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Theorem pclogsum 20454
Description: The logarithmic analogue of pcprod 12943. The sum of the logarithms of the primes dividing  A multiplied by their powers yields the logarithm of  A. (Contributed by Mario Carneiro, 15-Apr-2016.)
Assertion
Ref Expression
pclogsum  |-  ( A  e.  NN  ->  sum_ p  e.  ( ( 1 ... A )  i^i  Prime ) ( ( p  pCnt  A )  x.  ( log `  p ) )  =  ( log `  A
) )
Distinct variable group:    A, p

Proof of Theorem pclogsum
Dummy variables  m  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elin 3358 . . . . . 6  |-  ( p  e.  ( ( 1 ... A )  i^i 
Prime )  <->  ( p  e.  ( 1 ... A
)  /\  p  e.  Prime ) )
21baib 871 . . . . 5  |-  ( p  e.  ( 1 ... A )  ->  (
p  e.  ( ( 1 ... A )  i^i  Prime )  <->  p  e.  Prime ) )
32ifbid 3583 . . . 4  |-  ( p  e.  ( 1 ... A )  ->  if ( p  e.  (
( 1 ... A
)  i^i  Prime ) ,  ( log `  (
p ^ ( p 
pCnt  A ) ) ) ,  0 )  =  if ( p  e. 
Prime ,  ( log `  ( p ^ (
p  pCnt  A )
) ) ,  0 ) )
4 fvif 5540 . . . . 5  |-  ( log `  if ( p  e. 
Prime ,  ( p ^ ( p  pCnt  A ) ) ,  1 ) )  =  if ( p  e.  Prime ,  ( log `  (
p ^ ( p 
pCnt  A ) ) ) ,  ( log `  1
) )
5 log1 19939 . . . . . 6  |-  ( log `  1 )  =  0
6 ifeq2 3570 . . . . . 6  |-  ( ( log `  1 )  =  0  ->  if ( p  e.  Prime ,  ( log `  (
p ^ ( p 
pCnt  A ) ) ) ,  ( log `  1
) )  =  if ( p  e.  Prime ,  ( log `  (
p ^ ( p 
pCnt  A ) ) ) ,  0 ) )
75, 6ax-mp 8 . . . . 5  |-  if ( p  e.  Prime ,  ( log `  ( p ^ ( p  pCnt  A ) ) ) ,  ( log `  1
) )  =  if ( p  e.  Prime ,  ( log `  (
p ^ ( p 
pCnt  A ) ) ) ,  0 )
84, 7eqtri 2303 . . . 4  |-  ( log `  if ( p  e. 
Prime ,  ( p ^ ( p  pCnt  A ) ) ,  1 ) )  =  if ( p  e.  Prime ,  ( log `  (
p ^ ( p 
pCnt  A ) ) ) ,  0 )
93, 8syl6eqr 2333 . . 3  |-  ( p  e.  ( 1 ... A )  ->  if ( p  e.  (
( 1 ... A
)  i^i  Prime ) ,  ( log `  (
p ^ ( p 
pCnt  A ) ) ) ,  0 )  =  ( log `  if ( p  e.  Prime ,  ( p ^ (
p  pCnt  A )
) ,  1 ) ) )
109sumeq2i 12172 . 2  |-  sum_ p  e.  ( 1 ... A
) if ( p  e.  ( ( 1 ... A )  i^i 
Prime ) ,  ( log `  ( p ^ (
p  pCnt  A )
) ) ,  0 )  =  sum_ p  e.  ( 1 ... A
) ( log `  if ( p  e.  Prime ,  ( p ^ (
p  pCnt  A )
) ,  1 ) )
11 inss1 3389 . . . 4  |-  ( ( 1 ... A )  i^i  Prime )  C_  (
1 ... A )
12 simpr 447 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  p  e.  ( ( 1 ... A )  i^i  Prime ) )
1311, 12sseldi 3178 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  p  e.  ( 1 ... A
) )
14 elfznn 10819 . . . . . . . . . 10  |-  ( p  e.  ( 1 ... A )  ->  p  e.  NN )
1513, 14syl 15 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  p  e.  NN )
16 inss2 3390 . . . . . . . . . . 11  |-  ( ( 1 ... A )  i^i  Prime )  C_  Prime
1716, 12sseldi 3178 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  p  e.  Prime )
18 simpl 443 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  A  e.  NN )
1917, 18pccld 12903 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  (
p  pCnt  A )  e.  NN0 )
2015, 19nnexpcld 11266 . . . . . . . 8  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  (
p ^ ( p 
pCnt  A ) )  e.  NN )
2120nnrpd 10389 . . . . . . 7  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  (
p ^ ( p 
pCnt  A ) )  e.  RR+ )
2221relogcld 19974 . . . . . 6  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  ( log `  ( p ^
( p  pCnt  A
) ) )  e.  RR )
2322recnd 8861 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  ( log `  ( p ^
( p  pCnt  A
) ) )  e.  CC )
2423ralrimiva 2626 . . . 4  |-  ( A  e.  NN  ->  A. p  e.  ( ( 1 ... A )  i^i  Prime ) ( log `  (
p ^ ( p 
pCnt  A ) ) )  e.  CC )
25 fzfi 11034 . . . . . 6  |-  ( 1 ... A )  e. 
