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Theorem mulog2sumlem1 20699
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 11051 . . . . . 6  |-  ( ph  ->  ( 1 ... ( |_ `  A ) )  e.  Fin )
2 mulog2sumlem1.2 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR+ )
3 elfznn 10835 . . . . . . . . . 10  |-  ( m  e.  ( 1 ... ( |_ `  A
) )  ->  m  e.  NN )
43nnrpd 10405 . . . . . . . . 9  |-  ( m  e.  ( 1 ... ( |_ `  A
) )  ->  m  e.  RR+ )
5 rpdivcl 10392 . . . . . . . . 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 19990 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( log `  ( A  /  m
) )  e.  RR )
83adantl 452 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  m  e.  NN )
97, 8nndivred 9810 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  ( A  /  m ) )  /  m )  e.  RR )
101, 9fsumrecl 12223 . . . . 5  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  ( A  /  m ) )  /  m )  e.  RR )
112relogcld 19990 . . . . . . . 8  |-  ( ph  ->  ( log `  A
)  e.  RR )
1211resqcld 11287 . . . . . . 7  |-  ( ph  ->  ( ( log `  A
) ^ 2 )  e.  RR )
1312rehalfcld 9974 . . . . . 6  |-  ( ph  ->  ( ( ( log `  A ) ^ 2 )  /  2 )  e.  RR )
14 emre 20315 . . . . . . . 8  |-  gamma  e.  RR
15 remulcl 8838 . . . . . . . 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 10986 . . . . . . . . 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 20698 . . . . . . . . . . . 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 5591 . . . . . . . . 9  |-  ( ph  ->  F  =  ( x  e.  RR+  |->  ( F `
 x ) ) )
24 mulog2sumlem.1 . . . . . . . . 9  |-  ( ph  ->  F  ~~> r  L )
2523, 24eqbrtrrd 4061 . . . . . . . 8  |-  ( ph  ->  ( x  e.  RR+  |->  ( F `  x ) )  ~~> r  L )
2621ffvelrni 5680 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( F `
 x )  e.  RR )
2726adantl 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( F `  x )  e.  RR )
2818, 25, 27rlimrecl 12070 . . . . . . 7  |-  ( ph  ->  L  e.  RR )
2916, 28resubcld 9227 . . . . . 6  |-  ( ph  ->  ( ( gamma  x.  ( log `  A ) )  -  L )  e.  RR )
3013, 29readdcld 8878 . . . . 5  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( gamma  x.  ( log `  A ) )  -  L ) )  e.  RR )
3110, 30resubcld 9227 . . . 4  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  ( A  /  m ) )  /  m )  -  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( gamma  x.  ( log `  A ) )  -  L ) ) )  e.  RR )
3231recnd 8877 . . 3  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  ( A  /  m ) )  /  m )  -  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( gamma  x.  ( log `  A ) )  -  L ) ) )  e.  CC )
3332abscld 11934 . 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 10397 . . . . . . . 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 8877 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  A )  /  m )  e.  CC )
371, 36fsumcl 12222 . . . . 5  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  e.  CC )
3811recnd 8877 . . . . . 6  |-  ( ph  ->  ( log `  A
)  e.  CC )
39 readdcl 8836 . . . . . . . 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 8877 . . . . . 6  |-  ( ph  ->  ( ( log `  A
)  +  gamma )  e.  CC )
4238, 41mulcld 8871 . . . . 5  |-  ( ph  ->  ( ( log `  A
)  x.  ( ( log `  A )  +  gamma ) )  e.  CC )
4337, 42subcld 9173 . . . 4  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  A
)  /  m )  -  ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
) )  e.  CC )
4443abscld 11934 . . 3  |-  ( ph  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  -  ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
) ) )  e.  RR )
458nnrpd 10405 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  m  e.  RR+ )
4645relogcld 19990 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( log `  m )  e.  RR )
4746, 8nndivred 9810 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  m )  /  m )  e.  RR )
4847recnd 8877 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  m )  /  m )  e.  CC )
491, 48fsumcl 12222 . . . . 5  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  m
)  /  m )  e.  CC )
5013recnd 8877 . . . . . 6  |-  ( ph  ->  ( ( ( log `  A ) ^ 2 )  /  2 )  e.  CC )
5128recnd 8877 . . . . . 6  |-  ( ph  ->  L  e.  CC )
