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Theorem mulog2sumlem1 21220
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 11304 . . . . . 6  |-  ( ph  ->  ( 1 ... ( |_ `  A ) )  e.  Fin )
2 mulog2sumlem1.2 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR+ )
3 elfznn 11072 . . . . . . . . . 10  |-  ( m  e.  ( 1 ... ( |_ `  A
) )  ->  m  e.  NN )
43nnrpd 10639 . . . . . . . . 9  |-  ( m  e.  ( 1 ... ( |_ `  A
) )  ->  m  e.  RR+ )
5 rpdivcl 10626 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  ( A  /  m )  e.  RR+ )
62, 4, 5syl2an 464 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( A  /  m )  e.  RR+ )
76relogcld 20510 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( log `  ( A  /  m
) )  e.  RR )
83adantl 453 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  m  e.  NN )
97, 8nndivred 10040 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  ( A  /  m ) )  /  m )  e.  RR )
101, 9fsumrecl 12520 . . . . 5  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  ( A  /  m ) )  /  m )  e.  RR )
112relogcld 20510 . . . . . . . 8  |-  ( ph  ->  ( log `  A
)  e.  RR )
1211resqcld 11541 . . . . . . 7  |-  ( ph  ->  ( ( log `  A
) ^ 2 )  e.  RR )
1312rehalfcld 10206 . . . . . 6  |-  ( ph  ->  ( ( ( log `  A ) ^ 2 )  /  2 )  e.  RR )
14 emre 20836 . . . . . . . 8  |-  gamma  e.  RR
15 remulcl 9067 . . . . . . . 8  |-  ( (
gamma  e.  RR  /\  ( log `  A )  e.  RR )  ->  ( gamma  x.  ( log `  A
) )  e.  RR )
1614, 11, 15sylancr 645 . . . . . . 7  |-  ( ph  ->  ( gamma  x.  ( log `  A ) )  e.  RR )
17 rpsup 11239 . . . . . . . . 9  |-  sup ( RR+ ,  RR* ,  <  )  =  +oo
1817a1i 11 . . . . . . . 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 21219 . . . . . . . . . . . 12  |-  ( F : RR+ --> RR  /\  F  e.  dom  ~~> r  /\  (
( F  ~~> r  L  /\  A  e.  RR+  /\  _e  <_  A )  ->  ( abs `  ( ( F `
 A )  -  L ) )  <_ 
( ( log `  A
)  /  A ) ) )
2120simp1i 966 . . . . . . . . . . 11  |-  F : RR+
--> RR
2221a1i 11 . . . . . . . . . 10  |-  ( ph  ->  F : RR+ --> RR )
2322feqmptd 5771 . . . . . . . . 9  |-  ( ph  ->  F  =  ( x  e.  RR+  |->  ( F `
 x ) ) )
24 mulog2sumlem.1 . . . . . . . . 9  |-  ( ph  ->  F  ~~> r  L )
2523, 24eqbrtrrd 4226 . . . . . . . 8  |-  ( ph  ->  ( x  e.  RR+  |->  ( F `  x ) )  ~~> r  L )
2621ffvelrni 5861 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( F `
 x )  e.  RR )
2726adantl 453 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( F `  x )  e.  RR )
2818, 25, 27rlimrecl 12366 . . . . . . 7  |-  ( ph  ->  L  e.  RR )
2916, 28resubcld 9457 . . . . . 6  |-  ( ph  ->  ( ( gamma  x.  ( log `  A ) )  -  L )  e.  RR )
3013, 29readdcld 9107 . . . . 5  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( gamma  x.  ( log `  A ) )  -  L ) )  e.  RR )
3110, 30resubcld 9457 . . . 4  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  ( A  /  m ) )  /  m )  -  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( gamma  x.  ( log `  A ) )  -  L ) ) )  e.  RR )
3231recnd 9106 . . 3  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  ( A  /  m ) )  /  m )  -  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( gamma  x.  ( log `  A ) )  -  L ) ) )  e.  CC )
3332abscld 12230 . 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 10631 . . . . . . . 8  |-  ( ( ( log `  A
)  e.  RR  /\  m  e.  RR+ )  -> 
( ( log `  A
)  /  m )  e.  RR )
3511, 4, 34syl2an 464 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  A )  /  m )  e.  RR )
