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Theorem pclogsum 22576
Description: The logarithmic analogue of pcprod 13978. 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 3560 . . . . . 6  |-  ( p  e.  ( ( 1 ... A )  i^i 
Prime )  <->  ( p  e.  ( 1 ... A
)  /\  p  e.  Prime ) )
21baib 896 . . . . 5  |-  ( p  e.  ( 1 ... A )  ->  (
p  e.  ( ( 1 ... A )  i^i  Prime )  <->  p  e.  Prime ) )
32ifbid 3832 . . . 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 5723 . . . . 5  |-  ( log `  if ( p  e. 
Prime ,  ( p ^ ( p  pCnt  A ) ) ,  1 ) )  =  if ( p  e.  Prime ,  ( log `  (
p ^ ( p 
pCnt  A ) ) ) ,  ( log `  1
) )
5 log1 22056 . . . . . 6  |-  ( log `  1 )  =  0
6 ifeq2 3817 . . . . . 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 5 . . . . 5  |-  if ( p  e.  Prime ,  ( log `  ( p ^ ( p  pCnt  A ) ) ) ,  ( log `  1
) )  =  if ( p  e.  Prime ,  ( log `  (
p ^ ( p 
pCnt  A ) ) ) ,  0 )
84, 7eqtri 2463 . . . 4  |-  ( log `  if ( p  e. 
Prime ,  ( p ^ ( p  pCnt  A ) ) ,  1 ) )  =  if ( p  e.  Prime ,  ( log `  (
p ^ ( p 
pCnt  A ) ) ) ,  0 )
93, 8syl6eqr 2493 . . 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 13197 . 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 3591 . . . 4  |-  ( ( 1 ... A )  i^i  Prime )  C_  (
1 ... A )
12 simpr 461 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  p  e.  ( ( 1 ... A )  i^i  Prime ) )
1311, 12sseldi 3375 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  p  e.  ( 1 ... A
) )
14 elfznn 11499 . . . . . . . . . 10  |-  ( p  e.  ( 1 ... A )  ->  p  e.  NN )
1513, 14syl 16 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  p  e.  NN )
16 inss2 3592 . . . . . . . . . . 11  |-  ( ( 1 ... A )  i^i  Prime )  C_  Prime
1716, 12sseldi 3375 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  p  e.  Prime )
18 simpl 457 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  A  e.  NN )
1917, 18pccld 13938 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  (
p  pCnt  A )  e.  NN0 )
2015, 19nnexpcld 12050 . . . . . . . 8  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  (
p ^ ( p 
pCnt  A ) )  e.  NN )
2120nnrpd 11047 . . . . . . 7  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  (
p ^ ( p 
pCnt  A ) )  e.  RR+ )
2221relogcld 22094 . . . . . 6  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  ( log `  ( p ^
( p  pCnt  A
) ) )  e.  RR )
2322recnd 9433 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  ( log `  ( p ^
( p  pCnt  A
) ) )  e.  CC )
2423ralrimiva 2820 . . . 4  |-  ( A  e.  NN  ->  A. p  e.  ( ( 1 ... A )  i^i  Prime ) ( log `  (
p ^ ( p 
pCnt  A ) ) )  e.  CC )
25 fzfi 11815 . . . . . 6  |-  ( 1 ... A )  e. 
Fin
2625olci 391 . . . . 5  |-  ( ( 1 ... A ) 
C_  ( ZZ>= `  1
)  \/  ( 1 ... A )  e. 