Fin
2625olci 380 . . . . 5  |-  ( ( 1 ... A ) 
C_  ( ZZ>= `  1
)  \/  ( 1 ... A )  e. 
Fin )
27 sumss2 12199 . . . . 5  |-  ( ( ( ( ( 1 ... A )  i^i 
Prime )  C_  ( 1 ... A )  /\  A. p  e.  ( ( 1 ... A )  i^i  Prime ) ( log `  ( p ^ (
p  pCnt  A )
) )  e.  CC )  /\  ( ( 1 ... A )  C_  ( ZZ>= `  1 )  \/  ( 1 ... A
)  e.  Fin )
)  ->  sum_ p  e.  ( ( 1 ... A )  i^i  Prime ) ( log `  (
p ^ ( p 
pCnt  A ) ) )  =  sum_ p  e.  ( 1 ... A ) if ( p  e.  ( ( 1 ... A )  i^i  Prime ) ,  ( log `  (
p ^ ( p 
pCnt  A ) ) ) ,  0 ) )
2826, 27mpan2 652 . . . 4  |-  ( ( ( ( 1 ... A )  i^i  Prime ) 
C_  ( 1 ... A )  /\  A. p  e.  ( (
1 ... A )  i^i 
Prime ) ( log `  (
p ^ ( p 
pCnt  A ) ) )  e.  CC )  ->  sum_ p  e.  ( ( 1 ... A )  i^i  Prime ) ( log `  ( p ^ (
p  pCnt  A )
) )  =  sum_ p  e.  ( 1 ... A ) if ( p  e.  ( ( 1 ... A )  i^i  Prime ) ,  ( log `  ( p ^ ( p  pCnt  A ) ) ) ,  0 ) )
2911, 24, 28sylancr 644 . . 3  |-  ( A  e.  NN  ->  sum_ p  e.  ( ( 1 ... A )  i^i  Prime ) ( log `  (
p ^ ( p 
pCnt  A ) ) )  =  sum_ p  e.  ( 1 ... A ) if ( p  e.  ( ( 1 ... A )  i^i  Prime ) ,  ( log `  (
p ^ ( p 
pCnt  A ) ) ) ,  0 ) )
3015nnrpd 10389 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  p  e.  RR+ )
3119nn0zd 10115 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  (
p  pCnt  A )  e.  ZZ )
32 relogexp 19949 . . . . 5  |-  ( ( p  e.  RR+  /\  (
p  pCnt  A )  e.  ZZ )  ->  ( log `  ( p ^
( p  pCnt  A
) ) )  =  ( ( p  pCnt  A )  x.  ( log `  p ) ) )
3330, 31, 32syl2anc 642 . . . 4  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  ( log `  ( p ^
( p  pCnt  A
) ) )  =  ( ( p  pCnt  A )  x.  ( log `  p ) ) )
3433sumeq2dv 12176 . . 3  |-  ( A  e.  NN  ->  sum_ p  e.  ( ( 1 ... A )  i^i  Prime ) ( log `  (
p ^ ( p 
pCnt  A ) ) )  =  sum_ p  e.  ( ( 1 ... A
)  i^i  Prime ) ( ( p  pCnt  A
)  x.  ( log `  p ) ) )
3529, 34eqtr3d 2317 . 2  |-  ( A  e.  NN  ->  sum_ p  e.  ( 1 ... A
) if ( p  e.  ( ( 1 ... A )  i^i 
Prime ) ,  ( log `  ( p ^ (
p  pCnt  A )
) ) ,  0 )  =  sum_ p  e.  ( ( 1 ... A )  i^i  Prime ) ( ( p  pCnt  A )  x.  ( log `  p ) ) )
3614adantl 452 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  ->  p  e.  NN )
37 eleq1 2343 . . . . . . . 8  |-  ( n  =  p  ->  (
n  e.  Prime  <->  p  e.  Prime ) )
38 id 19 . . . . . . . . 9  |-  ( n  =  p  ->  n  =  p )
39 oveq1 5865 . . . . . . . . 9  |-  ( n  =  p  ->  (
n  pCnt  A )  =  ( p  pCnt  A ) )
4038, 39oveq12d 5876 . . . . . . . 8  |-  ( n  =  p  ->  (
n ^ ( n 
pCnt  A ) )  =  ( p ^ (
p  pCnt  A )
) )
41 eqidd 2284 . . . . . . . 8  |-  ( n  =  p  ->  1  =  1 )
4237, 40, 41ifbieq12d 3587 . . . . . . 7  |-  ( n  =  p  ->  if ( n  e.  Prime ,  ( n ^ (
n  pCnt  A )
) ,  1 )  =  if ( p  e.  Prime ,  ( p ^ ( p  pCnt  A ) ) ,  1 ) )
4342fveq2d 5529 . . . . . 6  |-  ( n  =  p  ->  ( log `  if ( n  e.  Prime ,  ( n ^ ( n  pCnt  A ) ) ,  1 ) )  =  ( log `  if ( p  e.  Prime ,  ( p ^ ( p 
pCnt  A ) ) ,  1 ) ) )
44 eqid 2283 . . . . . 6  |-  ( n  e.  NN  |->  ( log `  if ( n  e. 
