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Theorem dvdsflsumcom 22503
Description: A sum commutation from  sum_ n  <_  A ,  sum_ d  ||  n ,  B (
n ,  d ) to  sum_ d  <_  A ,  sum_ m  <_  A  /  d ,  B
( n ,  d m ). (Contributed by Mario Carneiro, 4-May-2016.)
Hypotheses
Ref Expression
dvdsflsumcom.1  |-  ( n  =  ( d  x.  m )  ->  B  =  C )
dvdsflsumcom.2  |-  ( ph  ->  A  e.  RR )
dvdsflsumcom.3  |-  ( (
ph  /\  ( n  e.  ( 1 ... ( |_ `  A ) )  /\  d  e.  {
x  e.  NN  |  x  ||  n } ) )  ->  B  e.  CC )
Assertion
Ref Expression
dvdsflsumcom  |-  ( ph  -> 
sum_ n  e.  (
1 ... ( |_ `  A ) ) sum_ d  e.  { x  e.  NN  |  x  ||  n } B  =  sum_ d  e.  ( 1 ... ( |_ `  A ) ) sum_ m  e.  ( 1 ... ( |_ `  ( A  /  d ) ) ) C )
Distinct variable groups:    m, d, n, x, A    B, m    C, n    ph, d, m, n
Allowed substitution hints:    ph( x)    B( x, n, d)    C( x, m, d)

Proof of Theorem dvdsflsumcom
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fzfid 11787 . . 3  |-  ( ph  ->  ( 1 ... ( |_ `  A ) )  e.  Fin )
2 fzfid 11787 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1 ... n )  e. 
Fin )
3 elfznn 11470 . . . . . 6  |-  ( n  e.  ( 1 ... ( |_ `  A
) )  ->  n  e.  NN )
43adantl 466 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  n  e.  NN )
5 sgmss 22419 . . . . 5  |-  ( n  e.  NN  ->  { x  e.  NN  |  x  ||  n }  C_  ( 1 ... n ) )
64, 5syl 16 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  { x  e.  NN  |  x  ||  n }  C_  ( 1 ... n ) )
7 ssfi 7525 . . . 4  |-  ( ( ( 1 ... n
)  e.  Fin  /\  { x  e.  NN  |  x  ||  n }  C_  ( 1 ... n
) )  ->  { x  e.  NN  |  x  ||  n }  e.  Fin )
82, 6, 7syl2anc 661 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  { x  e.  NN  |  x  ||  n }  e.  Fin )
9 nnre 10321 . . . . . . . . . . . 12  |-  ( d  e.  NN  ->  d  e.  RR )
109ad2antrl 727 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  -> 
d  e.  RR )
114adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  ->  n  e.  NN )
1211nnred 10329 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  ->  n  e.  RR )
13 dvdsflsumcom.2 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  RR )
1413ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  ->  A  e.  RR )
15 nnz 10660 . . . . . . . . . . . . 13  |-  ( d  e.  NN  ->  d  e.  ZZ )
16 dvdsle 13570 . . . . . . . . . . . . 13  |-  ( ( d  e.  ZZ  /\  n  e.  NN )  ->  ( d  ||  n  ->  d  <_  n )
)
1715, 4, 16syl2anr 478 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  d  e.  NN )  ->  (
d  ||  n  ->  d  <_  n ) )
1817impr 619 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  -> 
d  <_  n )
19 fznnfl 11693 . . . . . . . . . . . . . 14  |-  ( A  e.  RR  ->  (
n  e.  ( 1 ... ( |_ `  A ) )  <->  ( n  e.  NN  /\  n  <_  A ) ) )