5250, 51addcld 8870 . . . . 5  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  L
)  e.  CC )
5349, 52subcld 9173 . . . 4  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  L ) )  e.  CC )
5453abscld 11934 . . 3  |-  ( ph  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  L ) ) )  e.  RR )
5544, 54readdcld 8878 . 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 9831 . . 3  |-  2  e.  RR
5711, 2rerpdivcld 10433 . . 3  |-  ( ph  ->  ( ( log `  A
)  /  A )  e.  RR )
58 remulcl 8838 . . 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 19962 . . . . . . . . . . 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 5889 . . . . . . . . 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 8877 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( log `  m )  e.  CC )
6545rpcnne0d 10415 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( m  e.  CC  /\  m  =/=  0 ) )
66 divsubdir 9472 . . . . . . . . . 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 2328 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  ( A  /  m ) )  /  m )  =  ( ( ( log `  A
)  /  m )  -  ( ( log `  m )  /  m
) ) )
6968sumeq2dv 12192 . . . . . . 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 12266 . . . . . . 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 2328 . . . . . 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 8838 . . . . . . . . . . . . 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 8878 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( log `  A
)  x.  gamma )
)  e.  RR )
7574recnd 8877 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( log `  A
)  x.  gamma )
)  e.  CC )
7675, 50pncand 9174 . . . . . . . . 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 8865 . . . . . . . . . . . . 13  |-  gamma  e.  CC
7877a1i 10 . . . . . . . . . . . 12  |-  ( ph  -> 
gamma  e.  CC )
7938, 38, 78adddid 8875 . . . . . . . . . . 11  |-  ( ph  ->  ( ( log `  A
)  x.  ( ( log `  A )  +  gamma ) )  =  ( ( ( log `  A )  x.  ( log `  A ) )  +  ( ( log `  A )  x.  gamma ) ) )
8012recnd 8877 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( log `  A
) ^ 2 )  e.  CC )
81802halvesd 9973 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( ( log `  A
) ^ 2 )  /  2 ) )  =  ( ( log `  A ) ^ 2 ) )
8238sqvald 11258 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( log `  A
) ^ 2 )  =  ( ( log `  A )  x.  ( log `  A ) ) )
8381, 82eqtrd 2328 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( ( log `  A
) ^ 2 )  /  2 ) )  =  ( ( log `  A )  x.  ( log `  A ) ) )
8483oveq1d 5889 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  ( ( ( log `  A ) ^ 2 )  /  2 ) )  +  ( ( log `  A )  x.  gamma ) )  =  ( ( ( log `  A )  x.  ( log `  A ) )  +  ( ( log `  A )  x.  gamma ) ) )
8573recnd 8877 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( log `  A
)  x.  gamma )  e.  CC )
8650, 50, 85add32d 9050 . . . . . . . . . . 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 2334 . . . . . . . . . 10  |-  ( ph  ->  ( ( log `  A
)  x.  ( ( log `  A )  +  gamma ) )  =  ( ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  ( ( log `  A
)  x.  gamma )
)  +  ( ( ( log `  A
) ^ 2 )  /  2 ) ) )
8887oveq1d 5889 . . . . . . . . 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 8839 . . . . . . . . . . 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 5890 . . . . . . . . 9  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  (
gamma  x.  ( log `  A
) ) )  =  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( log `  A
)  x.  gamma )
) )
9276, 88, 913eqtr4rd 2339 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  (
gamma  x.  ( log `  A
) ) )  =  ( ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
)  -  ( ( ( log `  A
) ^ 2 )  /  2 ) ) )
9392oveq1d 5889 . . . . . . 7  |-  ( ph  ->  ( ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  ( gamma  x.  ( log `  A ) ) )  -  L )  =  ( ( ( ( log `  A
)  x.  ( ( log `  A )  +  gamma ) )  -  ( ( ( log `  A ) ^ 2 )  /  2 ) )  -  L ) )
9490, 85eqeltrd 2370 . . . . . . . 8  |-  ( ph  ->  ( gamma  x.  ( log `  A ) )  e.  CC )