Prime ,  ( n ^ ( n  pCnt  A ) ) ,  1 ) ) )  =  ( n  e.  NN  |->  ( log `  if ( n  e.  Prime ,  ( n ^ ( n 
pCnt  A ) ) ,  1 ) ) )
45 fvex 5539 . . . . . 6  |-  ( log `  if ( p  e. 
Prime ,  ( p ^ ( p  pCnt  A ) ) ,  1 ) )  e.  _V
4643, 44, 45fvmpt 5602 . . . . 5  |-  ( p  e.  NN  ->  (
( n  e.  NN  |->  ( log `  if ( n  e.  Prime ,  ( n ^ ( n 
pCnt  A ) ) ,  1 ) ) ) `
 p )  =  ( log `  if ( p  e.  Prime ,  ( p ^ (
p  pCnt  A )
) ,  1 ) ) )
4736, 46syl 15 . . . 4  |-  ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  ->  ( ( n  e.  NN  |->  ( log `  if ( n  e. 
Prime ,  ( n ^ ( n  pCnt  A ) ) ,  1 ) ) ) `  p )  =  ( log `  if ( p  e.  Prime ,  ( p ^ ( p 
pCnt  A ) ) ,  1 ) ) )
48 elnnuz 10264 . . . . 5  |-  ( A  e.  NN  <->  A  e.  ( ZZ>= `  1 )
)
4948biimpi 186 . . . 4  |-  ( A  e.  NN  ->  A  e.  ( ZZ>= `  1 )
)
5036adantr 451 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  /\  p  e.  Prime )  ->  p  e.  NN )
51 simpr 447 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  /\  p  e.  Prime )  ->  p  e.  Prime )
52 simpll 730 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  /\  p  e.  Prime )  ->  A  e.  NN )
5351, 52pccld 12903 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  /\  p  e.  Prime )  ->  ( p  pCnt  A )  e.  NN0 )
5450, 53nnexpcld 11266 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  /\  p  e.  Prime )  ->  ( p ^
( p  pCnt  A
) )  e.  NN )
55 1nn 9757 . . . . . . . . 9  |-  1  e.  NN
5655a1i 10 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  /\  -.  p  e. 
