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Theorem mulog2sumlem1 20631
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
StepHypRef Expression
1 fzfid 10987 . . . . . 6  |-  ( ph  ->  ( 1 ... ( |_ `  A ) )  e.  Fin )
2 mulog2sumlem1.2 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR+ )
3 elfznn 10771 . . . . . . . . . 10  |-  ( m  e.  ( 1 ... ( |_ `  A
) )  ->  m  e.  NN )
43nnrpd 10342 . . . . . . . . 9  |-  ( m  e.  ( 1 ... ( |_ `  A
) )  ->  m  e.  RR+ )
5 rpdivcl 10329 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  ( A  /  m )  e.  RR+ )
62, 4, 5syl2an 465 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( A  /  m )  e.  RR+ )
76relogcld 19922 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( log `  ( A  /  m
) )  e.  RR )
83adantl 454 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  m  e.  NN )
97, 8nndivred 9748 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  ( A  /  m ) )  /  m )  e.  RR )
101, 9fsumrecl 12158 . . . . 5  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  ( A  /  m ) )  /  m )  e.  RR )
112relogcld 19922 . . . . . . . 8  |-  ( ph  ->  ( log `  A
)  e.  RR )
1211resqcld 11223 . . . . . . 7  |-  ( ph  ->  ( ( log `  A
) ^ 2 )  e.  RR )
1312rehalfcld 9911 . . . . . 6  |-  ( ph  ->  ( ( ( log `  A ) ^ 2 )  /  2 )  e.  RR )
14 emre 20247 . . . . . . . 8  |-  gamma  e.  RR
15 remulcl 8776 . . . . . . . 8  |-  ( (
gamma  e.  RR  /\  ( log `  A )  e.  RR )  ->  ( gamma  x.  ( log `  A
) )  e.  RR )
1614, 11, 15sylancr 647 . . . . . . 7  |-  ( ph  ->  ( gamma  x.  ( log `  A ) )  e.  RR )
17 rpsup 10922 . . . . . . . . 9  |-  sup ( RR+ ,  RR* ,  <  )  =  +oo
1817a1i 12 . . . . . . . 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 20630 . . . . . . . . . . . 12  |-  ( F : RR+ --> RR  /\  F  e.  dom  ~~> r  /\  (
( F  ~~> r  L  /\  A  e.  RR+  /\  _e  <_  A )  ->  ( abs `  ( ( F `
 A )  -  L ) )  <_ 
( ( log `  A
)  /  A ) ) )
2120simp1i 969 . . . . . . . . . . 11  |-  F : RR+
--> RR
2221a1i 12 . . . . . . . . . 10  |-  ( ph  ->  F : RR+ --> RR )
2322feqmptd 5495 . . . . . . . . 9  |-  ( ph  ->  F  =  ( x  e.  RR+  |->  ( F `
 x ) ) )
24 mulog2sumlem.1 . . . . . . . . 9  |-  ( ph  ->  F  ~~> r  L )
2523, 24eqbrtrrd 4005 . . . . . . . 8  |-  ( ph  ->  ( x  e.  RR+  |->  ( F `  x ) )  ~~> r  L )
2621ffvelrni 5584 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( F `
 x )  e.  RR )
2726adantl 454 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( F `  x )  e.  RR )
2818, 25, 27rlimrecl 12005 . . . . . . 7  |-  ( ph  ->  L  e.  RR )
2916, 28resubcld 9165 . . . . . 6  |-  ( ph  ->  ( ( gamma  x.  ( log `  A ) )  -  L )  e.  RR )
3013, 29readdcld 8816 . . . . 5  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( gamma  x.  ( log `  A ) )  -  L ) )  e.  RR )
3110, 30resubcld 9165 . . . 4  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  ( A  /  m ) )  /  m )  -  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( gamma  x.  ( log `  A ) )  -  L ) ) )  e.  RR )
3231recnd 8815 . . 3  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  ( A  /  m ) )  /  m )  -  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( gamma  x.  ( log `  A ) )  -  L ) ) )  e.  CC )
3332abscld 11869 . 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 10334 . . . . . . . 8  |-  ( ( ( log `  A
)  e.  RR  /\  m  e.  RR+ )  -> 
( ( log `  A
)  /  m )  e.  RR )
3511, 4, 34syl2an 465 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  A )  /  m )  e.  RR )
3635recnd 8815 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  A )  /  m )  e.  CC )
371, 36fsumcl 12157 . . . . 5  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  e.  CC )
