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Theorem chtprm 23625
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 10901 . . . . . 6  |-  ( A  e.  ZZ  ->  ( A  +  1 )  e.  ZZ )
21adantr 463 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  ZZ )
3 zre 10864 . . . . 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 23582 . . . 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 23575 . . . . . 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 11926 . . . . . . . 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 6286 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( 2 ... ( |_ `  ( A  + 
1 ) ) )  =  ( 2 ... ( A  +  1 ) ) )
1211ineq1d 3685 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( |_ `  ( A  +  1 ) ) )  i^i  Prime )  =  ( ( 2 ... ( A  + 
1 ) )  i^i 
Prime ) )
138, 12eqtrd 2495 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 0 [,] ( A  +  1 ) )  i^i  Prime )  =  ( ( 2 ... ( A  + 
1 ) )  i^i 
Prime ) )
1413sumeq1d 13605 . . 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 2495 . 2  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( theta `  ( A  +  1 ) )  =  sum_ p  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
( log `  p
) )
16 zre 10864 . . . . . . . 8  |-  ( A  e.  ZZ  ->  A  e.  RR )
1716adantr 463 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  RR )
1817ltp1d 10471 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  <  ( A  +  1 ) )
1917, 4ltnled 9721 . . . . . 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 3704 . . . . . . 7  |-  ( ( 2 ... A )  i^i  Prime )  C_  (
2 ... A )
2221sseli 3485 . . . . . 6  |-  ( ( A  +  1 )  e.  ( ( 2 ... A )  i^i 
Prime )  ->  ( A  +  1 )  e.  ( 2 ... A
) )
23 elfzle2 11693 . . . . . 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 4076 . . . 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 10892 . . . . . . 7  |-  2  e.  ZZ
29 zcn 10865 . . . . . . . . . . 11  |-  ( A  e.  ZZ  ->  A  e.  CC )
3029adantr 463 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  CC )
31 ax-1cn 9539 . . . . . . . . . 10  |-  1  e.  CC
32 pncan 9817 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  + 
1 )  -  1 )  =  A )
3330, 31, 32sylancl 660 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( A  + 
1 )  -  1 )  =  A )
34 prmuz2 14319 . . . . . . . . . . 11  |-  ( ( A  +  1 )  e.  Prime  ->  ( A  +  1 )  e.  ( ZZ>= `  2 )
)
3534adantl 464 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  ( ZZ>= ` 
2 ) )
36 uz2m1nn 11157 . . . . . . . . . 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 2543 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  NN )
39 nnuz 11117 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
40 2m1e1 10646 . . . . . . . . . 10  |-  ( 2  -  1 )  =  1
4140fveq2i 5851 . . . . . . . . 9  |-  ( ZZ>= `  ( 2  -  1 ) )  =  (
ZZ>= `  1 )
4239, 41eqtr4i 2486 . . . . . . . 8  |-  NN  =  ( ZZ>= `  ( 2  -  1 ) )
4338, 42syl6eleq 2552 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  ( ZZ>= `  ( 2  -  1 ) ) )
44 fzsuc2 11741 . . . . . . 7  |-  ( ( 2  e.  ZZ  /\  A  e.  ( ZZ>= `  ( 2  -  1 ) ) )  -> 
( 2 ... ( A  +  1 ) )  =  ( ( 2 ... A )  u.  { ( A  +  1 ) } ) )
4528, 43, 44sylancr 661 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( 2 ... ( A  +  1 ) )  =  ( ( 2 ... A )  u.  { ( A  +  1 ) } ) )
4645ineq1d 3685 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  =  ( ( ( 2 ... A )  u.  { ( A  +  1 ) } )  i^i  Prime )
)
47 indir 3743 . . . . 5  |-  ( ( ( 2 ... A
)  u.  { ( A  +  1 ) } )  i^i  Prime )  =  ( ( ( 2 ... A )  i^i  Prime )  u.  ( { ( A  + 
1 ) }  i^i  Prime
) )
4846, 47syl6eq 2511 . . . 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 459 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  Prime )
5049snssd 4161 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  { ( A  + 
1 ) }  C_  Prime )
51 df-ss 3475 . . . . . 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 3644 . . . 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 2495 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  =  ( ( ( 2 ... A )  i^i  Prime )  u.  {
( A  +  1 ) } ) )
55 fzfid 12065 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( 2 ... ( A  +  1 ) )  e.  Fin )
56 inss1 3704 . . . 4  |-  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  C_  (
2 ... ( A  + 
1 ) )
57 ssfi 7733 . . . 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 660 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  e.  Fin )
59 inss2 3705 . . . . . . . 8  |-  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  C_  Prime
