MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mulog2sumlem1 Unicode version

Theorem mulog2sumlem1 20683
Description: Asymptotic formula for  sum_ n  <_  x ,  log (
x  /  n )  /  n  =  ( 1  /  2 ) log ^ 2 ( x )  +  gamma  x.  log x  -  L  +  O ( log x  /  x ), with explicit constants. Equation 10.2.7 of [Shapiro], p. 407. (Contributed by Mario Carneiro, 18-May-2016.)
Hypotheses
Ref Expression
logdivsum.1  |-  F  =  ( y  e.  RR+  |->  ( sum_ i  e.  ( 1 ... ( |_
`  y ) ) ( ( log `  i
)  /  i )  -  ( ( ( log `  y ) ^ 2 )  / 
2 ) ) )
mulog2sumlem.1  |-  ( ph  ->  F  ~~> r  L )
mulog2sumlem1.2  |-  ( ph  ->  A  e.  RR+ )
mulog2sumlem1.3  |-  ( ph  ->  _e  <_  A )
Assertion
Ref Expression
mulog2sumlem1  |-  ( ph  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  ( A  /  m ) )  /  m )  -  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( gamma  x.  ( log `  A ) )  -  L ) ) ) )  <_  (
2  x.  ( ( log `  A )  /  A ) ) )
Distinct variable groups:    i, m, y, A    ph, m
Allowed substitution hints:    ph( y, i)    F( y, i, m)    L( y, i, m)

Proof of Theorem mulog2sumlem1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fzfid 11035 . . . . . 6  |-  ( ph  ->  ( 1 ... ( |_ `  A ) )  e.  Fin )
2 mulog2sumlem1.2 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR+ )
3 elfznn 10819 . . . . . . . . . 10  |-  ( m  e.  ( 1 ... ( |_ `  A
) )  ->  m  e.  NN )
43nnrpd 10389 . . . . . . . . 9  |-  ( m  e.  ( 1 ... ( |_ `  A
) )  ->  m  e.  RR+ )
5 rpdivcl 10376 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  ( A  /  m )  e.  RR+ )
62, 4, 5syl2an 463 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( A  /  m )  e.  RR+ )
76relogcld 19974 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( log `  ( A  /  m
) )  e.  RR )
83adantl 452 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  m  e.  NN )
97, 8nndivred 9794 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  ( A  /  m ) )  /  m )  e.  RR )
101, 9fsumrecl 12207 . . . . 5  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  ( A  /  m ) )  /  m )  e.  RR )
112relogcld 19974 . . . . . . . 8  |-  ( ph  ->  ( log `  A
)  e.  RR )
1211resqcld 11271 . . . . . . 7  |-  ( ph  ->  ( ( log `  A
) ^ 2 )  e.  RR )
1312rehalfcld 9958 . . . . . 6  |-  ( ph  ->  ( ( ( log `  A ) ^ 2 )  /  2 )  e.  RR )
14 emre 20299 . . . . . . . 8  |-  gamma  e.  RR
15 remulcl 8822 . . . . . . . 8  |-  ( (
gamma  e.  RR  /\  ( log `  A )  e.  RR )  ->  ( gamma  x.  ( log `  A
) )  e.  RR )
1614, 11, 15sylancr 644 . . . . . . 7  |-  ( ph  ->  ( gamma  x.  ( log `  A ) )  e.  RR )
17 rpsup 10970 . . . . . . . . 9  |-  sup ( RR+ ,  RR* ,  <  )  =  +oo
1817a1i 10 . . . . . . . 8  |-  ( ph  ->  sup ( RR+ ,  RR* ,  <  )  =  +oo )
19 logdivsum.1 . . . . . . . . . . . . 13  |-  F  =  ( y  e.  RR+  |->  ( sum_ i  e.  ( 1 ... ( |_
`  y ) ) ( ( log `  i
)  /  i )  -  ( ( ( log `  y ) ^ 2 )  / 
2 ) ) )
2019logdivsum 20682 . . . . . . . . . . . 12  |-  ( F : RR+ --> RR  /\  F  e.  dom  ~~> r  /\  (
( F  ~~> r  L  /\  A  e.  RR+  /\  _e  <_  A )  ->  ( abs `  ( ( F `
 A )  -  L ) )  <_ 
( ( log `  A
)  /  A ) ) )
2120simp1i 964 . . . . . . . . . . 11  |-  F : RR+
--> RR
2221a1i 10 . . . . . . . . . 10  |-  ( ph  ->  F : RR+ --> RR )
2322feqmptd 5575 . . . . . . . . 9  |-  ( ph  ->  F  =  ( x  e.  RR+  |->  ( F `
 x ) ) )
24 mulog2sumlem.1 . . . . . . . . 9  |-  ( ph  ->  F  ~~> r  L )
2523, 24eqbrtrrd 4045 . . . . . . . 8  |-  ( ph  ->  ( x  e.  RR+  |->  ( F `  x ) )  ~~> r  L )
2621ffvelrni 5664 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( F `
 x )  e.  RR )