3811recnd 8815 . . . . . 6  |-  ( ph  ->  ( log `  A
)  e.  CC )
39 readdcl 8774 . . . . . . . 8  |-  ( ( ( log `  A
)  e.  RR  /\  gamma  e.  RR )  ->  (
( log `  A
)  +  gamma )  e.  RR )
4011, 14, 39sylancl 646 . . . . . . 7  |-  ( ph  ->  ( ( log `  A
)  +  gamma )  e.  RR )
4140recnd 8815 . . . . . 6  |-  ( ph  ->  ( ( log `  A
)  +  gamma )  e.  CC )
4238, 41mulcld 8809 . . . . 5  |-  ( ph  ->  ( ( log `  A
)  x.  ( ( log `  A )  +  gamma ) )  e.  CC )
4337, 42subcld 9111 . . . 4  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  A
)  /  m )  -  ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
) )  e.  CC )
4443abscld 11869 . . 3  |-  ( ph  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  -  ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
) ) )  e.  RR )
458nnrpd 10342 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  m  e.  RR+ )
4645relogcld 19922 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( log `  m )  e.  RR )
4746, 8nndivred 9748 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  m )  /  m )  e.  RR )
4847recnd 8815 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  m )  /  m )  e.  CC )
491, 48fsumcl 12157 . . . . 5  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  m
)  /  m )  e.  CC )
5013recnd 8815 . . . . . 6  |-  ( ph  ->  ( ( ( log `  A ) ^ 2 )  /  2 )  e.  CC )
5128recnd 8815 . . . . . 6  |-  ( ph  ->  L  e.  CC )
5250, 51addcld 8808 . . . . 5  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  L
)  e.  CC )
5349, 52subcld 9111 . . . 4  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  L ) )  e.  CC )
5453abscld 11869 . . 3  |-  ( ph  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  L ) ) )  e.  RR )
5544, 54readdcld 8816 . 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 9769 . . 3  |-  2  e.  RR
5711, 2rerpdivcld 10370 . . 3  |-  ( ph  ->  ( ( log `  A
)  /  A )  e.  RR )
58 remulcl 8776 . . 3  |-  ( ( 2  e.  RR  /\  ( ( log `  A
)  /  A )  e.  RR )  -> 
( 2  x.  (
( log `  A
)  /  A ) )  e.  RR )
5956, 57, 58sylancr 647 . 2  |-  ( ph  ->  ( 2  x.  (
( log `  A
)  /  A ) )  e.  RR )
60 relogdiv 19894 . . . . . . . . . . 11  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  ( log `  ( A  /  m ) )  =  ( ( log `  A
)  -  ( log `  m ) ) )
612, 4, 60syl2an 465 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( log `  ( A  /  m
) )  =  ( ( log `  A
)  -  ( log `  m ) ) )
6261oveq1d 5793 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  ( A  /  m ) )  /  m )  =  ( ( ( log `  A
)  -  ( log `  m ) )  /  m ) )
6338adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( log `  A )  e.  CC )
6446recnd 8815 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( log `  m )  e.  CC )
6545rpcnne0d 10352 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( m  e.  CC  /\  m  =/=  0 ) )
66 divsubdir 9410 . . . . . . . . . 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 1187 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (
( log `  A
)  -  ( log `  m ) )  /  m )  =  ( ( ( log `  A
)  /  m )  -  ( ( log `  m )  /  m
) ) )
6862, 67eqtrd 2288 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  ( A  /  m ) )  /  m )  =  ( ( ( log `  A
)  /  m )  -  ( ( log `  m )  /  m
) ) )
6968sumeq2dv 12127 . . . . . . 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 12201 . . . . . . 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 2288 . . . . . 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 8776 . . . . . . . . . . . . 13  |-  ( ( ( log `  A
)  e.  RR  /\  gamma  e.  RR )  ->  (
( log `  A
)  x.  gamma )  e.  RR )
7311, 14, 72sylancl 646 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( log `  A
