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Theorem chtprm 20391
Description: The Chebyshev function at a prime. (Contributed by Mario Carneiro, 22-Sep-2014.)
Assertion
Ref Expression
chtprm  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( theta `  ( A  +  1 ) )  =  ( ( theta `  A )  +  ( log `  ( A  +  1 ) ) ) )

Proof of Theorem chtprm
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 peano2z 10060 . . . . . 6  |-  ( A  e.  ZZ  ->  ( A  +  1 )  e.  ZZ )
21adantr 451 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  ZZ )
3 zre 10028 . . . . 5  |-  ( ( A  +  1 )  e.  ZZ  ->  ( A  +  1 )  e.  RR )
42, 3syl 15 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  RR )
5 chtval 20348 . . . 4  |-  ( ( A  +  1 )  e.  RR  ->  ( theta `  ( A  + 
1 ) )  = 
sum_ p  e.  (
( 0 [,] ( A  +  1 ) )  i^i  Prime )
( log `  p
) )
64, 5syl 15 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( theta `  ( A  +  1 ) )  =  sum_ p  e.  ( ( 0 [,] ( A  +  1 ) )  i^i  Prime )
( log `  p
) )
7 ppisval 20341 . . . . . 6  |-  ( ( A  +  1 )  e.  RR  ->  (
( 0 [,] ( A  +  1 ) )  i^i  Prime )  =  ( ( 2 ... ( |_ `  ( A  +  1
) ) )  i^i 
Prime ) )
84, 7syl 15 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 0 [,] ( A  +  1 ) )  i^i  Prime )  =  ( ( 2 ... ( |_ `  ( A  +  1
) ) )  i^i 
Prime ) )
9 flid 10939 . . . . . . . 8  |-  ( ( A  +  1 )  e.  ZZ  ->  ( |_ `  ( A  + 
1 ) )  =  ( A  +  1 ) )
102, 9syl 15 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( |_ `  ( A  +  1 ) )  =  ( A  +  1 ) )
1110oveq2d 5874 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( 2 ... ( |_ `  ( A  + 
1 ) ) )  =  ( 2 ... ( A  +  1 ) ) )
1211ineq1d 3369 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( |_ `  ( A  +  1 ) ) )  i^i  Prime )  =  ( ( 2 ... ( A  + 
1 ) )  i^i 
Prime ) )
138, 12eqtrd 2315 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 0 [,] ( A  +  1 ) )  i^i  Prime )  =  ( ( 2 ... ( A  + 
1 ) )  i^i 
Prime ) )
1413sumeq1d 12174 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  -> 
sum_ p  e.  (
( 0 [,] ( A  +  1 ) )  i^i  Prime )
( log `  p
)  =  sum_ p  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime ) ( log `  p
) )
156, 14eqtrd 2315 . 2  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( theta `  ( A  +  1 ) )  =  sum_ p  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
( log `  p
) )
16 zre 10028 . . . . . . . 8  |-  ( A  e.  ZZ  ->  A  e.  RR )
1716adantr 451 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  RR )
1817ltp1d 9687 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  <  ( A  +  1 ) )
1917, 4ltnled 8966 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  <  ( A  +  1 )  <->  -.  ( A  +  1 )  <_  A )
)
2018, 19mpbid 201 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  -.  ( A  + 
1 )  <_  A
)
21 inss1 3389 . . . . . . 7  |-  ( ( 2 ... A )  i^i  Prime )  C_  (
2 ... A )
2221sseli 3176 . . . . . 6  |-  ( ( A  +  1 )  e.  ( ( 2 ... A )  i^i 
Prime )  ->  ( A  +  1 )  e.  ( 2 ... A
) )
23 elfzle2 10800 . . . . . 6  |-  ( ( A  +  1 )  e.  ( 2 ... A )  ->  ( A  +  1 )  <_  A )
2422, 23syl 15 . . . . 5  |-  ( ( A  +  1 )  e.  ( ( 2 ... A )  i^i 
Prime )  ->  ( A  +  1 )  <_  A )
2520, 24nsyl 113 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  -.  ( A  + 
1 )  e.  ( ( 2 ... A
)  i^i  Prime ) )
26 disjsn 3693 . . . 4  |-  ( ( ( ( 2 ... A )  i^i  Prime )  i^i  { ( A  +  1 ) } )  =  (/)  <->  -.  ( A  +  1 )  e.  ( ( 2 ... A )  i^i 
Prime ) )
2725, 26sylibr 203 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( ( 2 ... A )  i^i 
Prime )  i^i  { ( A  +  1 ) } )  =  (/) )
28 2z 10054 . . . . . . 7  |-  2  e.  ZZ
29 zcn 10029 . . . . . . . . . . 11  |-  ( A  e.  ZZ  ->  A  e.  CC )
3029adantr 451 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  CC )
31 ax-1cn 8795 . . . . . . . . . 10  |-  1  e.  CC
32 pncan 9057 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  + 
1 )  -  1 )  =  A )