Fin )
27 sumss2 13224 . . . . 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 671 . . . 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 663 . . 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 11047 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  p  e.  RR+ )
3119nn0zd 10766 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  (
p  pCnt  A )  e.  ZZ )
32 relogexp 22066 . . . . 5  |-  ( ( p  e.  RR+  /\  (
p  pCnt  A )  e.  ZZ )  ->  ( log `  ( p ^
( p  pCnt  A
) ) )  =  ( ( p  pCnt  A )  x.  ( log `  p ) ) )
3330, 31, 32syl2anc 661 . . . 4  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  ( log `  ( p ^
( p  pCnt  A
) ) )  =  ( ( p  pCnt  A )  x.  ( log `  p ) ) )
3433sumeq2dv 13201 . . 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 2477 . 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 466 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  ->  p  e.  NN )
37 eleq1 2503 . . . . . . . 8  |-  ( n  =  p  ->  (
n  e.  Prime  <->  p  e.  Prime ) )
38 id 22 . . . . . . . . 9  |-  ( n  =  p  ->  n  =  p )
39 oveq1 6119 . . . . . . . . 9  |-  ( n  =  p  ->  (
n  pCnt  A )  =  ( p  pCnt  A ) )
4038, 39oveq12d 6130 . . . . . . . 8  |-  ( n  =  p  ->  (
n ^ ( n 
pCnt  A ) )  =  ( p ^ (
p  pCnt  A )
) )
4137, 40ifbieq1d 3833 . . . . . . 7  |-  ( n  =  p  ->  if ( n  e.  Prime ,  ( n ^ (
n  pCnt  A )
) ,  1 )  =  if ( p  e.  Prime ,  ( p ^ ( p  pCnt  A ) ) ,  1 ) )
4241fveq2d 5716 . . . . . 6  |-  ( n  =  p  ->  ( log `  if ( n  e.  Prime ,  ( n ^ ( n  pCnt  A ) ) ,  1 ) )  =  ( log `  if ( p  e.  Prime ,  ( p ^ ( p 
pCnt  A ) ) ,  1 ) ) )
43 eqid 2443 . . . . . 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 ) ) )
44 fvex 5722 . . . . . 6  |-  ( log `  if ( p  e. 
Prime ,  ( p ^ ( p  pCnt  A ) ) ,  1 ) )  e.  _V
4542, 43, 44fvmpt 5795 . . . . 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 ) ) )
4636, 45syl 16 . . . 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 ) ) )
47 elnnuz 10918 . . . . 5  |-  ( A  e.  NN  <->  A  e.  ( ZZ>= `  1 )
)
4847biimpi 194 . . . 4  |-  ( A  e.  NN  ->  A  e.  ( ZZ>= `  1 )
)
4936adantr 465 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  /\  p  e.  Prime )  ->  p  e.  NN )
50 simpr 461 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  /\  p  e.  Prime )  ->  p  e.  Prime )
51 simpll 753 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  /\  p  e.  Prime )  ->  A  e.  NN )
5250, 51pccld 13938 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  /\  p  e.  Prime )  ->  ( p  pCnt  A )  e.  NN0 )
5349, 52nnexpcld 12050 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  /\  p  e.  Prime )  ->  ( p ^
( p  pCnt  A
) )  e.  NN )
54 1nn 10354 . . . . . . . . 9  |-  1  e.  NN
5554a1i 11 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  /\  -.  p  e. 
Prime )  ->  1  e.  NN )
5653, 55ifclda 3842 . . . . . . 7  |-  ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  ->  if ( p  e.  Prime ,  ( p ^ ( p  pCnt  A ) ) ,  1 )  e.  NN )
5756nnrpd 11047 . . . . . 6  |-  ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  ->  if ( p  e.  Prime ,  ( p ^ ( p  pCnt  A ) ) ,  1 )  e.  RR+ )
5857relogcld 22094 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  ->  ( log `  if ( p  e.  Prime ,  ( p ^ (
p  pCnt  A )
) ,  1 ) )  e.  RR )
5958recnd 9433 . . . 4  |-  ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  ->  ( log `  if ( p  e.  Prime ,  ( p ^ (
p  pCnt  A )
) ,  1 ) )  e.  CC )
6046, 48, 59fsumser 13228 . . 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 ) )
61 rpmulcl 11033 . . . . 5  |-  ( ( p  e.  RR+  /\  m  e.  RR+ )  ->  (
p  x.  m )  e.  RR+ )
6261adantl 466 . . . 4  |-  ( ( A  e.  NN  /\  ( p  e.  RR+  /\  m  e.  RR+ ) )  -> 
( p  x.  m
)  e.  RR+ )
63 eqid 2443 . . . . . . 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 ) )
64 ovex 6137 . . . . . . . 8  |-  ( p ^ ( p  pCnt  A ) )  e.  _V
65 1ex 9402 . . . . . . . 8  |-  1  e.  _V
6664, 65ifex 3879 . . . . . . 7  |-  if ( p  e.  Prime ,  ( p ^ ( p 
pCnt  A ) ) ,  1 )  e.  _V
6741, 63, 66fvmpt 5795 . . . . . 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 ) )