)  x.  gamma )  e.  RR )
7413, 73readdcld 8816 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( log `  A
)  x.  gamma )
)  e.  RR )
7574recnd 8815 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( log `  A
)  x.  gamma )
)  e.  CC )
7675, 50pncand 9112 . . . . . . . . 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 8803 . . . . . . . . . . . . 13  |-  gamma  e.  CC
7877a1i 12 . . . . . . . . . . . 12  |-  ( ph  -> 
gamma  e.  CC )
7938, 38, 78adddid 8813 . . . . . . . . . . 11  |-  ( ph  ->  ( ( log `  A
)  x.  ( ( log `  A )  +  gamma ) )  =  ( ( ( log `  A )  x.  ( log `  A ) )  +  ( ( log `  A )  x.  gamma ) ) )
8012recnd 8815 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( log `  A
) ^ 2 )  e.  CC )
81802halvesd 9910 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( ( log `  A
) ^ 2 )  /  2 ) )  =  ( ( log `  A ) ^ 2 ) )
8238sqvald 11194 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( log `  A
) ^ 2 )  =  ( ( log `  A )  x.  ( log `  A ) ) )
8381, 82eqtrd 2288 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( ( log `  A
) ^ 2 )  /  2 ) )  =  ( ( log `  A )  x.  ( log `  A ) ) )
8483oveq1d 5793 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  ( ( ( log `  A ) ^ 2 )  /  2 ) )  +  ( ( log `  A )  x.  gamma ) )  =  ( ( ( log `  A )  x.  ( log `  A ) )  +  ( ( log `  A )  x.  gamma ) ) )
8573recnd 8815 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( log `  A
)  x.  gamma )  e.  CC )
8650, 50, 85add32d 8988 . . . . . . . . . . 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 2294 . . . . . . . . . 10  |-  ( ph  ->  ( ( log `  A
)  x.  ( ( log `  A )  +  gamma ) )  =  ( ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  ( ( log `  A
)  x.  gamma )
)  +  ( ( ( log `  A
) ^ 2 )  /  2 ) ) )
8887oveq1d 5793 . . . . . . . . 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 8777 . . . . . . . . . . 11  |-  ( (
gamma  e.  CC  /\  ( log `  A )  e.  CC )  ->  ( gamma  x.  ( log `  A
) )  =  ( ( log `  A
)  x.  gamma )
)
9077, 38, 89sylancr 647 . . . . . . . . . 10  |-  ( ph  ->  ( gamma  x.  ( log `  A ) )  =  ( ( log `  A )  x.  gamma ) )
9190oveq2d 5794 . . . . . . . . 9  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  (
gamma  x.  ( log `  A
) ) )  =  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( log `  A
)  x.  gamma )
) )
9276, 88, 913eqtr4rd 2299 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  (
gamma  x.  ( log `  A
) ) )  =  ( ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
)  -  ( ( ( log `  A
) ^ 2 )  /  2 ) ) )
9392oveq1d 5793 . . . . . . 7  |-  ( ph  ->  ( ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  ( gamma  x.  ( log `  A ) ) )  -  L )  =  ( ( ( ( log `  A
)  x.  ( ( log `  A )  +  gamma ) )  -  ( ( ( log `  A ) ^ 2 )  /  2 ) )  -  L ) )
9490, 85eqeltrd 2330 . . . . . . . 8  |-  ( ph  ->  ( gamma  x.  ( log `  A ) )  e.  CC )
9550, 94, 51addsubassd 9131 . . . . . . 7  |-  ( ph  ->  ( ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  ( gamma  x.  ( log `  A ) ) )  -  L )  =  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  ( ( gamma  x.  ( log `  A ) )  -  L ) ) )
9642, 50, 51subsub4d 9142 . . . . . . 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 2296 . . . . . 6  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( gamma  x.  ( log `  A ) )  -  L ) )  =  ( ( ( log `  A )  x.  ( ( log `  A )  +  gamma ) )  -  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  L ) ) )
