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Theorem pclogsum 24006
Description: The logarithmic analogue of pcprod 14803. 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 3655 . . . . . 6  |-  ( p  e.  ( ( 1 ... A )  i^i 
Prime )  <->  ( p  e.  ( 1 ... A
)  /\  p  e.  Prime ) )
21baib 911 . . . . 5  |-  ( p  e.  ( 1 ... A )  ->  (
p  e.  ( ( 1 ... A )  i^i  Prime )  <->  p  e.  Prime ) )
32ifbid 3937 . . . 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 5892 . . . . 5  |-  ( log `  if ( p  e. 
Prime ,  ( p ^ ( p  pCnt  A ) ) ,  1 ) )  =  if ( p  e.  Prime ,  ( log `  (
p ^ ( p 
pCnt  A ) ) ) ,  ( log `  1
) )
5 log1 23400 . . . . . 6  |-  ( log `  1 )  =  0
6 ifeq2 3920 . . . . . 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 2458 . . . 4  |-  ( log `  if ( p  e. 
Prime ,  ( p ^ ( p  pCnt  A ) ) ,  1 ) )  =  if ( p  e.  Prime ,  ( log `  (
p ^ ( p 
pCnt  A ) ) ) ,  0 )
93, 8syl6eqr 2488 . . 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 13743 . 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 3688 . . . 4  |-  ( ( 1 ... A )  i^i  Prime )  C_  (
1 ... A )
12 simpr 462 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  p  e.  ( ( 1 ... A )  i^i  Prime ) )
1311, 12sseldi 3468 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  p  e.  ( 1 ... A
) )
14 elfznn 11826 . . . . . . . . . 10  |-  ( p  e.  ( 1 ... A )  ->  p  e.  NN )
1513, 14syl 17 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  p  e.  NN )
16 inss2 3689 . . . . . . . . . . 11  |-  ( ( 1 ... A )  i^i  Prime )  C_  Prime
1716, 12sseldi 3468 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  p  e.  Prime )
18 simpl 458 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  A  e.  NN )
1917, 18pccld 14763 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  (
p  pCnt  A )  e.  NN0 )
2015, 19nnexpcld 12434 . . . . . . . 8  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  (
p ^ ( p 
pCnt  A ) )  e.  NN )
2120nnrpd 11339 . . . . . . 7  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  (
p ^ ( p 
pCnt  A ) )  e.  RR+ )
2221relogcld 23437 . . . . . 6  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  ( log `  ( p ^
( p  pCnt  A
) ) )  e.  RR )
2322recnd 9668 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  ( log `  ( p ^
( p  pCnt  A
) ) )  e.  CC )
2423ralrimiva 2846 . . . 4  |-  ( A  e.  NN  ->  A. p  e.  ( ( 1 ... A )  i^i  Prime ) ( log `  (
p ^ ( p 
pCnt  A ) ) )  e.  CC )
25 fzfi 12182 . . . . . 6  |-  ( 1 ... A )  e. 
Fin
2625olci 392 . . . . 5  |-  ( ( 1 ... A ) 
C_  ( ZZ>= `  1
)  \/  ( 1 ... A )  e. 
Fin )
27 sumss2 13770 . . . . 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 675 . . . 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 667 . . 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 11339 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  p  e.  RR+ )
3119nn0zd 11038 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  (
p  pCnt  A )  e.  ZZ )
32 relogexp 23410 . . . . 5  |-  ( ( p  e.  RR+  /\  (
p  pCnt  A )  e.  ZZ )  ->  ( log `  ( p ^
( p  pCnt  A
) ) )  =  ( ( p  pCnt  A )  x.  ( log `  p ) ) )
3330, 31, 32syl2anc 665 . . . 4  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  ( log `  ( p ^
( p  pCnt  A
) ) )  =  ( ( p  pCnt  A )  x.  ( log `  p ) ) )
3433sumeq2dv 13747 . . 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 2472 . 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 467 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  ->  p  e.  NN )
37 eleq1 2501 . . . . . . . 8  |-  ( n  =  p  ->  (
n  e.  Prime  <->  p  e.  Prime ) )
38 id 23 . . . . . . . . 9  |-  ( n  =  p  ->  n  =  p )
39 oveq1 6312 . . . . . . . . 9  |-  ( n  =  p  ->  (
n  pCnt  A )  =  ( p  pCnt  A ) )
4038, 39oveq12d 6323 . . . . . . . 8  |-  ( n  =  p  ->  (
n ^ ( n 
pCnt  A ) )  =  ( p ^ (
p  pCnt  A )
) )
4137, 40ifbieq1d 3938 . . . . . . 7  |-  ( n  =  p  ->  if ( n  e.  Prime ,  ( n ^ (
n  pCnt  A )
) ,  1 )  =  if ( p  e.  Prime ,  ( p ^ ( p  pCnt  A ) ) ,  1 ) )
4241fveq2d 5885 . . . . . 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 2429 . . . . . 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 5891 . . . . . 6  |-  ( log `  if ( p  e. 