2726adantl 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( F `  x )  e.  RR )
2818, 25, 27rlimrecl 12054 . . . . . . 7  |-  ( ph  ->  L  e.  RR )
2916, 28resubcld 9211 . . . . . 6  |-  ( ph  ->  ( ( gamma  x.  ( log `  A ) )  -  L )  e.  RR )
3013, 29readdcld 8862 . . . . 5  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( gamma  x.  ( log `  A ) )  -  L ) )  e.  RR )
3110, 30resubcld 9211 . . . 4  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  ( A  /  m ) )  /  m )  -  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( gamma  x.  ( log `  A ) )  -  L ) ) )  e.  RR )
3231recnd 8861 . . 3  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  ( A  /  m ) )  /  m )  -  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( gamma  x.  ( log `  A ) )  -  L ) ) )  e.  CC )
3332abscld 11918 . 2  |-  ( ph  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  ( A  /  m ) )  /  m )  -  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( gamma  x.  ( log `  A ) )  -  L ) ) ) )  e.  RR )
34 rerpdivcl 10381 . . . . . . . 8  |-  ( ( ( log `  A
)  e.  RR  /\  m  e.  RR+ )  -> 
( ( log `  A
)  /  m )  e.  RR )
3511, 4, 34syl2an 463 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  A )  /  m )  e.  RR )
3635recnd 8861 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  A )  /  m )  e.  CC )
371, 36fsumcl 12206 . . . . 5  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  e.  CC )
3811recnd 8861 . . . . . 6  |-  ( ph  ->  ( log `  A
)  e.  CC )
39 readdcl 8820 . . . . . . . 8  |-  ( ( ( log `  A
)  e.  RR  /\  gamma  e.  RR )  ->  (
( log `  A
)  +  gamma )  e.  RR )
4011, 14, 39sylancl 643 . . . . . . 7  |-  ( ph  ->  ( ( log `  A
)  +  gamma )  e.  RR )
4140recnd 8861 . . . . . 6  |-  ( ph  ->  ( ( log `  A
)  +  gamma )  e.  CC )
4238, 41mulcld 8855 . . . . 5  |-  ( ph  ->  ( ( log `  A
)  x.  ( ( log `  A )  +  gamma ) )  e.  CC )
4337, 42subcld 9157 . . . 4  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  A
)  /  m )  -  ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
) )  e.  CC )
4443abscld 11918 . . 3  |-  ( ph  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  -  ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
) ) )  e.  RR )
458nnrpd 10389 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  m  e.  RR+ )
4645relogcld 19974 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( log `  m )  e.  RR )
4746, 8nndivred 9794 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  m )  /  m )  e.  RR )
4847recnd 8861 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  m )  /  m )  e.  CC )
491, 48fsumcl 12206 . . . . 5  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  m
)  /  m )  e.  CC )
5013recnd 8861 . . . . . 6  |-  ( ph  ->  ( ( ( log `  A ) ^ 2 )  /  2 )  e.  CC )
5128recnd 8861 . . . . . 6  |-  ( ph  ->  L  e.  CC )
5250, 51addcld 8854 . . . . 5  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  L
)  e.  CC )
5349, 52subcld 9157 . . . 4  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  L ) )  e.  CC )
5453abscld 11918 . . 3  |-  ( ph  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  L ) ) )  e.  RR )
5544, 54readdcld 8862 . 2  |-  ( ph  ->  ( ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  -  ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
) ) )  +  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  L ) ) ) )  e.  RR )
56 2re 9815 . . 3  |-  2  e.  RR
5711, 2rerpdivcld 10417 . . 3  |-  ( ph  ->  ( ( log `  A
)  /  A )  e.  RR )
58 remulcl 8822 . . 3  |-  ( ( 2  e.  RR  /\  ( ( log `  A