9550, 94, 51addsubassd 9193 . . . . . . 7  |-  ( ph  ->  ( ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  ( gamma  x.  ( log `  A ) ) )  -  L )  =  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  ( ( gamma  x.  ( log `  A ) )  -  L ) ) )
9642, 50, 51subsub4d 9204 . . . . . . 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 2336 . . . . . 6  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( gamma  x.  ( log `  A ) )  -  L ) )  =  ( ( ( log `  A )  x.  ( ( log `  A )  +  gamma ) )  -  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  L ) ) )
9871, 97oveq12d 5892 . . . . 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 9222 . . . . 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 2328 . . . 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 5545 . . 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 11956 . . 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 4059 . 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 20320 . . . . . . 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 9807 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1  /  m )  e.  RR )
1071, 106fsumrecl 12223 . . . . . . . . . 10  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( 1  /  m )  e.  RR )
108107, 40resubcld 9227 . . . . . . . . 9  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( 1  /  m
)  -  ( ( log `  A )  +  gamma ) )  e.  RR )
109108recnd 8877 . . . . . . . 8  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( 1  /  m
)  -  ( ( log `  A )  +  gamma ) )  e.  CC )
110109abscld 11934 . . . . . . 7  |-  ( ph  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( 1  /  m )  -  ( ( log `  A )  +  gamma ) ) )  e.  RR )
1112rprecred 10417 . . . . . . 7  |-  ( ph  ->  ( 1  /  A
)  e.  RR )
112 0re 8854 . . . . . . . . 9  |-  0  e.  RR
113112a1i 10 . . . . . . . 8  |-  ( ph  ->  0  e.  RR )
114 1re 8853 . . . . . . . . 9  |-  1  e.  RR
115114a1i 10 . . . . . . . 8  |-  ( ph  ->  1  e.  RR )
116 0lt1 9312 . . . . . . . . 9  |-  0  <  1
117116a1i 10 . . . . . . . 8  |-  ( ph  ->  0  <  1 )
118 loge 19956 . . . . . . . . 9  |-  ( log `  _e )  =  1
119 mulog2sumlem1.3 . . . . . . . . . 10  |-  ( ph  ->  _e  <_  A )
120 epr 12502 . . . . . . . . . . 11  |-  _e  e.  RR+
121 logleb 19973 . . . . . . . . . . 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 4073 . . . . . . . 8  |-  ( ph  ->  1  <_  ( log `  A ) )
125113, 115, 11, 117, 124ltletrd 8992 . . . . . . 7  |-  ( ph  ->  0  <  ( log `  A ) )
126 lemul2 9625 . . . . . . 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 10408 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  m  e.  CC )
13045rpne0d 10411 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  m  =/=  0 )
13163, 129, 130divrecd 9555 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  A )  /  m )  =  ( ( log `  A
)  x.  ( 1  /  m ) ) )
132131sumeq2dv 12192 . . . . . . . . . 10  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  =  sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  A
)  x.  ( 1  /  m ) ) )
133106recnd 8877 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1  /  m )  e.  CC )
1341, 38, 133fsummulc2 12262 . . . . . . . . . 10  |-  ( ph  ->  ( ( log `  A
)  x.  sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( 1  /  m
) )  =  sum_ m  e.  ( 1 ... ( |_ `  A
) ) ( ( log `  A )  x.  ( 1  /  m ) ) )
135132, 134eqtr4d 2331 . . . . . . . . 9  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  =  ( ( log `  A )  x.  sum_ m  e.  ( 1 ... ( |_ `  A
) ) ( 1  /  m ) ) )
136135oveq1d 5889 . . . . . . . 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 12222 . . . . . . . . 9  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( 1  /  m )  e.  CC )
13838, 137, 41subdid 9251 . . . . . . . 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 2331 . . . . . . 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 5545 . . . . . 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 9173 . . . . . . 7  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( 1  /  m
)  -  ( ( log `  A )  +  gamma ) )  e.  CC )
14238, 141absmuld 11952 . . . . . 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 8983 . . . . . . . 8  |-  ( ph  ->  0  <_  ( log `  A ) )
