 18958  df-top 19907  df-bases 19908  df-topon 19909  df-topsp 19910  df-cld 20020  df-ntr 20021  df-cls 20022  df-nei 20100  df-lp 20138  df-perf 20139  df-cn 20229  df-cnp 20230  df-haus 20317  df-cmp 20388  df-tx 20563  df-hmeo 20756  df-fil 20847  df-fm 20939  df-flim 20940  df-flf 20941  df-xms 21321  df-ms 21322  df-tms 21323  df-cncf 21896  df-limc 22807  df-dv 22808  df-log 23492
This theorem is referenced by:  logfacrlim  24138
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