)  /  A )  e.  RR )  -> 
( 2  x.  (
( log `  A
)  /  A ) )  e.  RR )
5956, 57, 58sylancr 644 . 2  |-  ( ph  ->  ( 2  x.  (
( log `  A
)  /  A ) )  e.  RR )
60 relogdiv 19946 . . . . . . . . . . 11  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  ( log `  ( A  /  m ) )  =  ( ( log `  A
)  -  ( log `  m ) ) )
612, 4, 60syl2an 463 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( log `  ( A  /  m
) )  =  ( ( log `  A
)  -  ( log `  m ) ) )
6261oveq1d 5873 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  ( A  /  m ) )  /  m )  =  ( ( ( log `  A
)  -  ( log `  m ) )  /  m ) )
6338adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( log `  A )  e.  CC )
6446recnd 8861 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( log `  m )  e.  CC )
6545rpcnne0d 10399 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( m  e.  CC  /\  m  =/=  0 ) )
66 divsubdir 9456 . . . . . . . . . 10  |-  ( ( ( log `  A
)  e.  CC  /\  ( log `  m )  e.  CC  /\  (
m  e.  CC  /\  m  =/=  0 ) )  ->  ( ( ( log `  A )  -  ( log `  m
) )  /  m
)  =  ( ( ( log `  A
)  /  m )  -  ( ( log `  m )  /  m
) ) )
6763, 64, 65, 66syl3anc 1182 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (
( log `  A
)  -  ( log `  m ) )  /  m )  =  ( ( ( log `  A
)  /  m )  -  ( ( log `  m )  /  m
) ) )
6862, 67eqtrd 2315 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  ( A  /  m ) )  /  m )  =  ( ( ( log `  A
)  /  m )  -  ( ( log `  m )  /  m
) ) )
6968sumeq2dv 12176 . . . . . . 7  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  ( A  /  m ) )  /  m )  = 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( ( log `  A
)  /  m )  -  ( ( log `  m )  /  m
) ) )
701, 36, 48fsumsub 12250 . . . . . . 7  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( ( log `  A
)  /  m )  -  ( ( log `  m )  /  m
) )  =  (
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  -  sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  m
)  /  m ) ) )
7169, 70eqtrd 2315 . . . . . 6  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  ( A  /  m ) )  /  m )  =  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  A
)  /  m )  -  sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  m
)  /  m ) ) )
72 remulcl 8822 . . . . . . . . . . . . 13  |-  ( ( ( log `  A
)  e.  RR  /\  gamma  e.  RR )  ->  (
( log `  A
)  x.  gamma )  e.  RR )
7311, 14, 72sylancl 643 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( log `  A
)  x.  gamma )  e.  RR )
7413, 73readdcld 8862 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( log `  A
)  x.  gamma )
)  e.  RR )
7574recnd 8861 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( log `  A
)  x.  gamma )
)  e.  CC )
7675, 50pncand 9158 . . . . . . . . 9  |-  ( ph  ->  ( ( ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  ( ( log `  A
)  x.  gamma )
)  +  ( ( ( log `  A
) ^ 2 )  /  2 ) )  -  ( ( ( log `  A ) ^ 2 )  / 
2 ) )  =  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( log `  A
)  x.  gamma )
) )
7714recni 8849 . . . . . . . . . . . . 13  |-  gamma  e.  CC
7877a1i 10 . . . . . . . . . . . 12  |-  ( ph  -> 
gamma  e.  CC )
7938, 38, 78adddid 8859 . . . . . . . . . . 11  |-  ( ph  ->  ( ( log `  A
)  x.  ( ( log `  A )  +  gamma ) )  =  ( ( ( log `  A )  x.  ( log `  A ) )  +  ( ( log `  A )  x.  gamma ) ) )
8012recnd 8861 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( log `  A
) ^ 2 )  e.  CC )
81802halvesd 9957 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( ( log `  A
) ^ 2 )  /  2 ) )  =  ( ( log `  A ) ^ 2 ) )
