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Theorem chtprm 22503
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 10698 . . . . . 6  |-  ( A  e.  ZZ  ->  ( A  +  1 )  e.  ZZ )
21adantr 465 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  ZZ )
3 zre 10662 . . . . 5  |-  ( ( A  +  1 )  e.  ZZ  ->  ( A  +  1 )  e.  RR )
42, 3syl 16 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  RR )
5 chtval 22460 . . . 4  |-  ( ( A  +  1 )  e.  RR  ->  ( theta `  ( A  + 
1 ) )  = 
sum_ p  e.  (
( 0 [,] ( A  +  1 ) )  i^i  Prime )
( log `  p
) )
64, 5syl 16 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( theta `  ( A  +  1 ) )  =  sum_ p  e.  ( ( 0 [,] ( A  +  1 ) )  i^i  Prime )
( log `  p
) )
7 ppisval 22453 . . . . . 6  |-  ( ( A  +  1 )  e.  RR  ->  (
( 0 [,] ( A  +  1 ) )  i^i  Prime )  =  ( ( 2 ... ( |_ `  ( A  +  1
) ) )  i^i 
Prime ) )
84, 7syl 16 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 0 [,] ( A  +  1 ) )  i^i  Prime )  =  ( ( 2 ... ( |_ `  ( A  +  1
) ) )  i^i 
Prime ) )
9 flid 11669 . . . . . . . 8  |-  ( ( A  +  1 )  e.  ZZ  ->  ( |_ `  ( A  + 
1 ) )  =  ( A  +  1 ) )
102, 9syl 16 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( |_ `  ( A  +  1 ) )  =  ( A  +  1 ) )
1110oveq2d 6119 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( 2 ... ( |_ `  ( A  + 
1 ) ) )  =  ( 2 ... ( A  +  1 ) ) )
1211ineq1d 3563 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( |_ `  ( A  +  1 ) ) )  i^i  Prime )  =  ( ( 2 ... ( A  + 
1 ) )  i^i 
Prime ) )
138, 12eqtrd 2475 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 0 [,] ( A  +  1 ) )  i^i  Prime )  =  ( ( 2 ... ( A  + 
1 ) )  i^i 
Prime ) )
1413sumeq1d 13190 . . 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 2475 . 2  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( theta `  ( A  +  1 ) )  =  sum_ p  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
( log `  p
) )
16 zre 10662 . . . . . . . 8  |-  ( A  e.  ZZ  ->  A  e.  RR )
1716adantr 465 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  RR )
1817ltp1d 10275 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  <  ( A  +  1 ) )
1917, 4ltnled 9533 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  <  ( A  +  1 )  <->  -.  ( A  +  1 )  <_  A )
)
2018, 19mpbid 210 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  -.  ( A  + 
1 )  <_  A
)
21 inss1 3582 . . . . . . 7  |-  ( ( 2 ... A )  i^i  Prime )  C_  (
2 ... A )
2221sseli 3364 . . . . . 6  |-  ( ( A  +  1 )  e.  ( ( 2 ... A )  i^i 
Prime )  ->  ( A  +  1 )  e.  ( 2 ... A
) )
23 elfzle2 11467 . . . . . 6  |-  ( ( A  +  1 )  e.  ( 2 ... A )  ->  ( A  +  1 )  <_  A )
2422, 23syl 16 . . . . 5  |-  ( ( A  +  1 )  e.  ( ( 2 ... A )  i^i 
Prime )  ->  ( A  +  1 )  <_  A )
2520, 24nsyl 121 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  -.  ( A  + 
1 )  e.  ( ( 2 ... A
)  i^i  Prime ) )
26 disjsn 3948 . . . 4  |-  ( ( ( ( 2 ... A )  i^i  Prime )  i^i  { ( A  +  1 ) } )  =  (/)  <->  -.  ( A  +  1 )  e.  ( ( 2 ... A )  i^i 
Prime ) )
2725, 26sylibr 212 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( ( 2 ... A )  i^i 
Prime )  i^i  { ( A  +  1 ) } )  =  (/) )
28 2z 10690 . . . . . . 7  |-  2  e.  ZZ
29 zcn 10663 . . . . . . . . . . 11  |-  ( A  e.  ZZ  ->  A  e.  CC )
3029adantr 465 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  CC )
31 ax-1cn 9352 . . . . . . . . . 10  |-  1  e.  CC
32 pncan 9628 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  + 
1 )  -  1 )  =  A )
3330, 31, 32sylancl 662 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( A  + 
1 )  -  1 )  =  A )
34 prmuz2 13793 . . . . . . . . . . 11  |-  ( ( A  +  1 )  e.  Prime  ->  ( A  +  1 )  e.  ( ZZ>= `  2 )
)
3534adantl 466 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  ( ZZ>= ` 
2 ) )
36 uz2m1nn 10941 . . . . . . . . . 10  |-  ( ( A  +  1 )  e.  ( ZZ>= `  2
)  ->  ( ( A  +  1 )  -  1 )  e.  NN )
3735, 36syl 16 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( A  + 
1 )  -  1 )  e.  NN )
3833, 37eqeltrrd 2518 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  NN )
39 nnuz 10908 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
40 2m1e1 10448 . . . . . . . . . 10  |-  ( 2  -  1 )  =  1
4140fveq2i 5706 . . . . . . . . 9  |-  ( ZZ>= `  ( 2  -  1 ) )  =  (