14411, 143absidd 11921 . . . . . . 7  |-  ( ph  ->  ( abs `  ( log `  A ) )  =  ( log `  A
) )
145144oveq1d 5889 . . . . . 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 2332 . . . . 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 10408 . . . . . 6  |-  ( ph  ->  A  e.  CC )
1482rpne0d 10411 . . . . . 6  |-  ( ph  ->  A  =/=  0 )
14938, 147, 148divrecd 9555 . . . . 5  |-  ( ph  ->  ( ( log `  A
)  /  A )  =  ( ( log `  A )  x.  (
1  /  A ) ) )
150128, 146, 1493brtr4d 4069 . . . 4  |-  ( ph  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  -  ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
) ) )  <_ 
( ( log `  A
)  /  A ) )
151 fveq2 5541 . . . . . . . . . . . . . 14  |-  ( i  =  m  ->  ( log `  i )  =  ( log `  m
) )
152 id 19 . . . . . . . . . . . . . 14  |-  ( i  =  m  ->  i  =  m )
153151, 152oveq12d 5892 . . . . . . . . . . . . 13  |-  ( i  =  m  ->  (
( log `  i
)  /  i )  =  ( ( log `  m )  /  m
) )
154153cbvsumv 12185 . . . . . . . . . . . 12  |-  sum_ i  e.  ( 1 ... ( |_ `  y ) ) ( ( log `  i
)  /  i )  =  sum_ m  e.  ( 1 ... ( |_
`  y ) ) ( ( log `  m
)  /  m )
155 fveq2 5541 . . . . . . . . . . . . . 14  |-  ( y  =  A  ->  ( |_ `  y )  =  ( |_ `  A
) )
156155oveq2d 5890 . . . . . . . . . . . . 13  |-  ( y  =  A  ->  (
1 ... ( |_ `  y ) )  =  ( 1 ... ( |_ `  A ) ) )
157156sumeq1d 12190 . . . . . . . . . . . 12  |-  ( y  =  A  ->  sum_ m  e.  ( 1 ... ( |_ `  y ) ) ( ( log `  m
)  /  m )  =  sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  m
)  /  m ) )
158154, 157syl5eq 2340 . . . . . . . . . . 11  |-  ( y  =  A  ->  sum_ i  e.  ( 1 ... ( |_ `  y ) ) ( ( log `  i
)  /  i )  =  sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  m
)  /  m ) )
159 fveq2 5541 . . . . . . . . . . . . 13  |-  ( y  =  A  ->  ( log `  y )  =  ( log `  A
) )
160159oveq1d 5889 . . . . . . . . . . . 12  |-  ( y  =  A  ->  (
( log `  y
) ^ 2 )  =  ( ( log `  A ) ^ 2 ) )
161160oveq1d 5889 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
( ( log `  y
) ^ 2 )  /  2 )  =  ( ( ( log `  A ) ^ 2 )  /  2 ) )
162158, 161oveq12d 5892 . . . . . . . . . 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 5899 . . . . . . . . . 10  |-  ( sum_ m  e.  ( 1 ... ( |_ `  A
) ) ( ( log `  m )  /  m )  -  ( ( ( log `  A ) ^ 2 )  /  2 ) )  e.  _V
164162, 19, 163fvmpt 5618 . . . . . . . . 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 5889 . . . . . . 7  |-  ( ph  ->  ( ( F `  A )  -  L
)  =  ( (
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( log `  A ) ^ 2 )  / 
2 ) )  -  L ) )
16749, 50, 51subsub4d 9204 . . . . . . 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 2328 . . . . . 6  |-  ( ph  ->  ( ( F `  A )  -  L
)  =  ( sum_ m  e.  ( 1 ... ( |_ `  A
) ) ( ( log `  m )  /  m )  -  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  L
) ) )
169168fveq2d 5545 . . . . 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 4061 . . . 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 9406 . . 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 8877 . . . 4  |-  ( ph  ->  ( ( log `  A
)  /  A )  e.  CC )
1751742timesd 9970 . . 3  |-  ( ph  ->  ( 2  x.  (
( log `  A
)  /  A ) )  =  ( ( ( log `  A
)  /  A )  +  ( ( log `  A )  /  A
) ) )
176173, 175breqtrrd 4065 . 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 8989 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 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   -->wf 5267   ` cfv 5271  (class class class)co 5874   supcsup 7209   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    +oocpnf 8880   RR*cxr 8882    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   NNcn 9762   2c2 9811   RR+crp 10370   ...cfz 10798   |_cfl 10940   ^cexp 11120   abscabs 11735    ~~> r crli 11975   sum_csu 12174   _eceu 12360   logclog 19928   gammacem 20302
This theorem is referenced by:  mulog2sumlem2  20700
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-e 12366  df-sin 12367  df-cos 12368  df-pi 12370  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-cmp 17130  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930  df-cxp 19931  df-em 20303
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