3330, 31, 32sylancl 643 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( A  + 
1 )  -  1 )  =  A )
34 prmuz2 12776 . . . . . . . . . . 11  |-  ( ( A  +  1 )  e.  Prime  ->  ( A  +  1 )  e.  ( ZZ>= `  2 )
)
3534adantl 452 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  ( ZZ>= ` 
2 ) )
36 uz2m1nn 10292 . . . . . . . . . 10  |-  ( ( A  +  1 )  e.  ( ZZ>= `  2
)  ->  ( ( A  +  1 )  -  1 )  e.  NN )
3735, 36syl 15 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( A  + 
1 )  -  1 )  e.  NN )
3833, 37eqeltrrd 2358 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  NN )
39 nnuz 10263 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
40 2cn 9816 . . . . . . . . . . 11  |-  2  e.  CC
41 1p1e2 9840 . . . . . . . . . . 11  |-  ( 1  +  1 )  =  2
4240, 31, 31, 41subaddrii 9135 . . . . . . . . . 10  |-  ( 2  -  1 )  =  1
4342fveq2i 5528 . . . . . . . . 9  |-  ( ZZ>= `  ( 2  -  1 ) )  =  (
ZZ>= `  1 )
4439, 43eqtr4i 2306 . . . . . . . 8  |-  NN  =  ( ZZ>= `  ( 2  -  1 ) )
4538, 44syl6eleq 2373 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  ( ZZ>= `  ( 2  -  1 ) ) )
46 fzsuc2 10842 . . . . . . 7  |-  ( ( 2  e.  ZZ  /\  A  e.  ( ZZ>= `  ( 2  -  1 ) ) )  -> 
( 2 ... ( A  +  1 ) )  =  ( ( 2 ... A )  u.  { ( A  +  1 ) } ) )
4728, 45, 46sylancr 644 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( 2 ... ( A  +  1 ) )  =  ( ( 2 ... A )  u.  { ( A  +  1 ) } ) )
4847ineq1d 3369 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  =  ( ( ( 2 ... A )  u.  { ( A  +  1 ) } )  i^i  Prime )
)
49 indir 3417 . . . . 5  |-  ( ( ( 2 ... A
)  u.  { ( A  +  1 ) } )  i^i  Prime )  =  ( ( ( 2 ... A )  i^i  Prime )  u.  ( { ( A  + 
1 ) }  i^i  Prime
) )
5048, 49syl6eq 2331 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  =  ( ( ( 2 ... A )  i^i  Prime )  u.  ( { ( A  + 
1 ) }  i^i  Prime
) ) )
51 simpr 447 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  Prime )
5251snssd 3760 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  { ( A  + 
1 ) }  C_  Prime )
53 df-ss 3166 . . . . . 6  |-  ( { ( A  +  1 ) }  C_  Prime  <->  ( { ( A  + 
1 ) }  i^i  Prime
)  =  { ( A  +  1 ) } )
5452, 53sylib 188 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( { ( A  +  1 ) }  i^i  Prime )  =  {
( A  +  1 ) } )
5554uneq2d 3329 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( ( 2 ... A )  i^i 
Prime )  u.  ( { ( A  + 
1 ) }  i^i  Prime
) )  =  ( ( ( 2 ... A )  i^i  Prime )  u.  { ( A  +  1 ) } ) )
5650, 55eqtrd 2315 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  =  ( ( ( 2 ... A )  i^i  Prime )  u.  {
( A  +  1 ) } ) )
57 fzfid 11035 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( 2 ... ( A  +  1 ) )  e.  Fin )
58 inss1 3389 . . . 4  |-  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  C_  (
2 ... ( A  + 
1 ) )
59 ssfi 7083 . . . 4  |-  ( ( ( 2 ... ( A  +  1 ) )  e.  Fin  /\  ( ( 2 ... ( A  +  1 ) )  i^i  Prime ) 
C_  ( 2 ... ( A  +  1 ) ) )  -> 
( ( 2 ... ( A  +  1 ) )  i^i  Prime )  e.  Fin )
6057, 58, 59sylancl 643 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  e.  Fin )
61 inss2 3390 . . . . . . . 8  |-  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  C_  Prime
62 simpr 447 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  ( A  +  1 )  e.  Prime )  /\  p  e.  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
)  ->  p  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime ) )
6361, 62sseldi 3178 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  ( A  +  1 )  e.  Prime )  /\  p  e.  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
)  ->  p  e.  Prime )
64 prmnn 12761 . . . . . . 7  |-  ( p  e.  Prime  ->  p  e.  NN )
6563, 64syl 15 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  ( A  +  1 )  e.  Prime )  /\  p  e.  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
)  ->  p  e.  NN )
6665nnrpd 10389 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  ( A  +  1 )  e.  Prime )  /\  p  e.  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