8238sqvald 11242 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( log `  A
) ^ 2 )  =  ( ( log `  A )  x.  ( log `  A ) ) )
8381, 82eqtrd 2315 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( ( log `  A
) ^ 2 )  /  2 ) )  =  ( ( log `  A )  x.  ( log `  A ) ) )
8483oveq1d 5873 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  ( ( ( log `  A ) ^ 2 )  /  2 ) )  +  ( ( log `  A )  x.  gamma ) )  =  ( ( ( log `  A )  x.  ( log `  A ) )  +  ( ( log `  A )  x.  gamma ) ) )
8573recnd 8861 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( log `  A
)  x.  gamma )  e.  CC )
8650, 50, 85add32d 9034 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  ( ( ( log `  A ) ^ 2 )  /  2 ) )  +  ( ( log `  A )  x.  gamma ) )  =  ( ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  ( ( log `  A
)  x.  gamma )
)  +  ( ( ( log `  A
) ^ 2 )  /  2 ) ) )
8779, 84, 863eqtr2d 2321 . . . . . . . . . 10  |-  ( ph  ->  ( ( log `  A
)  x.  ( ( log `  A )  +  gamma ) )  =  ( ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  ( ( log `  A
)  x.  gamma )
)  +  ( ( ( log `  A
) ^ 2 )  /  2 ) ) )
8887oveq1d 5873 . . . . . . . . 9  |-  ( ph  ->  ( ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
)  -  ( ( ( log `  A
) ^ 2 )  /  2 ) )  =  ( ( ( ( ( ( log `  A ) ^ 2 )  /  2 )  +  ( ( log `  A )  x.  gamma ) )  +  ( ( ( log `  A
) ^ 2 )  /  2 ) )  -  ( ( ( log `  A ) ^ 2 )  / 
2 ) ) )
89 mulcom 8823 . . . . . . . . . . 11  |-  ( (
gamma  e.  CC  /\  ( log `  A )  e.  CC )  ->  ( gamma  x.  ( log `  A
) )  =  ( ( log `  A
)  x.  gamma )
)
9077, 38, 89sylancr 644 . . . . . . . . . 10  |-  ( ph  ->  ( gamma  x.  ( log `  A ) )  =  ( ( log `  A )  x.  gamma ) )
9190oveq2d 5874 . . . . . . . . 9  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  (
gamma  x.  ( log `  A
) ) )  =  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( log `  A
)  x.  gamma )
) )
9276, 88, 913eqtr4rd 2326 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  (
gamma  x.  ( log `  A
) ) )  =  ( ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
)  -  ( ( ( log `  A
) ^ 2 )  /  2 ) ) )
9392oveq1d 5873 . . . . . . 7  |-  ( ph  ->  ( ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  ( gamma  x.  ( log `  A ) ) )  -  L )  =  ( ( ( ( log `  A
)  x.  ( ( log `  A )  +  gamma ) )  -  ( ( ( log `  A ) ^ 2 )  /  2 ) )  -  L ) )
9490, 85eqeltrd 2357 . . . . . . . 8  |-  ( ph  ->  ( gamma  x.  ( log `  A ) )  e.  CC )
9550, 94, 51addsubassd 9177 . . . . . . 7  |-  ( ph  ->  ( ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  ( gamma  x.  ( log `  A ) ) )  -  L )  =  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  ( ( gamma  x.  ( log `  A ) )  -  L ) ) )
9642, 50, 51subsub4d 9188 . . . . . . 7  |-  ( ph  ->  ( ( ( ( log `  A )  x.  ( ( log `  A )  +  gamma ) )  -  ( ( ( log `  A
) ^ 2 )  /  2 ) )  -  L )  =  ( ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
)  -  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  L ) ) )
9793, 95, 963eqtr3d 2323 . . . . . 6  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( gamma  x.  ( log `  A ) )  -  L ) )  =  ( ( ( log `  A )  x.  ( ( log `  A )  +  gamma ) )  -  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  L ) ) )
9871, 97oveq12d 5876 . . . . 5  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  ( A  /  m ) )  /  m )  -  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( gamma  x.  ( log `  A ) )  -  L ) ) )  =  ( (
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  -  sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  m
)  /  m ) )  -  ( ( ( log `  A