Prime ,  ( p ^ ( p  pCnt  A ) ) ,  1 ) )  e.  _V
4542, 43, 44fvmpt 5964 . . . . 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 17 . . . 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 11195 . . . . 5  |-  ( A  e.  NN  <->  A  e.  ( ZZ>= `  1 )
)
4847biimpi 197 . . . 4  |-  ( A  e.  NN  ->  A  e.  ( ZZ>= `  1 )
)
4936adantr 466 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  /\  p  e.  Prime )  ->  p  e.  NN )
50 simpr 462 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  /\  p  e.  Prime )  ->  p  e.  Prime )
51 simpll 758 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  /\  p  e.  Prime )  ->  A  e.  NN )
5250, 51pccld 14763 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  /\  p  e.  Prime )  ->  ( p  pCnt  A )  e.  NN0 )
5349, 52nnexpcld 12434 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  /\  p  e.  Prime )  ->  ( p ^
( p  pCnt  A
) )  e.  NN )
54 1nn 10620 . . . . . . . . 9  |-  1  e.  NN
5554a1i 11 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  /\  -.  p  e. 
Prime )  ->  1  e.  NN )
5653, 55ifclda 3947 . . . . . . 7  |-  ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  ->  if ( p  e.  Prime ,  ( p ^ ( p  pCnt  A ) ) ,  1 )  e.  NN )
5756nnrpd 11339 . . . . . 6  |-  ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  ->  if ( p  e.  Prime ,  ( p ^ ( p  pCnt  A ) ) ,  1 )  e.  RR+ )
5857relogcld 23437 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  ->  ( log `  if ( p  e.  Prime ,  ( p ^ (
p  pCnt  A )
) ,  1 ) )  e.  RR )
5958recnd 9668 . . . 4  |-  ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  ->  ( log `  if ( p  e.  Prime ,  ( p ^ (
p  pCnt  A )
) ,  1 ) )  e.  CC )
6046, 48, 59fsumser 13774 . . 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 11324 . . . . 5  |-  ( ( p  e.  RR+  /\  m  e.  RR+ )  ->  (
p  x.  m )  e.  RR+ )
6261adantl 467 . . . 4  |-  ( ( A  e.  NN  /\  ( p  e.  RR+  /\  m  e.  RR+ ) )  -> 
( p  x.  m
)  e.  RR+ )
63 eqid 2429 . . . . . . 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 6333 . . . . . . . 8  |-  ( p ^ ( p  pCnt  A ) )  e.  _V
65 1ex 9637 . . . . . . . 8  |-  1  e.  _V
6664, 65ifex 3983 . . . . . . 7  |-  if ( p  e.  Prime ,  ( p ^ ( p 
pCnt  A ) ) ,  1 )  e.  _V
6741, 63, 66fvmpt 5964 . . . . . 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 17 . . . . 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 23406 . . . . 5  |-  ( ( p  e.  RR+  /\  m  e.  RR+ )  ->  ( log `  ( p  x.  m ) )  =  ( ( log `  p
)  +  ( log `  m ) ) )
7170adantl 467 . . . 4  |-  ( ( A  e.  NN  /\  ( p  e.  RR+  /\  m  e.  RR+ ) )  -> 
( log `  (
p  x.  m ) )  =  ( ( log `  p )  +  ( log `  m
) ) )
7268fveq2d 5885 . . . . 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 2473 . . . 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 12257 . . 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 14803 . . . 4  |-  ( A  e.  NN  ->  (  seq 1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ (
n  pCnt  A )
) ,  1 ) ) ) `  A
)  =  A )
7675fveq2d 5885 . . 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 2476 . 2  |-  ( A  e.  NN  ->  sum_ p  e.  ( 1 ... A
) ( log `  if ( p  e.  Prime ,  ( p ^ (
p  pCnt  A )
) ,  1 ) )  =  ( log `  A ) )
7810, 35, 773eqtr3a 2494 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 369    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782    i^i cin 3441    C_ wss 3442   ifcif 3915    |-> cmpt 4484   ` cfv 5601  (class class class)co 6305   Fincfn 7577   CCcc 9536   0cc0 9538   1c1 9539    + caddc 9541    x. cmul 9543   NNcn 10609   ZZcz 10937   ZZ>=cuz 11159   RR+crp 11302   ...cfz 11782    seqcseq 12210   ^cexp 12269   sum_csu 13730   Primecprime 14593    pCnt cpc 14749   logclog 23369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617  ax-mulf 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-fi 7931  df-sup 7962  df-inf 7963  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-shft 13109  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-limsup 13504  df-clim 13530  df-rlim 13531  df-sum 13731  df-ef 14099  df-sin 14101  df-cos 14102  df-pi 14104  df-dvds 14284  df-gcd 14443  df-prm 14594  df-pc 14750  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-starv 15167  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-hom 15176  df-cco 15177  df-rest 15280  df-topn 15281  df-0g 15299  df-gsum 15300  df-topgen 15301  df-pt 15302  df-prds 15305  df-xrs 15359  df-qtop 15364  df-imas 15365  df-xps 15367  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-mulg 16627  df-cntz 16922  df-cmn 17367  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-fbas 18902  df-fg 18903  df-cnfld 18906  df-top 19852  df-bases 19853  df-topon 19854  df-topsp 19855  df-cld 19965  df-ntr 19966  df-cls 19967  df-nei 20045  df-lp 20083  df-perf 20084  df-cn 20174  df-cnp 20175  df-haus 20262  df-tx 20508  df-hmeo 20701  df-fil 20792  df-fm 20884  df-flim 20885  df-flf 20886  df-xms 21266  df-ms 21267  df-tms 21268  df-cncf 21806  df-limc 22698  df-dv 22699  df-log 23371
This theorem is referenced by:  vmasum  24007  chebbnd1lem1  24170
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