60 simpr 459 . . . . . . . 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 3487 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  ( A  +  1 )  e.  Prime )  /\  p  e.  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
)  ->  p  e.  Prime )
62 prmnn 14304 . . . . . . 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 11257 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  ( A  +  1 )  e.  Prime )  /\  p  e.  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
)  ->  p  e.  RR+ )
6564relogcld 23176 . . . 4  |-  ( ( ( A  e.  ZZ  /\  ( A  +  1 )  e.  Prime )  /\  p  e.  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
)  ->  ( log `  p )  e.  RR )
6665recnd 9611 . . 3  |-  ( ( ( A  e.  ZZ  /\  ( A  +  1 )  e.  Prime )  /\  p  e.  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
)  ->  ( log `  p )  e.  CC )
6727, 54, 58, 66fsumsplit 13644 . 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 23582 . . . . 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 23575 . . . . . . 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 11926 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  ( |_ `  A )  =  A )
7372adantr 463 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( |_ `  A
)  =  A )
7473oveq2d 6286 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( 2 ... ( |_ `  A ) )  =  ( 2 ... A ) )
7574ineq1d 3685 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( |_ `  A
) )  i^i  Prime )  =  ( ( 2 ... A )  i^i 
Prime ) )
7671, 75eqtrd 2495 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 0 [,] A )  i^i  Prime )  =  ( ( 2 ... A )  i^i 
Prime ) )
7776sumeq1d 13605 . . . 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 2496 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  -> 
sum_ p  e.  (
( 2 ... A
)  i^i  Prime ) ( log `  p )  =  ( theta `  A
) )
79 prmnn 14304 . . . . 5  |-  ( ( A  +  1 )  e.  Prime  ->  ( A  +  1 )  e.  NN )
8079adantl 464 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  NN )
8180nnrpd 11257 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  RR+ )
8281relogcld 23176 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( log `  ( A  +  1 ) )  e.  RR )
8382recnd 9611 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( log `  ( A  +  1 ) )  e.  CC )
84 fveq2 5848 . . . . 5  |-  ( p  =  ( A  + 
1 )  ->  ( log `  p )  =  ( log `  ( A  +  1 ) ) )
8584sumsn 13645 . . . 4  |-  ( ( ( A  +  1 )  e.  NN  /\  ( log `  ( A  +  1 ) )  e.  CC )  ->  sum_ p  e.  { ( A  +  1 ) }  ( log `  p
)  =  ( log `  ( A  +  1 ) ) )
8680, 83, 85syl2anc 659 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  -> 
sum_ p  e.  { ( A  +  1 ) }  ( log `  p
)  =  ( log `  ( A  +  1 ) ) )
8778, 86oveq12d 6288 . 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 2499 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 367    = wceq 1398    e. wcel 1823    u. cun 3459    i^i cin 3460    C_ wss 3461   (/)c0 3783   {csn 4016   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Fincfn 7509   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    < clt 9617    <_ cle 9618    - cmin 9796   NNcn 10531   2c2 10581   ZZcz 10860   ZZ>=cuz 11082   [,]cicc 11535   ...cfz 11675   |_cfl 11908   sum_csu 13590   Primecprime 14301   logclog 23108   thetaccht 23562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ioc 11537  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12090  df-exp 12149  df-fac 12336  df-bc 12363  df-hash 12388  df-shft 12982  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-limsup 13376  df-clim 13393  df-rlim 13394  df-sum 13591  df-ef 13885  df-sin 13887  df-cos 13888  df-pi 13890  df-dvds 14071  df-prm 14302  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-hom 14808  df-cco 14809  df-rest 14912  df-topn 14913  df-0g 14931  df-gsum 14932  df-topgen 14933  df-pt 14934  df-prds 14937  df-xrs 14991  df-qtop 14996  df-imas 14997  df-xps 14999  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-mulg 16259  df-cntz 16554  df-cmn 16999  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-fbas 18611  df-fg 18612  df-cnfld 18616  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cld 19687  df-ntr 19688  df-cls 19689  df-nei 19766  df-lp 19804  df-perf 19805  df-cn 19895  df-cnp 19896  df-haus 19983  df-tx 20229  df-hmeo 20422  df-fil 20513  df-fm 20605  df-flim 20606  df-flf 20607  df-xms 20989  df-ms 20990  df-tms 20991  df-cncf 21548  df-limc 22436  df-dv 22437  df-log 23110  df-cht 23568
This theorem is referenced by:  cht2  23644  cht3  23645
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