2013, 19syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( n  e.  ( 1 ... ( |_
`  A ) )  <-> 
( n  e.  NN  /\  n  <_  A )
) )
2120simplbda 624 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  n  <_  A )
2221adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  ->  n  <_  A )
2310, 12, 14, 18, 22letrd 9520 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  -> 
d  <_  A )
2423ex 434 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (
d  e.  NN  /\  d  ||  n )  -> 
d  <_  A )
)
2524pm4.71rd 635 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (
d  e.  NN  /\  d  ||  n )  <->  ( d  <_  A  /\  ( d  e.  NN  /\  d  ||  n ) ) ) )
26 ancom 450 . . . . . . . . 9  |-  ( ( d  <_  A  /\  ( d  e.  NN  /\  d  ||  n ) )  <->  ( ( d  e.  NN  /\  d  ||  n )  /\  d  <_  A ) )
27 an32 796 . . . . . . . . 9  |-  ( ( ( d  e.  NN  /\  d  ||  n )  /\  d  <_  A
)  <->  ( ( d  e.  NN  /\  d  <_  A )  /\  d  ||  n ) )
2826, 27bitri 249 . . . . . . . 8  |-  ( ( d  <_  A  /\  ( d  e.  NN  /\  d  ||  n ) )  <->  ( ( d  e.  NN  /\  d  <_  A )  /\  d  ||  n ) )
2925, 28syl6bb 261 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (
d  e.  NN  /\  d  ||  n )  <->  ( (
d  e.  NN  /\  d  <_  A )  /\  d  ||  n ) ) )
30 fznnfl 11693 . . . . . . . . . 10  |-  ( A  e.  RR  ->  (
d  e.  ( 1 ... ( |_ `  A ) )  <->  ( d  e.  NN  /\  d  <_  A ) ) )
3113, 30syl 16 . . . . . . . . 9  |-  ( ph  ->  ( d  e.  ( 1 ... ( |_
`  A ) )  <-> 
( d  e.  NN  /\  d  <_  A )
) )
3231adantr 465 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( d  e.  ( 1 ... ( |_ `  A ) )  <-> 
( d  e.  NN  /\  d  <_  A )
) )
3332anbi1d 704 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (
d  e.  ( 1 ... ( |_ `  A ) )  /\  d  ||  n )  <->  ( (
d  e.  NN  /\  d  <_  A )  /\  d  ||  n ) ) )
3429, 33bitr4d 256 . . . . . 6  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (
d  e.  NN  /\  d  ||  n )  <->  ( d  e.  ( 1 ... ( |_ `  A ) )  /\  d  ||  n
) ) )
3534pm5.32da 641 . . . . 5  |-  ( ph  ->  ( ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  <->  ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( d  e.  ( 1 ... ( |_ `  A ) )  /\  d  ||  n
) ) ) )
36 an12 795 . . . . 5  |-  ( ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( d  e.  ( 1 ... ( |_
`  A ) )  /\  d  ||  n
) )  <->  ( d  e.  ( 1 ... ( |_ `  A ) )  /\  ( n  e.  ( 1 ... ( |_ `  A ) )  /\  d  ||  n
) ) )
3735, 36syl6bb 261 . . . 4  |-  ( ph  ->  ( ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  <->  ( d  e.  ( 1 ... ( |_ `  A ) )  /\  ( n  e.  ( 1 ... ( |_ `  A ) )  /\  d  ||  n
) ) ) )
38 breq1 4290 . . . . . 6  |-  ( x  =  d  ->  (
x  ||  n  <->  d  ||  n ) )