3635recnd 9106 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  A )  /  m )  e.  CC )
371, 36fsumcl 12519 . . . . 5  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  e.  CC )
3811recnd 9106 . . . . . 6  |-  ( ph  ->  ( log `  A
)  e.  CC )
39 readdcl 9065 . . . . . . . 8  |-  ( ( ( log `  A
)  e.  RR  /\  gamma  e.  RR )  ->  (
( log `  A
)  +  gamma )  e.  RR )
4011, 14, 39sylancl 644 . . . . . . 7  |-  ( ph  ->  ( ( log `  A
)  +  gamma )  e.  RR )
4140recnd 9106 . . . . . 6  |-  ( ph  ->  ( ( log `  A
)  +  gamma )  e.  CC )
4238, 41mulcld 9100 . . . . 5  |-  ( ph  ->  ( ( log `  A
)  x.  ( ( log `  A )  +  gamma ) )  e.  CC )
4337, 42subcld 9403 . . . 4  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  A
)  /  m )  -  ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
) )  e.  CC )
4443abscld 12230 . . 3  |-  ( ph  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  -  ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
) ) )  e.  RR )
458nnrpd 10639 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  m  e.  RR+ )
4645relogcld 20510 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( log `  m )  e.  RR )
4746, 8nndivred 10040 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  m )  /  m )  e.  RR )
4847recnd 9106 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  m )  /  m )  e.  CC )
491, 48fsumcl 12519 . . . . 5  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  m
)  /  m )  e.  CC )
5013recnd 9106 . . . . . 6  |-  ( ph  ->  ( ( ( log `  A ) ^ 2 )  /  2 )  e.  CC )
5128recnd 9106 . . . . . 6  |-  ( ph  ->  L  e.  CC )
5250, 51addcld 9099 . . . . 5  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  L
)  e.  CC )
5349, 52subcld 9403 . . . 4  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  L ) )  e.  CC )
5453abscld 12230 . . 3  |-  ( ph  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  L ) ) )  e.  RR )
5544, 54readdcld 9107 . 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 10061 . . 3  |-  2  e.  RR
5711, 2rerpdivcld 10667 . . 3  |-  ( ph  ->  ( ( log `  A
)  /  A )  e.  RR )
58 remulcl 9067 . . 3  |-  ( ( 2  e.  RR  /\  ( ( log `  A
)  /  A )  e.  RR )  -> 
( 2  x.  (
( log `  A
)  /  A ) )  e.  RR )
5956, 57, 58sylancr 645 . 2  |-  ( ph  ->  ( 2  x.  (
( log `  A
)  /  A ) )  e.  RR )
60 relogdiv 20479 . . . . . . . . . . 11  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  ( log `  ( A  /  m ) )  =  ( ( log `  A
)  -  ( log `  m ) ) )
612, 4, 60syl2an 464 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( log `  ( A  /  m
) )  =  ( ( log `  A
)  -  ( log `  m ) ) )
6261oveq1d 6088 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  ( A  /  m ) )  /  m )  =  ( ( ( log `  A
)  -  ( log `  m ) )  /  m ) )
6338adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( log `  A )  e.  CC )
6446recnd 9106 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( log `  m )  e.  CC )
6545rpcnne0d 10649 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( m  e.  CC  /\  m  =/=  0 ) )
66 divsubdir 9702 . . . . . . . . . 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 1184 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (
( log `  A
)  -  ( log `  m ) )  /  m )  =  ( ( ( log `  A
)  /  m )  -  ( ( log `  m )  /  m
) ) )
6862, 67eqtrd 2467 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  ( A  /  m ) )  /  m )  =  ( ( ( log `  A
)  /  m )  -  ( ( log `  m )  /  m
) ) )
