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Theorem pclogsum 23246
Description: The logarithmic analogue of pcprod 14273. 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 3687 . . . . . 6  |-  ( p  e.  ( ( 1 ... A )  i^i 
Prime )  <->  ( p  e.  ( 1 ... A
)  /\  p  e.  Prime ) )
21baib 901 . . . . 5  |-  ( p  e.  ( 1 ... A )  ->  (
p  e.  ( ( 1 ... A )  i^i  Prime )  <->  p  e.  Prime ) )
32ifbid 3961 . . . 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 5877 . . . . 5  |-  ( log `  if ( p  e. 
Prime ,  ( p ^ ( p  pCnt  A ) ) ,  1 ) )  =  if ( p  e.  Prime ,  ( log `  (
p ^ ( p 
pCnt  A ) ) ) ,  ( log `  1
) )
5 log1 22726 . . . . . 6  |-  ( log `  1 )  =  0
6 ifeq2 3944 . . . . . 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 2496 . . . 4  |-  ( log `  if ( p  e. 
Prime ,  ( p ^ ( p  pCnt  A ) ) ,  1 ) )  =  if ( p  e.  Prime ,  ( log `  (
p ^ ( p 
pCnt  A ) ) ) ,  0 )
93, 8syl6eqr 2526 . . 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 13484 . 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 3718 . . . 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 3502 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  p  e.  ( 1 ... A
) )
14 elfznn 11714 . . . . . . . . . 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 3719 . . . . . . . . . . 11  |-  ( ( 1 ... A )  i^i  Prime )  C_  Prime
1716, 12sseldi 3502 . . . . . . . . . 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 14233 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  (
p  pCnt  A )  e.  NN0 )
2015, 19nnexpcld 12299 . . . . . . . 8  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  (
p ^ ( p 
pCnt  A ) )  e.  NN )
2120nnrpd 11255 . . . . . . 7  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  (
p ^ ( p 
pCnt  A ) )  e.  RR+ )
2221relogcld 22764 . . . . . 6  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  ( log `  ( p ^
( p  pCnt  A
) ) )  e.  RR )
2322recnd 9622 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  ( log `  ( p ^
( p  pCnt  A
) ) )  e.  CC )
2423ralrimiva 2878 . . . 4  |-  ( A  e.  NN  ->  A. p  e.  ( ( 1 ... A )  i^i  Prime ) ( log `  (
p ^ ( p 
pCnt  A ) ) )  e.  CC )
25 fzfi 12050 . . . . . 6  |-  ( 1 ... A )  e. 
Fin
2625olci 391 . . . . 5  |-  ( ( 1 ... A ) 
C_  ( ZZ>= `  1
)  \/  ( 1 ... A )  e. 
Fin )
27 sumss2 13511 . . . . 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 11255 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  p  e.  RR+ )
3119nn0zd 10964 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( (
1 ... A )  i^i 
Prime ) )  ->  (
p  pCnt  A )  e.  ZZ )
32 relogexp 22736 . . . . 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 13488 . . 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 2510 . 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 2539 . . . . . . . 8  |-  ( n  =  p  ->  (
n  e.  Prime  <->  p  e.  Prime ) )
38 id 22 . . . . . . . . 9  |-  ( n  =  p  ->  n  =  p )
39 oveq1 6291 . . . . . . . . 9  |-  ( n  =  p  ->  (
n  pCnt  A )  =  ( p  pCnt  A ) )
4038, 39oveq12d 6302 . . . . . . . 8  |-  ( n  =  p  ->  (
n ^ ( n 
pCnt  A ) )  =  ( p ^ (
p  pCnt  A )
) )
4137, 40ifbieq1d 3962 . . . . . . 7  |-  ( n  =  p  ->  if ( n  e.  Prime ,  ( n ^ (
n  pCnt  A )
) ,  1 )  =  if ( p  e.  Prime ,  ( p ^ ( p  pCnt  A ) ) ,  1 ) )
4241fveq2d 5870 . . . . . 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 2467 . . . . . 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 5876 . . . . . 6  |-  ( log `  if ( p  e. 