9871, 97oveq12d 5796 . . . . 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 9160 . . . . 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 2288 . . . 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 5448 . . 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 11891 . . 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 4003 . 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 20252 . . . . . . 7  |-  ( A  e.  RR+  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( 1  /  m
)  -  ( ( log `  A )  +  gamma ) ) )  <_  ( 1  /  A ) )
1052, 104syl 17 . . . . . 6  |-  ( ph  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( 1  /  m )  -  ( ( log `  A )  +  gamma ) ) )  <_  (
1  /  A ) )
1068nnrecred 9745 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1  /  m )  e.  RR )
1071, 106fsumrecl 12158 . . . . . . . . . 10  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( 1  /  m )  e.  RR )
108107, 40resubcld 9165 . . . . . . . . 9  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( 1  /  m
)  -  ( ( log `  A )  +  gamma ) )  e.  RR )
109108recnd 8815 . . . . . . . 8  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( 1  /  m
)  -  ( ( log `  A )  +  gamma ) )  e.  CC )
110109abscld 11869 . . . . . . 7  |-  ( ph  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( 1  /  m )  -  ( ( log `  A )  +  gamma ) ) )  e.  RR )
1112rprecred 10354 . . . . . . 7  |-  ( ph  ->  ( 1  /  A
)  e.  RR )
112 0re 8792 . . . . . . . . 9  |-  0  e.  RR
113112a1i 12 . . . . . . . 8  |-  ( ph  ->  0  e.  RR )
114 1re 8791 . . . . . . . . 9  |-  1  e.  RR
115114a1i 12 . . . . . . . 8  |-  ( ph  ->  1  e.  RR )
116 0lt1 9250 . . . . . . . . 9  |-  0  <  1
117116a1i 12 . . . . . . . 8  |-  ( ph  ->  0  <  1 )
118 loge 19888 . . . . . . . . 9  |-  ( log `  _e )  =  1
119 mulog2sumlem1.3 . . . . . . . . . 10  |-  ( ph  ->  _e  <_  A )
120 epr 12434 . . . . . . . . . . 11  |-  _e  e.  RR+
121 logleb 19905 . . . . . . . . . . 11  |-  ( ( _e  e.  RR+  /\  A  e.  RR+ )  ->  (
_e  <_  A  <->  ( log `  _e )  <_  ( log `  A ) ) )
122120, 2, 121sylancr 647 . . . . . . . . . 10  |-  ( ph  ->  ( _e  <_  A  <->  ( log `  _e )  <_  ( log `  A
) ) )
123119, 122mpbid 203 . . . . . . . . 9  |-  ( ph  ->  ( log `  _e )  <_  ( log `  A
) )
124118, 123syl5eqbrr 4017 . . . . . . . 8  |-  ( ph  ->  1  <_  ( log `  A ) )
125113, 115, 11, 117, 124ltletrd 8930 . . . . . . 7  |-  ( ph  ->  0  <  ( log `  A ) )
126 lemul2 9563 . . . . . . 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 1191 . . . . . 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 203 . . . . 5  |-  ( ph  ->  ( ( log `  A
)  x.  ( abs `  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( 1  /  m
)  -  ( ( log `  A )  +  gamma ) ) ) )  <_  ( ( log `  A )  x.  ( 1  /  A
) ) )
12945rpcnd 10345 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  m  e.  CC )
13045rpne0d 10348 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  m  =/=  0 )
13163, 129, 130divrecd 9493 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  A )  /  m )  =  ( ( log `  A
)  x.  ( 1  /  m ) ) )
132131sumeq2dv 12127 . . . . . . . . . 10  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  =  sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  A
)  x.  ( 1  /  m ) ) )
133106recnd 8815 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1  /  m )  e.  CC )
1341, 38, 133fsummulc2 12197 . . . . . . . . . 10  |-  ( ph  ->  ( ( log `  A
)  x.  sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( 1  /  m
) )  =  sum_ m  e.  ( 1 ... ( |_ `  A
) ) ( ( log `  A )  x.  ( 1  /  m ) ) )
135132, 134eqtr4d 2291 . . . . . . . . 9  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  =  ( ( log `  A )  x.  sum_ m  e.  ( 1 ... ( |_ `  A
) ) ( 1  /  m ) ) )
136135oveq1d 5793 . . . . . . . 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 12157 . . . . . . . . 9  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( 1  /  m )  e.  CC )