)  x.  ( ( log `  A )  +  gamma ) )  -  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  L
) ) ) )
9937, 49, 42, 52sub4d 9206 . . . . 5  |-  ( ph  ->  ( ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  -  sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  m
)  /  m ) )  -  ( ( ( log `  A
)  x.  ( ( log `  A )  +  gamma ) )  -  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  L
) ) )  =  ( ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  -  ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
) )  -  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  L ) ) ) )
10098, 99eqtrd 2315 . . . 4  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  ( A  /  m ) )  /  m )  -  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( gamma  x.  ( log `  A ) )  -  L ) ) )  =  ( (
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  -  ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
) )  -  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  L ) ) ) )
101100fveq2d 5529 . . 3  |-  ( ph  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  ( A  /  m ) )  /  m )  -  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( gamma  x.  ( log `  A ) )  -  L ) ) ) )  =  ( abs `  ( (
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  -  ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
) )  -  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  L ) ) ) ) )
10243, 53abs2dif2d 11940 . . 3  |-  ( ph  ->  ( abs `  (
( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  A
)  /  m )  -  ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
) )  -  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  L ) ) ) )  <_  ( ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  -  ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
) ) )  +  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  L ) ) ) ) )
103101, 102eqbrtrd 4043 . 2  |-  ( ph  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  ( A  /  m ) )  /  m )  -  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( gamma  x.  ( log `  A ) )  -  L ) ) ) )  <_  (
( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  -  ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
) ) )  +  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  L ) ) ) ) )
104 harmonicbnd4 20304 . . . . . . 7  |-  ( A  e.  RR+  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( 1  /  m
)  -  ( ( log `  A )  +  gamma ) ) )  <_  ( 1  /  A ) )
1052, 104syl 15 . . . . . 6  |-  ( ph  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( 1  /  m )  -  ( ( log `  A )  +  gamma ) ) )  <_  (
1  /  A ) )
1068nnrecred 9791 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1  /  m )  e.  RR )
1071, 106fsumrecl 12207 . . . . . . . . . 10  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( 1  /  m )  e.  RR )
108107, 40resubcld 9211 . . . . . . . . 9  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( 1  /  m
)  -  ( ( log `  A )  +  gamma ) )  e.  RR )
109108recnd 8861 . . . . . . . 8  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( 1  /  m
)  -  ( ( log `  A )  +  gamma ) )  e.  CC )
110109abscld 11918 . . . . . . 7  |-  ( ph  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( 1  /  m )  -  ( ( log `  A )  +  gamma ) ) )  e.  RR )
1112rprecred 10401 . . . . . . 7  |-  ( ph  ->  ( 1  /  A
)  e.  RR )
112 0re 8838 . . . . . . . . 9  |-  0  e.  RR
113112a1i 10 . . . . . . . 8  |-  ( ph  ->  0  e.  RR )
114 1re 8837 . . . . . . . . 9  |-  1  e.  RR
115114a1i 10 . . . . . . . 8  |-  ( ph  ->  1  e.  RR )
116 0lt1 9296 . . . . . . . . 9  |-  0  <  1
