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Theorem chtprm 20444
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 10107 . . . . . 6  |-  ( A  e.  ZZ  ->  ( A  +  1 )  e.  ZZ )
21adantr 451 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  ZZ )
3 zre 10075 . . . . 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 20401 . . . 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 20394 . . . . . 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 10986 . . . . . . . 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 5916 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( 2 ... ( |_ `  ( A  + 
1 ) ) )  =  ( 2 ... ( A  +  1 ) ) )
1211ineq1d 3403 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( |_ `  ( A  +  1 ) ) )  i^i  Prime )  =  ( ( 2 ... ( A  + 
1 ) )  i^i 
Prime ) )
138, 12eqtrd 2348 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 0 [,] ( A  +  1 ) )  i^i  Prime )  =  ( ( 2 ... ( A  + 
1 ) )  i^i 
Prime ) )
1413sumeq1d 12221 . . 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 2348 . 2  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( theta `  ( A  +  1 ) )  =  sum_ p  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
( log `  p
) )
16 zre 10075 . . . . . . . 8  |-  ( A  e.  ZZ  ->  A  e.  RR )
1716adantr 451 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  RR )
1817ltp1d 9732 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  <  ( A  +  1 ) )
1917, 4ltnled 9011 . . . . . 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 3423 . . . . . . 7  |-  ( ( 2 ... A )  i^i  Prime )  C_  (
2 ... A )
2221sseli 3210 . . . . . 6  |-  ( ( A  +  1 )  e.  ( ( 2 ... A )  i^i 
Prime )  ->  ( A  +  1 )  e.  ( 2 ... A
) )
23 elfzle2 10847 . . . . . 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 3727 . . . 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 10101 . . . . . . 7  |-  2  e.  ZZ
29 zcn 10076 . . . . . . . . . . 11  |-  ( A  e.  ZZ  ->  A  e.  CC )
3029adantr 451 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  CC )
31 ax-1cn 8840 . . . . . . . . . 10  |-  1  e.  CC
32 pncan 9102 . . . . . . . . . 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 12823 . . . . . . . . . . 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 10339 . . . . . . . . . 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 2391 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  NN )
39 nnuz 10310 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
40 2cn 9861 . . . . . . . . . . 11  |-  2  e.  CC
41 1p1e2 9885 . . . . . . . . . . 11  |-  ( 1  +  1 )  =  2
4240, 31, 31, 41subaddrii 9180 . . . . . . . . . 10  |-  ( 2  -  1 )  =  1
4342fveq2i 5566 . . . . . . . . 9  |-  ( ZZ>= `  ( 2  -  1 ) )  =  (
ZZ>= `  1 )
4439, 43eqtr4i 2339 . . . . . . . 8  |-  NN  =  ( ZZ>= `  ( 2  -  1 ) )
4538, 44syl6eleq 2406 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  ( ZZ>= `  ( 2  -  1 ) ) )
46 fzsuc2 10889 . . . . . . 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 3403 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  =  ( ( ( 2 ... A )  u.  { ( A  +  1 ) } )  i^i  Prime )
)
49 indir 3451 . . . . 5  |-  ( ( ( 2 ... A
)  u.  { ( A  +  1 ) } )  i^i  Prime )  =  ( ( ( 2 ... A )  i^i  Prime )  u.  ( { ( A  + 
1 ) }  i^i  Prime
) )
5048, 49syl6eq 2364 . . . 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 3797 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  { ( A  + 
1 ) }  C_  Prime )
53 df-ss 3200 . . . . . 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 3363 . . . 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 2348 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  =  ( ( ( 2 ... A )  i^i  Prime )  u.  {
( A  +  1 ) } ) )
57 fzfid 11082 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( 2 ... ( A  +  1 ) )  e.  Fin )
58 inss1 3423 . . . 4  |-  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  C_  (
2 ... ( A  + 
1 ) )
59 ssfi 7126 . . . 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 3424 . . . . . . . 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 3212 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  ( A  +  1 )  e.  Prime )  /\  p  e.  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
)  ->  p  e.  Prime )
64 prmnn 12808 . . . . . . 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 10436 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  ( A  +  1 )  e.  Prime )  /\  p  e.  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
)  ->  p  e.  RR+ )
6766relogcld 20027 . . . 4  |-  ( ( ( A  e.  ZZ  /\  ( A  +  1 )  e.  Prime )  /\  p  e.  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