13838, 137, 41subdid 9189 . . . . . . . 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 2291 . . . . . . 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 5448 . . . . . 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 9111 . . . . . . 7  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( 1  /  m
)  -  ( ( log `  A )  +  gamma ) )  e.  CC )
14238, 141absmuld 11887 . . . . . 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 8921 . . . . . . . 8  |-  ( ph  ->  0  <_  ( log `  A ) )
14411, 143absidd 11856 . . . . . . 7  |-  ( ph  ->  ( abs `  ( log `  A ) )  =  ( log `  A
) )
145144oveq1d 5793 . . . . . 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 2292 . . . . 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 10345 . . . . . 6  |-  ( ph  ->  A  e.  CC )
1482rpne0d 10348 . . . . . 6  |-  ( ph  ->  A  =/=  0 )
14938, 147, 148divrecd 9493 . . . . 5  |-  ( ph  ->  ( ( log `  A
)  /  A )  =  ( ( log `  A )  x.  (
1  /  A ) ) )
150128, 146, 1493brtr4d 4013 . . . 4  |-  ( ph  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  -  ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
) ) )  <_ 
( ( log `  A
)  /  A ) )
151 fveq2 5444 . . . . . . . . . . . . . 14  |-  ( i  =  m  ->  ( log `  i )  =  ( log `  m
) )
152 id 21 . . . . . . . . . . . . . 14  |-  ( i  =  m  ->  i  =  m )
153151, 152oveq12d 5796 . . . . . . . . . . . . 13  |-  ( i  =  m  ->  (
( log `  i
)  /  i )  =  ( ( log `  m )  /  m
) )
154153cbvsumv 12120 . . . . . . . . . . . 12  |-  sum_ i  e.  ( 1 ... ( |_ `  y ) ) ( ( log `  i
)  /  i )  =  sum_ m  e.  ( 1 ... ( |_
`  y ) ) ( ( log `  m
)  /  m )
155 fveq2 5444 . . . . . . . . . . . . . 14  |-  ( y  =  A  ->  ( |_ `  y )  =  ( |_ `  A
) )
156155oveq2d 5794 . . . . . . . . . . . . 13  |-  ( y  =  A  ->  (
1 ... ( |_ `  y ) )  =  ( 1 ... ( |_ `  A ) ) )
157156sumeq1d 12125 . . . . . . . . . . . 12  |-  ( y  =  A  ->  sum_ m  e.  ( 1 ... ( |_ `  y ) ) ( ( log `  m
)  /  m )  =  sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  m
)  /  m ) )
158154, 157syl5eq 2300 . . . . . . . . . . 11  |-  ( y  =  A  ->  sum_ i  e.  ( 1 ... ( |_ `  y ) ) ( ( log `  i
)  /  i )  =  sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  m
)  /  m ) )
159 fveq2 5444 . . . . . . . . . . . . 13  |-  ( y  =  A  ->  ( log `  y )  =  ( log `  A
) )
160159oveq1d 5793 . . . . . . . . . . . 12  |-  ( y  =  A  ->  (
( log `  y
) ^ 2 )  =  ( ( log `  A ) ^ 2 ) )
161160oveq1d 5793 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
( ( log `  y
) ^ 2 )  /  2 )  =  ( ( ( log `  A ) ^ 2 )  /  2 ) )
162158, 161oveq12d 5796 . . . . . . . . . 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 5803 . . . . . . . . . 10  |-  ( sum_ m  e.  ( 1 ... ( |_ `  A
) ) ( ( log `  m )  /  m )  -  ( ( ( log `  A ) ^ 2 )  /  2 ) )  e.  _V
164162, 19, 163fvmpt 5522 . . . . . . . . 9  |-  ( A  e.  RR+  ->  ( F `
 A )  =  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( log `  A ) ^ 2 )  / 
2 ) ) )
1652, 164syl 17 . . . . . . . 8  |-  ( ph  ->  ( F `  A
)  =  ( sum_ m  e.  ( 1 ... ( |_ `  A
) ) ( ( log `  m )  /  m )  -  ( ( ( log `  A ) ^ 2 )  /  2 ) ) )
166165oveq1d 5793 . . . . . . 7  |-  ( ph  ->  ( ( F `  A )  -  L
)  =  ( (
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( log `  A ) ^ 2 )  / 
2 ) )  -  L ) )
16749, 50, 51subsub4d 9142 . . . . . . 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 2288 . . . . . 6  |-  ( ph  ->  ( ( F `  A )  -  L
)  =  ( sum_ m  e.  ( 1 ... ( |_ `  A
) ) ( ( log `  m )  /  m )  -  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  L
) ) )
169168fveq2d 5448 . . . . 5  |-  ( ph  ->  ( abs `  (