117116a1i 10 . . . . . . . 8  |-  ( ph  ->  0  <  1 )
118 loge 19940 . . . . . . . . 9  |-  ( log `  _e )  =  1
119 mulog2sumlem1.3 . . . . . . . . . 10  |-  ( ph  ->  _e  <_  A )
120 epr 12486 . . . . . . . . . . 11  |-  _e  e.  RR+
121 logleb 19957 . . . . . . . . . . 11  |-  ( ( _e  e.  RR+  /\  A  e.  RR+ )  ->  (
_e  <_  A  <->  ( log `  _e )  <_  ( log `  A ) ) )
122120, 2, 121sylancr 644 . . . . . . . . . 10  |-  ( ph  ->  ( _e  <_  A  <->  ( log `  _e )  <_  ( log `  A
) ) )
123119, 122mpbid 201 . . . . . . . . 9  |-  ( ph  ->  ( log `  _e )  <_  ( log `  A
) )
124118, 123syl5eqbrr 4057 . . . . . . . 8  |-  ( ph  ->  1  <_  ( log `  A ) )
125113, 115, 11, 117, 124ltletrd 8976 . . . . . . 7  |-  ( ph  ->  0  <  ( log `  A ) )
126 lemul2 9609 . . . . . . 7  |-  ( ( ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( 1  /  m )  -  ( ( log `  A )  +  gamma ) ) )  e.  RR  /\  ( 1  /  A
)  e.  RR  /\  ( ( log `  A
)  e.  RR  /\  0  <  ( log `  A
) ) )  -> 
( ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( 1  /  m )  -  ( ( log `  A )  +  gamma ) ) )  <_  (
1  /  A )  <-> 
( ( log `  A
)  x.  ( abs `  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( 1  /  m
)  -  ( ( log `  A )  +  gamma ) ) ) )  <_  ( ( log `  A )  x.  ( 1  /  A
) ) ) )
127110, 111, 11, 125, 126syl112anc 1186 . . . . . 6  |-  ( ph  ->  ( ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( 1  /  m )  -  ( ( log `  A )  +  gamma ) ) )  <_  (
1  /  A )  <-> 
( ( log `  A
)  x.  ( abs `  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( 1  /  m
)  -  ( ( log `  A )  +  gamma ) ) ) )  <_  ( ( log `  A )  x.  ( 1  /  A
) ) ) )
128105, 127mpbid 201 . . . . 5  |-  ( ph  ->  ( ( log `  A
)  x.  ( abs `  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( 1  /  m
)  -  ( ( log `  A )  +  gamma ) ) ) )  <_  ( ( log `  A )  x.  ( 1  /  A
) ) )
12945rpcnd 10392 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  m  e.  CC )
13045rpne0d 10395 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  m  =/=  0 )
13163, 129, 130divrecd 9539 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  A )  /  m )  =  ( ( log `  A
)  x.  ( 1  /  m ) ) )
132131sumeq2dv 12176 . . . . . . . . . 10  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  =  sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  A
)  x.  ( 1  /  m ) ) )
133106recnd 8861 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1  /  m )  e.  CC )
1341, 38, 133fsummulc2 12246 . . . . . . . . . 10  |-  ( ph  ->  ( ( log `  A
)  x.  sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( 1  /  m
) )  =  sum_ m  e.  ( 1 ... ( |_ `  A
) ) ( ( log `  A )  x.  ( 1  /  m ) ) )
135132, 134eqtr4d 2318 . . . . . . . . 9  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  =  ( ( log `  A )  x.  sum_ m  e.  ( 1 ... ( |_ `  A
) ) ( 1  /  m ) ) )
136135oveq1d 5873 . . . . . . . 8  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  A
)  /  m )  -  ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
) )  =  ( ( ( log `  A
)  x.  sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( 1  /  m
) )  -  (
( log `  A
)  x.  ( ( log `  A )  +  gamma ) ) ) )
1371, 133fsumcl 12206 . . . . . . . . 9  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( 1  /  m )  e.  CC )
13838, 137, 41subdid 9235 . . . . . . . 8  |-  ( ph  ->  ( ( log `  A
)  x.  ( sum_ m  e.  ( 1 ... ( |_ `  A
) ) ( 1  /  m )  -  ( ( log `  A
)  +  gamma )
) )  =  ( ( ( log `  A
)  x.  sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( 1  /  m
) )  -  (
( log `  A