)  ->  ( log `  p )  e.  RR )
6867recnd 8906 . . 3  |-  ( ( ( A  e.  ZZ  /\  ( A  +  1 )  e.  Prime )  /\  p  e.  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
)  ->  ( log `  p )  e.  CC )
6927, 56, 60, 68fsumsplit 12259 . 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 20401 . . . . 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 20394 . . . . . . 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 10986 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  ( |_ `  A )  =  A )
7574adantr 451 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( |_ `  A
)  =  A )
7675oveq2d 5916 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( 2 ... ( |_ `  A ) )  =  ( 2 ... A ) )
7776ineq1d 3403 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( |_ `  A
) )  i^i  Prime )  =  ( ( 2 ... A )  i^i 
Prime ) )
7873, 77eqtrd 2348 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 0 [,] A )  i^i  Prime )  =  ( ( 2 ... A )  i^i 
Prime ) )
7978sumeq1d 12221 . . . 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 2349 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  -> 
sum_ p  e.  (
( 2 ... A
)  i^i  Prime ) ( log `  p )  =  ( theta `  A
) )
81 prmnn 12808 . . . . 5  |-  ( ( A  +  1 )  e.  Prime  ->  ( A  +  1 )  e.  NN )
8281adantl 452 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  NN )
8382nnrpd 10436 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  RR+ )
8483relogcld 20027 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( log `  ( A  +  1 ) )  e.  RR )
8584recnd 8906 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( log `  ( A  +  1 ) )  e.  CC )
86 fveq2 5563 . . . . 5  |-  ( p  =  ( A  + 
1 )  ->  ( log `  p )  =  ( log `  ( A  +  1 ) ) )
8786sumsn 12260 . . . 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 5918 . 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 2352 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 1633    e. wcel 1701    u. cun 3184    i^i cin 3185    C_ wss 3186   (/)c0 3489   {csn 3674   class class class wbr 4060   ` cfv 5292  (class class class)co 5900   Fincfn 6906   CCcc 8780   RRcr 8781   0cc0 8782   1c1 8783    + caddc 8785    < clt 8912    <_ cle 8913    - cmin 9082   NNcn 9791   2c2 9840   ZZcz 10071   ZZ>=cuz 10277   [,]cicc 10706   ...cfz 10829   |_cfl 10971   sum_csu 12205   Primecprime 12805   logclog 19965   thetaccht 20381
This theorem is referenced by:  cht2  20463  cht3  20464
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-inf2 7387  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860  ax-addf 8861  ax-mulf 8862
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-of 6120  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-2o 6522  df-oadd 6525  df-er 6702  df-map 6817  df-pm 6818  df-ixp 6861  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-fi 7210  df-sup 7239  df-oi 7270  df-card 7617  df-cda 7839  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-10 9857  df-n0 10013  df-z 10072  df-dec 10172  df-uz 10278  df-q 10364  df-rp 10402  df-xneg 10499  df-xadd 10500  df-xmul 10501  df-ioo 10707  df-ioc 10708  df-ico 10709  df-icc 10710  df-fz 10830  df-fzo 10918  df-fl 10972  df-mod 11021  df-seq 11094  df-exp 11152  df-fac 11336  df-bc 11363  df-hash 11385  df-shft 11609  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768  df-limsup 11992  df-clim 12009  df-rlim 12010  df-sum 12206  df-ef 12396  df-sin 12398  df-cos 12399  df-pi 12401  df-dvds 12579  df-prm 12806  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-mulr 13269  df-starv 13270  df-sca 13271  df-vsca 13272  df-tset 13274  df-ple 13275  df-ds 13277  df-unif 13278  df-hom 13279  df-cco 13280  df-rest 13376  df-topn 13377  df-topgen 13393  df-pt 13394  df-prds 13397  df-xrs 13452  df-0g 13453  df-gsum 13454  df-qtop 13459  df-imas 13460  df-xps 13462  df-mre 13537  df-mrc 13538  df-acs 13540  df-mnd 14416  df-submnd 14465  df-mulg 14541  df-cntz 14842  df-cmn 15140  df-xmet 16425  df-met 16426  df-bl 16427  df-mopn 16428  df-fbas 16429  df-fg 16430  df-cnfld 16433  df-top 16692  df-bases 16694  df-topon 16695  df-topsp 16696  df-cld 16812  df-ntr 16813  df-cls 16814  df-nei 16891  df-lp 16924  df-perf 16925  df-cn 17013  df-cnp 17014  df-haus 17099  df-tx 17313  df-hmeo 17502  df-fil 17593  df-fm 17685  df-flim 17686  df-flf 17687  df-xms 17937  df-ms 17938  df-tms 17939  df-cncf 18434  df-limc 19269  df-dv 19270  df-log 19967  df-cht 20387
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