ZZ>= `  1 )
4239, 41eqtr4i 2466 . . . . . . . 8  |-  NN  =  ( ZZ>= `  ( 2  -  1 ) )
4338, 42syl6eleq 2533 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  ( ZZ>= `  ( 2  -  1 ) ) )
44 fzsuc2 11526 . . . . . . 7  |-  ( ( 2  e.  ZZ  /\  A  e.  ( ZZ>= `  ( 2  -  1 ) ) )  -> 
( 2 ... ( A  +  1 ) )  =  ( ( 2 ... A )  u.  { ( A  +  1 ) } ) )
4528, 43, 44sylancr 663 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( 2 ... ( A  +  1 ) )  =  ( ( 2 ... A )  u.  { ( A  +  1 ) } ) )
4645ineq1d 3563 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  =  ( ( ( 2 ... A )  u.  { ( A  +  1 ) } )  i^i  Prime )
)
47 indir 3610 . . . . 5  |-  ( ( ( 2 ... A
)  u.  { ( A  +  1 ) } )  i^i  Prime )  =  ( ( ( 2 ... A )  i^i  Prime )  u.  ( { ( A  + 
1 ) }  i^i  Prime
) )
4846, 47syl6eq 2491 . . . 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
) ) )
49 simpr 461 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  Prime )
5049snssd 4030 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  { ( A  + 
1 ) }  C_  Prime )
51 df-ss 3354 . . . . . 6  |-  ( { ( A  +  1 ) }  C_  Prime  <->  ( { ( A  + 
1 ) }  i^i  Prime
)  =  { ( A  +  1 ) } )
5250, 51sylib 196 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( { ( A  +  1 ) }  i^i  Prime )  =  {
( A  +  1 ) } )
5352uneq2d 3522 . . . 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 ) } ) )
5448, 53eqtrd 2475 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  =  ( ( ( 2 ... A )  i^i  Prime )  u.  {
( A  +  1 ) } ) )
55 fzfid 11807 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( 2 ... ( A  +  1 ) )  e.  Fin )
56 inss1 3582 . . . 4  |-  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  C_  (
2 ... ( A  + 
1 ) )
57 ssfi 7545 . . . 4  |-  ( ( ( 2 ... ( A  +  1 ) )  e.  Fin  /\  ( ( 2 ... ( A  +  1 ) )  i^i  Prime ) 
C_  ( 2 ... ( A  +  1 ) ) )  -> 
( ( 2 ... ( A  +  1 ) )  i^i  Prime )  e.  Fin )
5855, 56, 57sylancl 662 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  e.  Fin )
59 inss2 3583 . . . . . . . 8  |-  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  C_  Prime
60 simpr 461 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  ( A  +  1 )  e.  Prime )  /\  p  e.  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
)  ->  p  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime ) )
6159, 60sseldi 3366 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  ( A  +  1 )  e.  Prime )  /\  p  e.  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
)  ->  p  e.  Prime )
62 prmnn 13778 . . . . . . 7  |-  ( p  e.  Prime  ->  p  e.  NN )
6361, 62syl 16 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  ( A  +  1 )  e.  Prime )  /\  p  e.  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
)  ->  p  e.  NN )
6463nnrpd 11038 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  ( A  +  1 )  e.  Prime )  /\  p  e.  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
)  ->  p  e.  RR+ )
6564relogcld 22084 . . . 4  |-  ( ( ( A  e.  ZZ  /\  ( A  +  1 )  e.  Prime )  /\  p  e.  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
)  ->  ( log `  p )  e.  RR )
6665recnd 9424 . . 3  |-  ( ( ( A  e.  ZZ  /\  ( A  +  1 )  e.  Prime )  /\  p  e.  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
)  ->  ( log `  p )  e.  CC )
6727, 54, 58, 66fsumsplit 13228 . 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 ) ) )
68 chtval 22460 . . . . 5  |-  ( A  e.  RR  ->  ( theta `  A )  = 
sum_ p  e.  (
( 0 [,] A
)  i^i  Prime ) ( log `  p ) )
6917, 68syl 16 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( theta `  A )  =  sum_ p  e.  ( ( 0 [,] A
)  i^i  Prime ) ( log `  p ) )
70 ppisval 22453 . . . . . . 7  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  =  ( ( 2 ... ( |_ `  A
) )  i^i  Prime ) )
7117, 70syl 16 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 0 [,] A )  i^i  Prime )  =  ( ( 2 ... ( |_ `  A ) )  i^i 
Prime ) )
72 flid 11669 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  ( |_ `  A )  =  A )
7372adantr 465 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( |_ `  A
)  =  A )
7473oveq2d 6119 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( 2 ... ( |_ `  A ) )  =  ( 2 ... A ) )
7574ineq1d 3563 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( |_ `  A