6968sumeq2dv 12489 . . . . . . 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 12563 . . . . . . 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 2467 . . . . . 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 9067 . . . . . . . . . . . . 13  |-  ( ( ( log `  A
)  e.  RR  /\  gamma  e.  RR )  ->  (
( log `  A
)  x.  gamma )  e.  RR )
7311, 14, 72sylancl 644 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( log `  A
)  x.  gamma )  e.  RR )
7413, 73readdcld 9107 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( log `  A
)  x.  gamma )
)  e.  RR )
7574recnd 9106 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( log `  A
)  x.  gamma )
)  e.  CC )
7675, 50pncand 9404 . . . . . . . . 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 9094 . . . . . . . . . . . . 13  |-  gamma  e.  CC
7877a1i 11 . . . . . . . . . . . 12  |-  ( ph  -> 
gamma  e.  CC )
7938, 38, 78adddid 9104 . . . . . . . . . . 11  |-  ( ph  ->  ( ( log `  A
)  x.  ( ( log `  A )  +  gamma ) )  =  ( ( ( log `  A )  x.  ( log `  A ) )  +  ( ( log `  A )  x.  gamma ) ) )
8012recnd 9106 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( log `  A
) ^ 2 )  e.  CC )
81802halvesd 10205 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( ( log `  A
) ^ 2 )  /  2 ) )  =  ( ( log `  A ) ^ 2 ) )
8238sqvald 11512 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( log `  A
) ^ 2 )  =  ( ( log `  A )  x.  ( log `  A ) ) )
8381, 82eqtrd 2467 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( ( log `  A
) ^ 2 )  /  2 ) )  =  ( ( log `  A )  x.  ( log `  A ) ) )
8483oveq1d 6088 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  ( ( ( log `  A ) ^ 2 )  /  2 ) )  +  ( ( log `  A )  x.  gamma ) )  =  ( ( ( log `  A )  x.  ( log `  A ) )  +  ( ( log `  A )  x.  gamma ) ) )
8573recnd 9106 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( log `  A
)  x.  gamma )  e.  CC )
8650, 50, 85add32d 9280 . . . . . . . . . . 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 2473 . . . . . . . . . 10  |-  ( ph  ->  ( ( log `  A
)  x.  ( ( log `  A )  +  gamma ) )  =  ( ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  ( ( log `  A
)  x.  gamma )
)  +  ( ( ( log `  A
) ^ 2 )  /  2 ) ) )
8887oveq1d 6088 . . . . . . . . 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 9068 . . . . . . . . . . 11  |-  ( (
gamma  e.  CC  /\  ( log `  A )  e.  CC )  ->  ( gamma  x.  ( log `  A
) )  =  ( ( log `  A
)  x.  gamma )
)
9077, 38, 89sylancr 645 . . . . . . . . . 10  |-  ( ph  ->  ( gamma  x.  ( log `  A ) )  =  ( ( log `  A )  x.  gamma ) )
9190oveq2d 6089 . . . . . . . . 9  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  (
gamma  x.  ( log `  A
) ) )  =  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( log `  A
)  x.  gamma )
) )
9276, 88, 913eqtr4rd 2478 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  (
gamma  x.  ( log `  A
) ) )  =  ( ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
)  -  ( ( ( log `  A
) ^ 2 )  /  2 ) ) )
9392oveq1d 6088 . . . . . . 7  |-  ( ph  ->  ( ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  ( gamma  x.  ( log `  A ) ) )  -  L )  =  ( ( ( ( log `  A
)  x.  ( ( log `  A )  +  gamma ) )  -  ( ( ( log `  A ) ^ 2 )  /  2 ) )  -  L ) )
9490, 85eqeltrd 2509 . . . . . . . 8  |-  ( ph  ->  ( gamma  x.  ( log `  A ) )  e.  CC )
9550, 94, 51addsubassd 9423 . . . . . . 7  |-  ( ph  ->  ( ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  ( gamma  x.  ( log `  A ) ) )  -  L )  =  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  ( ( gamma  x.  ( log `  A ) )  -  L ) ) )