( F `  A
)  -  L ) )  =  ( abs `  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  L ) ) ) )
17020simp3i 971 . . . . . 6  |-  ( ( F  ~~> r  L  /\  A  e.  RR+  /\  _e  <_  A )  ->  ( abs `  ( ( F `
 A )  -  L ) )  <_ 
( ( log `  A
)  /  A ) )
17124, 2, 119, 170syl3anc 1187 . . . . 5  |-  ( ph  ->  ( abs `  (
( F `  A
)  -  L ) )  <_  ( ( log `  A )  /  A ) )
172169, 171eqbrtrrd 4005 . . . 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 9344 . . 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 8815 . . . 4  |-  ( ph  ->  ( ( log `  A
)  /  A )  e.  CC )
1751742timesd 9907 . . 3  |-  ( ph  ->  ( 2  x.  (
( log `  A
)  /  A ) )  =  ( ( ( log `  A
)  /  A )  +  ( ( log `  A )  /  A
) ) )
176173, 175breqtrrd 4009 . 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 8927 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 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   class class class wbr 3983    e. cmpt 4037   dom cdm 4647   -->wf 4655   ` cfv 4659  (class class class)co 5778   supcsup 7147   CCcc 8689   RRcr 8690   0cc0 8691   1c1 8692    + caddc 8694    x. cmul 8696    +oocpnf 8818   RR*cxr 8820    < clt 8821    <_ cle 8822    - cmin 8991    / cdiv 9377   NNcn 9700   2c2 9749   RR+crp 10307   ...cfz 10734   |_cfl 10876   ^cexp 11056   abscabs 11670    ~~> r crli 11910   sum_csu 12109   _eceu 12292   logclog 19860   gammacem 20234
This theorem is referenced by:  mulog2sumlem2  20632
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-inf2 7296  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768  ax-pre-sup 8769  ax-addf 8770  ax-mulf 8771
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-of 5998  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-2o 6434  df-oadd 6437  df-er 6614  df-map 6728  df-pm 6729  df-ixp 6772  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-fi 7119  df-sup 7148  df-oi 7179  df-card 7526  df-cda 7748  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378  df-n 9701  df-2 9758  df-3 9759  df-4 9760  df-5 9761  df-6 9762  df-7 9763  df-8 9764  df-9 9765  df-10 9766  df-n0 9919  df-z 9978  df-dec 10078  df-uz 10184  df-q 10270  df-rp 10308  df-xneg 10405  df-xadd 10406  df-xmul 10407  df-ioo 10612  df-ioc 10613  df-ico 10614  df-icc 10615  df-fz 10735  df-fzo 10823  df-fl 10877  df-mod 10926  df-seq 10999  df-exp 11057  df-fac 11241  df-bc 11268  df-hash 11290  df-shft 11513  df-cj 11535  df-re 11536  df-im 11537  df-sqr 11671  df-abs 11672  df-limsup 11896  df-clim 11913  df-rlim 11914  df-sum 12110  df-ef 12297  df-e 12298  df-sin 12299  df-cos 12300  df-pi 12302  df-struct 13098  df-ndx 13099  df-slot 13100  df-base 13101  df-sets 13102  df-ress 13103  df-plusg 13169  df-mulr 13170  df-starv 13171  df-sca 13172  df-vsca 13173  df-tset 13175  df-ple 13176  df-ds 13178  df-hom 13180  df-cco 13181  df-rest 13275  df-topn 13276  df-topgen 13292  df-pt 13293  df-prds 13296  df-xrs 13351  df-0g 13352  df-gsum 13353  df-qtop 13358  df-imas 13359  df-xps 13361  df-mre 13436  df-mrc 13437  df-acs 13439  df-mnd 14315  df-submnd 14364  df-mulg 14440  df-cntz 14741  df-cmn 15039  df-xmet 16321  df-met 16322  df-bl 16323  df-mopn 16324  df-cnfld 16326  df-top 16584  df-bases 16586  df-topon 16587  df-topsp 16588  df-cld 16704  df-ntr 16705  df-cls 16706  df-nei 16783  df-lp 16816  df-perf 16817  df-cn 16905  df-cnp 16906  df-haus 16991  df-cmp 17062  df-tx 17205  df-hmeo 17394  df-fbas 17468  df-fg 17469  df-fil 17489  df-fm 17581  df-flim 17582  df-flf 17583  df-xms 17833  df-ms 17834  df-tms 17835  df-cncf 18330  df-limc 19164  df-dv 19165  df-log 19862  df-cxp 19863  df-em 20235
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