)  x.  ( ( log `  A )  +  gamma ) ) ) )
139136, 138eqtr4d 2318 . . . . . . 7  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  A
)  /  m )  -  ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
) )  =  ( ( log `  A
)  x.  ( sum_ m  e.  ( 1 ... ( |_ `  A
) ) ( 1  /  m )  -  ( ( log `  A
)  +  gamma )
) ) )
140139fveq2d 5529 . . . . . 6  |-  ( ph  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  -  ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
) ) )  =  ( abs `  (
( log `  A
)  x.  ( sum_ m  e.  ( 1 ... ( |_ `  A
) ) ( 1  /  m )  -  ( ( log `  A
)  +  gamma )
) ) ) )
141137, 41subcld 9157 . . . . . . 7  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( 1  /  m
)  -  ( ( log `  A )  +  gamma ) )  e.  CC )
14238, 141absmuld 11936 . . . . . 6  |-  ( ph  ->  ( abs `  (
( log `  A
)  x.  ( sum_ m  e.  ( 1 ... ( |_ `  A
) ) ( 1  /  m )  -  ( ( log `  A
)  +  gamma )
) ) )  =  ( ( abs `  ( log `  A ) )  x.  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( 1  /  m )  -  ( ( log `  A )  +  gamma ) ) ) ) )
143113, 11, 125ltled 8967 . . . . . . . 8  |-  ( ph  ->  0  <_  ( log `  A ) )
14411, 143absidd 11905 . . . . . . 7  |-  ( ph  ->  ( abs `  ( log `  A ) )  =  ( log `  A
) )
145144oveq1d 5873 . . . . . 6  |-  ( ph  ->  ( ( abs `  ( log `  A ) )  x.  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( 1  /  m )  -  ( ( log `  A )  +  gamma ) ) ) )  =  ( ( log `  A
)  x.  ( abs `  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( 1  /  m
)  -  ( ( log `  A )  +  gamma ) ) ) ) )
146140, 142, 1453eqtrd 2319 . . . . 5  |-  ( ph  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  -  ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
) ) )  =  ( ( log `  A
)  x.  ( abs `  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( 1  /  m
)  -  ( ( log `  A )  +  gamma ) ) ) ) )
1472rpcnd 10392 . . . . . 6  |-  ( ph  ->  A  e.  CC )
1482rpne0d 10395 . . . . . 6  |-  ( ph  ->  A  =/=  0 )
14938, 147, 148divrecd 9539 . . . . 5  |-  ( ph  ->  ( ( log `  A
)  /  A )  =  ( ( log `  A )  x.  (
1  /  A ) ) )
150128, 146, 1493brtr4d 4053 . . . 4  |-  ( ph  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  -  ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
) ) )  <_ 
( ( log `  A
)  /  A ) )
151 fveq2 5525 . . . . . . . . . . . . . 14  |-  ( i  =  m  ->  ( log `  i )  =  ( log `  m
) )
152 id 19 . . . . . . . . . . . . . 14  |-  ( i  =  m  ->  i  =  m )
153151, 152oveq12d 5876 . . . . . . . . . . . . 13  |-  ( i  =  m  ->  (
( log `  i
)  /  i )  =  ( ( log `  m )  /  m
) )
154153cbvsumv 12169 . . . . . . . . . . . 12  |-  sum_ i  e.  ( 1 ... ( |_ `  y ) ) ( ( log `  i
)  /  i )  =  sum_ m  e.  ( 1 ... ( |_
`  y ) ) ( ( log `  m
)  /  m )
155 fveq2 5525 . . . . . . . . . . . . . 14  |-  ( y  =  A  ->  ( |_ `  y )  =  ( |_ `  A
) )
156155oveq2d 5874 . . . . . . . . . . . . 13  |-  ( y  =  A  ->  (
1 ... ( |_ `  y ) )  =  ( 1 ... ( |_ `  A ) ) )
157156sumeq1d 12174 . . . . . . . . . . . 12  |-  ( y  =  A  ->  sum_ m  e.  ( 1 ... ( |_ `  y ) ) ( ( log `  m
)  /  m )  =  sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  m
)  /  m ) )
158154, 157syl5eq 2327 . . . . . . . . . . 11  |-  ( y  =  A  ->  sum_ i  e.  ( 1 ... ( |_ `  y ) ) ( ( log `  i
)  /  i )  =  sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  m
)  /  m ) )
159 fveq2 5525 . . . . . . . . . . . . 13  |-  ( y  =  A  ->  ( log `  y )  =  ( log `  A
) )
160159oveq1d 5873 . . . . . . . . . . . 12  |-  ( y  =  A  ->  (
( log `  y
) ^ 2 )  =  ( ( log `  A ) ^ 2 ) )
161160oveq1d 5873 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