Prime )  ->  1  e.  NN )
5754, 56ifclda 3592 . . . . . . 7  |-  ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  ->  if ( p  e.  Prime ,  ( p ^ ( p  pCnt  A ) ) ,  1 )  e.  NN )
5857nnrpd 10389 . . . . . 6  |-  ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  ->  if ( p  e.  Prime ,  ( p ^ ( p  pCnt  A ) ) ,  1 )  e.  RR+ )
5958relogcld 19974 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  ->  ( log `  if ( p  e.  Prime ,  ( p ^ (
p  pCnt  A )
) ,  1 ) )  e.  RR )
6059recnd 8861 . . . 4  |-  ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  ->  ( log `  if ( p  e.  Prime ,  ( p ^ (
p  pCnt  A )
) ,  1 ) )  e.  CC )
6147, 49, 60fsumser 12203 . . 3  |-  ( A  e.  NN  ->  sum_ p  e.  ( 1 ... A
) ( log `  if ( p  e.  Prime ,  ( p ^ (
p  pCnt  A )
) ,  1 ) )  =  (  seq  1 (  +  , 
( n  e.  NN  |->  ( log `  if ( n  e.  Prime ,  ( n ^ ( n 
pCnt  A ) ) ,  1 ) ) ) ) `  A ) )
62 rpmulcl 10375 . . . . 5  |-  ( ( p  e.  RR+  /\  m  e.  RR+ )  ->  (
p  x.  m )  e.  RR+ )
6362adantl 452 . . . 4  |-  ( ( A  e.  NN  /\  ( p  e.  RR+  /\  m  e.  RR+ ) )  -> 
( p  x.  m
)  e.  RR+ )
64 eqid 2283 . . . . . . 7  |-  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ ( n 
pCnt  A ) ) ,  1 ) )  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ (
n  pCnt  A )
) ,  1 ) )
65 ovex 5883 . . . . . . . 8  |-  ( p ^ ( p  pCnt  A ) )  e.  _V
66 1ex 8833 . . . . . . . 8  |-  1  e.  _V
6765, 66ifex 3623 . . . . . . 7  |-  if ( p  e.  Prime ,  ( p ^ ( p 
pCnt  A ) ) ,  1 )  e.  _V
6842, 64, 67fvmpt 5602 . . . . . 6  |-  ( p  e.  NN  ->  (
( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ (
n  pCnt  A )
) ,  1 ) ) `  p )  =  if ( p  e.  Prime ,  ( p ^ ( p  pCnt  A ) ) ,  1 ) )
6936, 68syl 15 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  ->  ( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ ( n 
pCnt  A ) ) ,  1 ) ) `  p )  =  if ( p  e.  Prime ,  ( p ^ (
p  pCnt  A )
) ,  1 ) )
7069, 58eqeltrd 2357 . . . 4  |-  ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  ->  ( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ ( n 
pCnt  A ) ) ,  1 ) ) `  p )  e.  RR+ )
71 relogmul 19945 . . . . 5  |-  ( ( p  e.  RR+  /\  m  e.  RR+ )  ->  ( log `  ( p  x.  m ) )  =  ( ( log `  p
)  +  ( log `  m ) ) )
7271adantl 452 . . . 4  |-  ( ( A  e.  NN  /\  ( p  e.  RR+  /\  m  e.  RR+ ) )  -> 
( log `  (
p  x.  m ) )  =  ( ( log `  p )  +  ( log `  m
) ) )
7369fveq2d 5529 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  ->  ( log `  (
( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ (
n  pCnt  A )
) ,  1 ) ) `  p ) )  =  ( log `  if ( p  e. 
Prime ,  ( p ^ ( p  pCnt  A ) ) ,  1 ) ) )
7473, 47eqtr4d 2318 . . . 4  |-  ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  ->  ( log `  (
( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ (
n  pCnt  A )
) ,  1 ) ) `  p ) )  =  ( ( n  e.  NN  |->  ( log `  if ( n  e.  Prime ,  ( n ^ ( n 
pCnt  A ) ) ,  1 ) ) ) `
 p ) )
7563, 70, 49, 72, 74seqhomo 11093 . . 3  |-  ( A  e.  NN  ->  ( log `  (  seq  1
(  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ (
n  pCnt  A )
) ,  1 ) ) ) `  A
) )  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  ( log `  if ( n  e.  Prime ,  ( n ^ (
n  pCnt  A )
) ,  1 ) ) ) ) `  A ) )
7664pcprod 12943 . . . 4  |-  ( A  e.  NN  ->  (  seq  1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ (
n  pCnt  A )
) ,  1 ) ) ) `  A
)  =  A )
7776fveq2d 5529 . . 3  |-  ( A  e.  NN  ->  ( log `  (  seq  1
(  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ (
n  pCnt  A )
) ,  1 ) ) ) `  A
) )  =  ( log `  A ) )
7861, 75, 773eqtr2d 2321 . 2  |-  ( A  e.  NN  ->  sum_ p  e.  ( 1 ... A
) ( log `  if ( p  e.  Prime ,  ( p ^ (
p  pCnt  A )
) ,  1 ) )  =  ( log `  A ) )
7910, 35, 783eqtr3a 2339 1  |-  ( A  e.  NN  ->  sum_ p  e.  ( ( 1 ... A )  i^i  Prime ) ( ( p  pCnt  A )  x.  ( log `  p ) )  =  ( log `  A
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    i^i cin 3151    C_ wss 3152   ifcif 3565    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   Fincfn 6863   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742   NNcn 9746   ZZcz 10024   ZZ>=cuz 10230   RR+crp 10354   ...cfz 10782    seq cseq 11046   ^cexp 11104   sum_csu 12158   Primecprime 12758    pCnt cpc 12889   logclog 19912
This theorem is referenced by:  vmasum  20455  chebbnd1lem1  20618
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914
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