9642, 50, 51subsub4d 9434 . . . . . . 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 2475 . . . . . 6  |-  ( ph  ->  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  ( ( gamma  x.  ( log `  A ) )  -  L ) )  =  ( ( ( log `  A )  x.  ( ( log `  A )  +  gamma ) )  -  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  L ) ) )
9871, 97oveq12d 6091 . . . . 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 9452 . . . . 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 2467 . . . 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 5724 . . 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 12252 . . 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 4224 . 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 20841 . . . . . . 7  |-  ( A  e.  RR+  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( 1  /  m
)  -  ( ( log `  A )  +  gamma ) ) )  <_  ( 1  /  A ) )
1052, 104syl 16 . . . . . 6  |-  ( ph  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( 1  /  m )  -  ( ( log `  A )  +  gamma ) ) )  <_  (
1  /  A ) )
1068nnrecred 10037 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1  /  m )  e.  RR )
1071, 106fsumrecl 12520 . . . . . . . . . 10  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( 1  /  m )  e.  RR )
108107, 40resubcld 9457 . . . . . . . . 9  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( 1  /  m
)  -  ( ( log `  A )  +  gamma ) )  e.  RR )
109108recnd 9106 . . . . . . . 8  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( 1  /  m
)  -  ( ( log `  A )  +  gamma ) )  e.  CC )
110109abscld 12230 . . . . . . 7  |-  ( ph  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( 1  /  m )  -  ( ( log `  A )  +  gamma ) ) )  e.  RR )
1112rprecred 10651 . . . . . . 7  |-  ( ph  ->  ( 1  /  A
)  e.  RR )
112 0re 9083 . . . . . . . . 9  |-  0  e.  RR
113112a1i 11 . . . . . . . 8  |-  ( ph  ->  0  e.  RR )
114 1re 9082 . . . . . . . . 9  |-  1  e.  RR
115114a1i 11 . . . . . . . 8  |-  ( ph  ->  1  e.  RR )
116 0lt1 9542 . . . . . . . . 9  |-  0  <  1
117116a1i 11 . . . . . . . 8  |-  ( ph  ->  0  <  1 )
118 loge 20473 . . . . . . . . 9  |-  ( log `  _e )  =  1
119 mulog2sumlem1.3 . . . . . . . . . 10  |-  ( ph  ->  _e  <_  A )
120 epr 12799 . . . . . . . . . . 11  |-  _e  e.  RR+
121 logleb 20490 . . . . . . . . . . 11  |-  ( ( _e  e.  RR+  /\  A  e.  RR+ )  ->  (
_e  <_  A  <->  ( log `  _e )  <_  ( log `  A ) ) )
122120, 2, 121sylancr 645 . . . . . . . . . 10  |-  ( ph  ->  ( _e  <_  A  <->  ( log `  _e )  <_  ( log `  A
) ) )
123119, 122mpbid 202 . . . . . . . . 9  |-  ( ph  ->  ( log `  _e )  <_  ( log `  A
) )
124118, 123syl5eqbrr 4238 . . . . . . . 8  |-  ( ph  ->  1  <_  ( log `  A ) )
125113, 115, 11, 117, 124ltletrd 9222 . . . . . . 7  |-  ( ph  ->  0  <  ( log `  A ) )
126 lemul2 9855 . . . . . . 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 1188 . . . . . 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 202 . . . . 5  |-  ( ph  ->  ( ( log `  A
)  x.  ( abs `  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( 1  /  m
)  -  ( ( log `  A )  +  gamma ) ) ) )  <_  ( ( log `  A )  x.  ( 1  /  A
) ) )
12945rpcnd 10642 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  m  e.  CC )
13045rpne0d 10645 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  m  =/=  0 )
13163, 129, 130divrecd 9785 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( ( log `  A )  /  m )  =  ( ( log `  A
)  x.  ( 1  /  m ) ) )
132131sumeq2dv 12489 . . . . . . . . . 10  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  =  sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  A