( ( log `  y
) ^ 2 )  /  2 )  =  ( ( ( log `  A ) ^ 2 )  /  2 ) )
162158, 161oveq12d 5876 . . . . . . . . . 10  |-  ( y  =  A  ->  ( sum_ i  e.  ( 1 ... ( |_ `  y ) ) ( ( log `  i
)  /  i )  -  ( ( ( log `  y ) ^ 2 )  / 
2 ) )  =  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( log `  A ) ^ 2 )  / 
2 ) ) )
163 ovex 5883 . . . . . . . . . 10  |-  ( sum_ m  e.  ( 1 ... ( |_ `  A
) ) ( ( log `  m )  /  m )  -  ( ( ( log `  A ) ^ 2 )  /  2 ) )  e.  _V
164162, 19, 163fvmpt 5602 . . . . . . . . 9  |-  ( A  e.  RR+  ->  ( F `
 A )  =  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( log `  A ) ^ 2 )  / 
2 ) ) )
1652, 164syl 15 . . . . . . . 8  |-  ( ph  ->  ( F `  A
)  =  ( sum_ m  e.  ( 1 ... ( |_ `  A
) ) ( ( log `  m )  /  m )  -  ( ( ( log `  A ) ^ 2 )  /  2 ) ) )
166165oveq1d 5873 . . . . . . 7  |-  ( ph  ->  ( ( F `  A )  -  L
)  =  ( (
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( log `  A ) ^ 2 )  / 
2 ) )  -  L ) )
16749, 50, 51subsub4d 9188 . . . . . . 7  |-  ( ph  ->  ( ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( log `  A ) ^ 2 )  / 
2 ) )  -  L )  =  (
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  L ) ) )
168166, 167eqtrd 2315 . . . . . 6  |-  ( ph  ->  ( ( F `  A )  -  L
)  =  ( sum_ m  e.  ( 1 ... ( |_ `  A
) ) ( ( log `  m )  /  m )  -  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  L
) ) )
169168fveq2d 5529 . . . . 5  |-  ( ph  ->  ( abs `  (
( F `  A
)  -  L ) )  =  ( abs `  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  L ) ) ) )
17020simp3i 966 . . . . . 6  |-  ( ( F  ~~> r  L  /\  A  e.  RR+  /\  _e  <_  A )  ->  ( abs `  ( ( F `
 A )  -  L ) )  <_ 
( ( log `  A
)  /  A ) )
17124, 2, 119, 170syl3anc 1182 . . . . 5  |-  ( ph  ->  ( abs `  (
( F `  A
)  -  L ) )  <_  ( ( log `  A )  /  A ) )
172169, 171eqbrtrrd 4045 . . . 4  |-  ( ph  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  L ) ) )  <_  ( ( log `  A )  /  A
) )
17344, 54, 57, 57, 150, 172le2addd 9390 . . 3  |-  ( ph  ->  ( ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  -  ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
) ) )  +  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  L ) ) ) )  <_  ( (
( log `  A
)  /  A )  +  ( ( log `  A )  /  A
) ) )
17457recnd 8861 . . . 4  |-  ( ph  ->  ( ( log `  A
)  /  A )  e.  CC )
1751742timesd 9954 . . 3  |-  ( ph  ->  ( 2  x.  (
( log `  A
)  /  A ) )  =  ( ( ( log `  A
)  /  A )  +  ( ( log `  A )  /  A
) ) )
176173, 175breqtrrd 4049 . 2  |-  ( ph  ->  ( ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  -  ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
) ) )  +  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  L ) ) ) )  <_  ( 2  x.  ( ( log `  A )  /  A
) ) )
17733, 55, 59, 103, 176letrd 8973 1  |-  ( ph  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  ( A  /  m ) )  /  m )  -  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( gamma  x.  ( log `  A ) )  -  L ) ) ) )  <_  (
2  x.  ( ( log `  A )  /  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023    e. cmpt 4077   dom cdm 4689   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   2c2 9795   RR+crp 10354   ...cfz 10782   |_cfl 10924   ^cexp 11104   abscabs 11719    ~~> r crli 11959   sum_csu 12158   _eceu 12344   logclog 19912   gammacem 20286
This theorem is referenced by:  mulog2sumlem2  20684
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-e 12350  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  df-cxp 19915  df-em 20287
  Copyright terms: Public domain W3C validator