6836, 67syl 16 . . . . 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 ) )
6968, 57eqeltrd 2517 . . . 4  |-  ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  ->  ( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ ( n 
pCnt  A ) ) ,  1 ) ) `  p )  e.  RR+ )
70 relogmul 22062 . . . . 5  |-  ( ( p  e.  RR+  /\  m  e.  RR+ )  ->  ( log `  ( p  x.  m ) )  =  ( ( log `  p
)  +  ( log `  m ) ) )
7170adantl 466 . . . 4  |-  ( ( A  e.  NN  /\  ( p  e.  RR+  /\  m  e.  RR+ ) )  -> 
( log `  (
p  x.  m ) )  =  ( ( log `  p )  +  ( log `  m
) ) )
7268fveq2d 5716 . . . . 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 ) ) )
7372, 46eqtr4d 2478 . . . 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 ) )
7462, 69, 48, 71, 73seqhomo 11874 . . 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 ) )
7563pcprod 13978 . . . 4  |-  ( A  e.  NN  ->  (  seq 1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ (
n  pCnt  A )
) ,  1 ) ) ) `  A
)  =  A )
7675fveq2d 5716 . . 3  |-  ( A  e.  NN  ->  ( log `  (  seq 1
(  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ (
n  pCnt  A )
) ,  1 ) ) ) `  A
) )  =  ( log `  A ) )
7760, 74, 763eqtr2d 2481 . 2  |-  ( A  e.  NN  ->  sum_ p  e.  ( 1 ... A
) ( log `  if ( p  e.  Prime ,  ( p ^ (
p  pCnt  A )
) ,  1 ) )  =  ( log `  A ) )
7810, 35, 773eqtr3a 2499 1  |-  ( A  e.  NN  ->  sum_ p  e.  ( ( 1 ... A )  i^i  Prime ) ( ( p  pCnt  A )  x.  ( log `  p ) )  =  ( log `  A
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2736    i^i cin 3348    C_ wss 3349   ifcif 3812    e. cmpt 4371   ` cfv 5439  (class class class)co 6112   Fincfn 7331   CCcc 9301   0cc0 9303   1c1 9304    + caddc 9306    x. cmul 9308   NNcn 10343   ZZcz 10667   ZZ>=cuz 10882   RR+crp 11012   ...cfz 11458    seqcseq 11827   ^cexp 11886   sum_csu 13184   Primecprime 13784    pCnt cpc 13924   logclog 22028
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 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381  ax-addf 9382  ax-mulf 9383
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 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-of 6341  df-om 6498  df-1st 6598  df-2nd 6599  df-supp 6712  df-recs 6853  df-rdg 6887  df-1o 6941  df-2o 6942  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-ixp 7285  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-fsupp 7642  df-fi 7682  df-sup 7712  df-oi 7745  df-card 8130  df-cda 8358  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-9 10408  df-10 10409  df-n0 10601  df-z 10668  df-dec 10777  df-uz 10883  df-q 10975  df-rp 11013  df-xneg 11110  df-xadd 11111  df-xmul 11112  df-ioo 11325  df-ioc 11326  df-ico 11327  df-icc 11328  df-fz 11459  df-fzo 11570  df-fl 11663  df-mod 11730  df-seq 11828  df-exp 11887  df-fac 12073  df-bc 12100  df-hash 12125  df-shft 12577  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-limsup 12970  df-clim 12987  df-rlim 12988  df-sum 13185  df-ef 13374  df-sin 13376  df-cos 13377  df-pi 13379  df-dvds 13557  df-gcd 13712  df-prm 13785  df-pc 13925  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-mulr 14273  df-starv 14274  df-sca 14275  df-vsca 14276  df-ip 14277  df-tset 14278  df-ple 14279  df-ds 14281  df-unif 14282  df-hom 14283  df-cco 14284  df-rest 14382  df-topn 14383  df-0g 14401  df-gsum 14402  df-topgen 14403  df-pt 14404  df-prds 14407  df-xrs 14461  df-qtop 14466  df-imas 14467  df-xps 14469  df-mre 14545  df-mrc 14546  df-acs 14548  df-mnd 15436  df-submnd 15486  df-mulg 15569  df-cntz 15856  df-cmn 16300  df-psmet 17831  df-xmet 17832  df-met 17833  df-bl 17834  df-mopn 17835  df-fbas 17836  df-fg 17837  df-cnfld 17841  df-top 18525  df-bases 18527  df-topon 18528  df-topsp 18529  df-cld 18645  df-ntr 18646  df-cls 18647  df-nei 18724  df-lp 18762  df-perf 18763  df-cn 18853  df-cnp 18854  df-haus 18941  df-tx 19157  df-hmeo 19350  df-fil 19441  df-fm 19533  df-flim 19534  df-flf 19535  df-xms 19917  df-ms 19918  df-tms 19919  df-cncf 20476  df-limc 21363  df-dv 21364  df-log 22030
This theorem is referenced by:  vmasum  22577  chebbnd1lem1  22740
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