)  ->  p  e.  RR+ )
6766relogcld 19974 . . . 4  |-  ( ( ( A  e.  ZZ  /\  ( A  +  1 )  e.  Prime )  /\  p  e.  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
)  ->  ( log `  p )  e.  RR )
6867recnd 8861 . . 3  |-  ( ( ( A  e.  ZZ  /\  ( A  +  1 )  e.  Prime )  /\  p  e.  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
)  ->  ( log `  p )  e.  CC )
6927, 56, 60, 68fsumsplit 12212 . 2  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  -> 
sum_ p  e.  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
( log `  p
)  =  ( sum_ p  e.  ( ( 2 ... A )  i^i 
Prime ) ( log `  p
)  +  sum_ p  e.  { ( A  + 
1 ) }  ( log `  p ) ) )
70 chtval 20348 . . . . 5  |-  ( A  e.  RR  ->  ( theta `  A )  = 
sum_ p  e.  (
( 0 [,] A
)  i^i  Prime ) ( log `  p ) )
7117, 70syl 15 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( theta `  A )  =  sum_ p  e.  ( ( 0 [,] A
)  i^i  Prime ) ( log `  p ) )
72 ppisval 20341 . . . . . . 7  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  =  ( ( 2 ... ( |_ `  A
) )  i^i  Prime ) )
7317, 72syl 15 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 0 [,] A )  i^i  Prime )  =  ( ( 2 ... ( |_ `  A ) )  i^i 
Prime ) )
74 flid 10939 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  ( |_ `  A )  =  A )
7574adantr 451 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( |_ `  A
)  =  A )
7675oveq2d 5874 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( 2 ... ( |_ `  A ) )  =  ( 2 ... A ) )
7776ineq1d 3369 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( |_ `  A
) )  i^i  Prime )  =  ( ( 2 ... A )  i^i 
Prime ) )
7873, 77eqtrd 2315 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 0 [,] A )  i^i  Prime )  =  ( ( 2 ... A )  i^i 
Prime ) )
7978sumeq1d 12174 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  -> 
sum_ p  e.  (
( 0 [,] A
)  i^i  Prime ) ( log `  p )  =  sum_ p  e.  ( ( 2 ... A
)  i^i  Prime ) ( log `  p ) )
8071, 79eqtr2d 2316 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  -> 
sum_ p  e.  (
( 2 ... A
)  i^i  Prime ) ( log `  p )  =  ( theta `  A
) )
81 prmnn 12761 . . . . 5  |-  ( ( A  +  1 )  e.  Prime  ->  ( A  +  1 )  e.  NN )
8281adantl 452 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  NN )
8382nnrpd 10389 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  RR+ )
8483relogcld 19974 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( log `  ( A  +  1 ) )  e.  RR )
8584recnd 8861 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( log `  ( A  +  1 ) )  e.  CC )
86 fveq2 5525 . . . . 5  |-  ( p  =  ( A  + 
1 )  ->  ( log `  p )  =  ( log `  ( A  +  1 ) ) )
8786sumsn 12213 . . . 4  |-  ( ( ( A  +  1 )  e.  NN  /\  ( log `  ( A  +  1 ) )  e.  CC )  ->  sum_ p  e.  { ( A  +  1 ) }  ( log `  p
)  =  ( log `  ( A  +  1 ) ) )
8882, 85, 87syl2anc 642 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  -> 
sum_ p  e.  { ( A  +  1 ) }  ( log `  p
)  =  ( log `  ( A  +  1 ) ) )
8980, 88oveq12d 5876 . 2  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( sum_ p  e.  ( ( 2 ... A
)  i^i  Prime ) ( log `  p )  +  sum_ p  e.  {
( A  +  1 ) }  ( log `  p ) )  =  ( ( theta `  A
)  +  ( log `  ( A  +  1 ) ) ) )
9015, 69, 893eqtrd 2319 1  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( theta `  ( A  +  1 ) )  =  ( ( theta `  A )  +  ( log `  ( A  +  1 ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Fincfn 6863   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    < clt 8867    <_ cle 8868    - cmin 9037   NNcn 9746   2c2 9795   ZZcz 10024   ZZ>=cuz 10230   [,]cicc 10659   ...cfz 10782   |_cfl 10924   sum_csu 12158   Primecprime 12758   logclog 19912   thetaccht 20328
This theorem is referenced by:  cht2  20410  cht3  20411
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-prm 12759  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  df-cht 20334
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