)  x.  ( 1  /  m ) ) )
133106recnd 9106 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1  /  m )  e.  CC )
1341, 38, 133fsummulc2 12559 . . . . . . . . . 10  |-  ( ph  ->  ( ( log `  A
)  x.  sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( 1  /  m
) )  =  sum_ m  e.  ( 1 ... ( |_ `  A
) ) ( ( log `  A )  x.  ( 1  /  m ) ) )
135132, 134eqtr4d 2470 . . . . . . . . 9  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  =  ( ( log `  A )  x.  sum_ m  e.  ( 1 ... ( |_ `  A
) ) ( 1  /  m ) ) )
136135oveq1d 6088 . . . . . . . 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 12519 . . . . . . . . 9  |-  ( ph  -> 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( 1  /  m )  e.  CC )
13838, 137, 41subdid 9481 . . . . . . . 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 2470 . . . . . . 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 5724 . . . . . 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 9403 . . . . . . 7  |-  ( ph  ->  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( 1  /  m
)  -  ( ( log `  A )  +  gamma ) )  e.  CC )
14238, 141absmuld 12248 . . . . . 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 9213 . . . . . . . 8  |-  ( ph  ->  0  <_  ( log `  A ) )
14411, 143absidd 12217 . . . . . . 7  |-  ( ph  ->  ( abs `  ( log `  A ) )  =  ( log `  A
) )
145144oveq1d 6088 . . . . . 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 2471 . . . . 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 10642 . . . . . 6  |-  ( ph  ->  A  e.  CC )
1482rpne0d 10645 . . . . . 6  |-  ( ph  ->  A  =/=  0 )
14938, 147, 148divrecd 9785 . . . . 5  |-  ( ph  ->  ( ( log `  A
)  /  A )  =  ( ( log `  A )  x.  (
1  /  A ) ) )
150128, 146, 1493brtr4d 4234 . . . 4  |-  ( ph  ->  ( abs `  ( sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( ( log `  A
)  /  m )  -  ( ( log `  A )  x.  (
( log `  A
)  +  gamma )
) ) )  <_ 
( ( log `  A
)  /  A ) )
151 fveq2 5720 . . . . . . . . . . . . . 14  |-  ( i  =  m  ->  ( log `  i )  =  ( log `  m
) )
152 id 20 . . . . . . . . . . . . . 14  |-  ( i  =  m  ->  i  =  m )
153151, 152oveq12d 6091 . . . . . . . . . . . . 13  |-  ( i  =  m  ->  (
( log `  i
)  /  i )  =  ( ( log `  m )  /  m
) )
154153cbvsumv 12482 . . . . . . . . . . . 12  |-  sum_ i  e.  ( 1 ... ( |_ `  y ) ) ( ( log `  i
)  /  i )  =  sum_ m  e.  ( 1 ... ( |_
`  y ) ) ( ( log `  m
)  /  m )
155 fveq2 5720 . . . . . . . . . . . . . 14  |-  ( y  =  A  ->  ( |_ `  y )  =  ( |_ `  A
) )
156155oveq2d 6089 . . . . . . . . . . . . 13  |-  ( y  =  A  ->  (
1 ... ( |_ `  y ) )  =  ( 1 ... ( |_ `  A ) ) )
157156sumeq1d 12487 . . . . . . . . . . . 12  |-  ( y  =  A  ->  sum_ m  e.  ( 1 ... ( |_ `  y ) ) ( ( log `  m
)  /  m )  =  sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  m
)  /  m ) )
158154, 157syl5eq 2479 . . . . . . . . . . 11  |-  ( y  =  A  ->  sum_ i  e.  ( 1 ... ( |_ `  y ) ) ( ( log `  i
)  /  i )  =  sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  m
)  /  m ) )
159 fveq2 5720 . . . . . . . . . . . . 13  |-  ( y  =  A  ->  ( log `  y )  =  ( log `  A
) )
160159oveq1d 6088 . . . . . . . . . . . 12  |-  ( y  =  A  ->  (
( log `  y
) ^ 2 )  =  ( ( log `  A ) ^ 2 ) )
161160oveq1d 6088 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
( ( log `  y
) ^ 2 )  /  2 )  =  ( ( ( log `  A ) ^ 2 )  /  2 ) )
162158, 161oveq12d 6091 . . . . . . . . . 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 6098 . . . . . . . . . 10  |-  ( sum_ m  e.  ( 1 ... ( |_ `  A