) )  i^i  Prime )  =  ( ( 2 ... A )  i^i 
Prime ) )
7671, 75eqtrd 2475 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 0 [,] A )  i^i  Prime )  =  ( ( 2 ... A )  i^i 
Prime ) )
7776sumeq1d 13190 . . . 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 ) )
7869, 77eqtr2d 2476 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  -> 
sum_ p  e.  (
( 2 ... A
)  i^i  Prime ) ( log `  p )  =  ( theta `  A
) )
79 prmnn 13778 . . . . 5  |-  ( ( A  +  1 )  e.  Prime  ->  ( A  +  1 )  e.  NN )
8079adantl 466 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  NN )
8180nnrpd 11038 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  RR+ )
8281relogcld 22084 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( log `  ( A  +  1 ) )  e.  RR )
8382recnd 9424 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( log `  ( A  +  1 ) )  e.  CC )
84 fveq2 5703 . . . . 5  |-  ( p  =  ( A  + 
1 )  ->  ( log `  p )  =  ( log `  ( A  +  1 ) ) )
8584sumsn 13229 . . . 4  |-  ( ( ( A  +  1 )  e.  NN  /\  ( log `  ( A  +  1 ) )  e.  CC )  ->  sum_ p  e.  { ( A  +  1 ) }  ( log `  p
)  =  ( log `  ( A  +  1 ) ) )
8680, 83, 85syl2anc 661 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  -> 
sum_ p  e.  { ( A  +  1 ) }  ( log `  p
)  =  ( log `  ( A  +  1 ) ) )
8778, 86oveq12d 6121 . 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 ) ) ) )
8815, 67, 873eqtrd 2479 1  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( theta `  ( A  +  1 ) )  =  ( ( theta `  A )  +  ( log `  ( A  +  1 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    u. cun 3338    i^i cin 3339    C_ wss 3340   (/)c0 3649   {csn 3889   class class class wbr 4304   ` cfv 5430  (class class class)co 6103   Fincfn 7322   CCcc 9292   RRcr 9293   0cc0 9294   1c1 9295    + caddc 9297    < clt 9430    <_ cle 9431    - cmin 9607   NNcn 10334   2c2 10383   ZZcz 10658   ZZ>=cuz 10873   [,]cicc 11315   ...cfz 11449   |_cfl 11652   sum_csu 13175   Primecprime 13775   logclog 22018   thetaccht 22440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372  ax-addf 9373  ax-mulf 9374
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-iin 4186  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-of 6332  df-om 6489  df-1st 6589  df-2nd 6590  df-supp 6703  df-recs 6844  df-rdg 6878  df-1o 6932  df-2o 6933  df-oadd 6936  df-er 7113  df-map 7228  df-pm 7229  df-ixp 7276  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-fsupp 7633  df-fi 7673  df-sup 7703  df-oi 7736  df-card 8121  df-cda 8349  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-4 10394  df-5 10395  df-6 10396  df-7 10397  df-8 10398  df-9 10399  df-10 10400  df-n0 10592  df-z 10659  df-dec 10768  df-uz 10874  df-q 10966  df-rp 11004  df-xneg 11101  df-xadd 11102  df-xmul 11103  df-ioo 11316  df-ioc 11317  df-ico 11318  df-icc 11319  df-fz 11450  df-fzo 11561  df-fl 11654  df-mod 11721  df-seq 11819  df-exp 11878  df-fac 12064  df-bc 12091  df-hash 12116  df-shft 12568  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-limsup 12961  df-clim 12978  df-rlim 12979  df-sum 13176  df-ef 13365  df-sin 13367  df-cos 13368  df-pi 13370  df-dvds 13548  df-prm 13776  df-struct 14188  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-mulr 14264  df-starv 14265  df-sca 14266  df-vsca 14267  df-ip 14268  df-tset 14269  df-ple 14270  df-ds 14272  df-unif 14273  df-hom 14274  df-cco 14275  df-rest 14373  df-topn 14374  df-0g 14392  df-gsum 14393  df-topgen 14394  df-pt 14395  df-prds 14398  df-xrs 14452  df-qtop 14457  df-imas 14458  df-xps 14460  df-mre 14536  df-mrc 14537  df-acs 14539  df-mnd 15427  df-submnd 15477  df-mulg 15560  df-cntz 15847  df-cmn 16291  df-psmet 17821  df-xmet 17822  df-met 17823  df-bl 17824  df-mopn 17825  df-fbas 17826  df-fg 17827  df-cnfld 17831  df-top 18515  df-bases 18517  df-topon 18518  df-topsp 18519  df-cld 18635  df-ntr 18636  df-cls 18637  df-nei 18714  df-lp 18752  df-perf 18753  df-cn 18843  df-cnp 18844  df-haus 18931  df-tx 19147  df-hmeo 19340  df-fil 19431  df-fm 19523  df-flim 19524  df-flf 19525  df-xms 19907  df-ms 19908  df-tms 19909  df-cncf 20466  df-limc 21353  df-dv 21354  df-log 22020  df-cht 22446
This theorem is referenced by:  cht2  22522  cht3  22523
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