Prime ,  ( p ^ ( p  pCnt  A ) ) ,  1 ) )  e.  _V
4542, 43, 44fvmpt 5950 . . . . 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 11118 . . . . 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 14233 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  /\  p  e.  Prime )  ->  ( p  pCnt  A )  e.  NN0 )
5349, 52nnexpcld 12299 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  /\  p  e.  Prime )  ->  ( p ^
( p  pCnt  A
) )  e.  NN )
54 1nn 10547 . . . . . . . . 9  |-  1  e.  NN
5554a1i 11 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  /\  -.  p  e. 
Prime )  ->  1  e.  NN )
5653, 55ifclda 3971 . . . . . . 7  |-  ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  ->  if ( p  e.  Prime ,  ( p ^ ( p  pCnt  A ) ) ,  1 )  e.  NN )
5756nnrpd 11255 . . . . . 6  |-  ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  ->  if ( p  e.  Prime ,  ( p ^ ( p  pCnt  A ) ) ,  1 )  e.  RR+ )
5857relogcld 22764 . . . . 5  |-  ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  ->  ( log `  if ( p  e.  Prime ,  ( p ^ (
p  pCnt  A )
) ,  1 ) )  e.  RR )
5958recnd 9622 . . . 4  |-  ( ( A  e.  NN  /\  p  e.  ( 1 ... A ) )  ->  ( log `  if ( p  e.  Prime ,  ( p ^ (
p  pCnt  A )
) ,  1 ) )  e.  CC )
6046, 48, 59fsumser 13515 . . 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 11241 . . . . 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 2467 . . . . . . 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 6309 . . . . . . . 8  |-  ( p ^ ( p  pCnt  A ) )  e.  _V
65 1ex 9591 . . . . . . . 8  |-  1  e.  _V
6664, 65ifex 4008 . . . . . . 7  |-  if ( p  e.  Prime ,  ( p ^ ( p 
pCnt  A ) ) ,  1 )  e.  _V
6741, 63, 66fvmpt 5950 . . . . . 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 2555 . . . 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 22732 . . . . 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 5870 . . . . 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 2511 . . . 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 12122 . . 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 14273 . . . 4  |-  ( A  e.  NN  ->  (  seq 1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ (
n  pCnt  A )
) ,  1 ) ) ) `  A
)  =  A )
7675fveq2d 5870 . . 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 2514 . 2  |-  ( A  e.  NN  ->  sum_ p  e.  ( 1 ... A
) ( log `  if ( p  e.  Prime ,  ( p ^ (
p  pCnt  A )
) ,  1 ) )  =  ( log `  A ) )
7810, 35, 773eqtr3a 2532 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 1379    e. wcel 1767   A.wral 2814    i^i cin 3475    C_ wss 3476   ifcif 3939    |-> cmpt 4505   ` cfv 5588  (class class class)co 6284   Fincfn 7516   CCcc 9490   0cc0 9492   1c1 9493    + caddc 9495    x. cmul 9497   NNcn 10536   ZZcz 10864   ZZ>=cuz 11082   RR+crp 11220   ...cfz 11672    seqcseq 12075   ^cexp 12134   sum_csu 13471   Primecprime 14076    pCnt cpc 14219   logclog 22698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-ioc 11534  df-ico 11535  df-icc 11536  df-fz 11673  df-fzo 11793  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-fac 12322  df-bc 12349  df-hash 12374  df-shft 12863  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-limsup 13257  df-clim 13274  df-rlim 13275  df-sum 13472  df-ef 13665  df-sin 13667  df-cos 13668  df-pi 13670  df-dvds 13848  df-gcd 14004  df-prm 14077  df-pc 14220  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-rest 14678  df-topn 14679  df-0g 14697  df-gsum 14698  df-topgen 14699  df-pt 14700  df-prds 14703  df-xrs 14757  df-qtop 14762  df-imas 14763  df-xps 14765  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-submnd 15787  df-mulg 15870  df-cntz 16160  df-cmn 16606  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-fbas 18215  df-fg 18216  df-cnfld 18220  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cld 19314  df-ntr 19315  df-cls 19316  df-nei 19393  df-lp 19431  df-perf 19432  df-cn 19522  df-cnp 19523  df-haus 19610  df-tx 19826  df-hmeo 20019  df-fil 20110  df-fm 20202  df-flim 20203  df-flf 20204  df-xms 20586  df-ms 20587  df-tms 20588  df-cncf 21145  df-limc 22033  df-dv 22034  df-log 22700
This theorem is referenced by:  vmasum  23247  chebbnd1lem1  23410
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