) ) ( ( log `  m )  /  m )  -  ( ( ( log `  A ) ^ 2 )  /  2 ) )  e.  _V
164162, 19, 163fvmpt 5798 . . . . . . . . 9  |-  ( A  e.  RR+  ->  ( F `
 A )  =  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( log `  A ) ^ 2 )  / 
2 ) ) )
1652, 164syl 16 . . . . . . . 8  |-  ( ph  ->  ( F `  A
)  =  ( sum_ m  e.  ( 1 ... ( |_ `  A
) ) ( ( log `  m )  /  m )  -  ( ( ( log `  A ) ^ 2 )  /  2 ) ) )
166165oveq1d 6088 . . . . . . 7  |-  ( ph  ->  ( ( F `  A )  -  L
)  =  ( (
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( log `  A ) ^ 2 )  / 
2 ) )  -  L ) )
16749, 50, 51subsub4d 9434 . . . . . . 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 2467 . . . . . 6  |-  ( ph  ->  ( ( F `  A )  -  L
)  =  ( sum_ m  e.  ( 1 ... ( |_ `  A
) ) ( ( log `  m )  /  m )  -  ( ( ( ( log `  A ) ^ 2 )  / 
2 )  +  L
) ) )
169168fveq2d 5724 . . . . 5  |-  ( ph  ->  ( abs `  (
( F `  A
)  -  L ) )  =  ( abs `  ( sum_ m  e.  ( 1 ... ( |_
`  A ) ) ( ( log `  m
)  /  m )  -  ( ( ( ( log `  A
) ^ 2 )  /  2 )  +  L ) ) ) )
17020simp3i 968 . . . . . 6  |-  ( ( F  ~~> r  L  /\  A  e.  RR+  /\  _e  <_  A )  ->  ( abs `  ( ( F `
 A )  -  L ) )  <_ 
( ( log `  A
)  /  A ) )
17124, 2, 119, 170syl3anc 1184 . . . . 5  |-  ( ph  ->  ( abs `  (
( F `  A
)  -  L ) )  <_  ( ( log `  A )  /  A ) )
172169, 171eqbrtrrd 4226 . . . 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 9636 . . 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 9106 . . . 4  |-  ( ph  ->  ( ( log `  A
)  /  A )  e.  CC )
1751742timesd 10202 . . 3  |-  ( ph  ->  ( 2  x.  (
( log `  A
)  /  A ) )  =  ( ( ( log `  A
)  /  A )  +  ( ( log `  A )  /  A
) ) )
176173, 175breqtrrd 4230 . 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 9219 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204    e. cmpt 4258   dom cdm 4870   -->wf 5442   ` cfv 5446  (class class class)co 6073   supcsup 7437   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    +oocpnf 9109   RR*cxr 9111    < clt 9112    <_ cle 9113    - cmin 9283    / cdiv 9669   NNcn 9992   2c2 10041   RR+crp 10604   ...cfz 11035   |_cfl 11193   ^cexp 11374   abscabs 12031    ~~> r crli 12271   sum_csu 12471   _eceu 12657   logclog 20444   gammacem 20822
This theorem is referenced by:  mulog2sumlem2  21221
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-ioc 10913  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-fac 11559  df-bc 11586  df-hash 11611  df-shft 11874  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-limsup 12257  df-clim 12274  df-rlim 12275  df-sum 12472  df-ef 12662  df-e 12663  df-sin 12664  df-cos 12665  df-pi 12667  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-hom 13545  df-cco 13546  df-rest 13642  df-topn 13643  df-topgen 13659  df-pt 13660  df-prds 13663  df-xrs 13718  df-0g 13719  df-gsum 13720  df-qtop 13725  df-imas 13726  df-xps 13728  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-mulg 14807  df-cntz 15108  df-cmn 15406  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-fbas 16691  df-fg 16692  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-ntr 17076  df-cls 17077  df-nei 17154  df-lp 17192  df-perf 17193  df-cn 17283  df-cnp 17284  df-haus 17371  df-cmp 17442  df-tx 17586  df-hmeo 17779  df-fil 17870  df-fm 17962  df-flim 17963  df-flf 17964  df-xms 18342  df-ms 18343  df-tms 18344  df-cncf 18900  df-limc 19745  df-dv 19746  df-log 20446  df-cxp 20447  df-em 20823
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