3938elrab 3112 . . . . 5  |-  ( d  e.  { x  e.  NN  |  x  ||  n }  <->  ( d  e.  NN  /\  d  ||  n ) )
4039anbi2i 694 . . . 4  |-  ( ( n  e.  ( 1 ... ( |_ `  A ) )  /\  d  e.  { x  e.  NN  |  x  ||  n } )  <->  ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( d  e.  NN  /\  d  ||  n ) ) )
41 breq2 4291 . . . . . 6  |-  ( x  =  n  ->  (
d  ||  x  <->  d  ||  n ) )
4241elrab 3112 . . . . 5  |-  ( n  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  d  ||  x } 
<->  ( n  e.  ( 1 ... ( |_
`  A ) )  /\  d  ||  n
) )
4342anbi2i 694 . . . 4  |-  ( ( d  e.  ( 1 ... ( |_ `  A ) )  /\  n  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  d  ||  x } )  <->  ( d  e.  ( 1 ... ( |_ `  A ) )  /\  ( n  e.  ( 1 ... ( |_ `  A ) )  /\  d  ||  n
) ) )
4437, 40, 433bitr4g 288 . . 3  |-  ( ph  ->  ( ( n  e.  ( 1 ... ( |_ `  A ) )  /\  d  e.  {
x  e.  NN  |  x  ||  n } )  <-> 
( d  e.  ( 1 ... ( |_
`  A ) )  /\  n  e.  {
x  e.  ( 1 ... ( |_ `  A ) )  |  d  ||  x }
) ) )
45 dvdsflsumcom.3 . . 3  |-  ( (
ph  /\  ( n  e.  ( 1 ... ( |_ `  A ) )  /\  d  e.  {
x  e.  NN  |  x  ||  n } ) )  ->  B  e.  CC )
461, 1, 8, 44, 45fsumcom2 13233 . 2  |-  ( ph  -> 
sum_ n  e.  (
1 ... ( |_ `  A ) ) sum_ d  e.  { x  e.  NN  |  x  ||  n } B  =  sum_ d  e.  ( 1 ... ( |_ `  A ) ) sum_ n  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  d  ||  x } B )
47 dvdsflsumcom.1 . . . 4  |-  ( n  =  ( d  x.  m )  ->  B  =  C )
48 fzfid 11787 . . . 4  |-  ( (
ph  /\  d  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1 ... ( |_ `  ( A  /  d
) ) )  e. 
Fin )
4913adantr 465 . . . . 5  |-  ( (
ph  /\  d  e.  ( 1 ... ( |_ `  A ) ) )  ->  A  e.  RR )
5031simprbda 623 . . . . 5  |-  ( (
ph  /\  d  e.  ( 1 ... ( |_ `  A ) ) )  ->  d  e.  NN )
51 eqid 2438 . . . . 5  |-  ( y  e.  ( 1 ... ( |_ `  ( A  /  d ) ) )  |->  ( d  x.  y ) )  =  ( y  e.  ( 1 ... ( |_
`  ( A  / 
d ) ) ) 
|->  ( d  x.  y
) )
5249, 50, 51dvdsflf1o 22502 . . . 4  |-  ( (
ph  /\  d  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( y  e.  ( 1 ... ( |_ `  ( A  / 
d ) ) ) 
|->  ( d  x.  y
) ) : ( 1 ... ( |_
`  ( A  / 
d ) ) ) -1-1-onto-> { x  e.  ( 1 ... ( |_ `  A ) )  |  d  ||  x }
)
53 oveq2 6094 . . . . . 6  |-  ( y  =  m  ->  (
d  x.  y )  =  ( d  x.  m ) )
54 ovex 6111 . . . . . 6  |-  ( d  x.  m )  e. 
_V
5553, 51, 54fvmpt 5769 . . . . 5  |-  ( m  e.  ( 1 ... ( |_ `  ( A  /  d ) ) )  ->  ( (
y  e.  ( 1 ... ( |_ `  ( A  /  d
) ) )  |->  ( d  x.  y ) ) `  m )  =  ( d  x.  m ) )
5655adantl 466 . . . 4  |-  ( ( ( ph  /\  d  e.  ( 1 ... ( |_ `  A ) ) )  /\  m  e.  ( 1 ... ( |_ `  ( A  / 
d ) ) ) )  ->  ( (
y  e.  ( 1 ... ( |_ `  ( A  /  d
) ) )  |->  ( d  x.  y ) ) `  m )  =  ( d  x.  m ) )
5744biimpar 485 . . . . . 6  |-  ( (
ph  /\  ( d  e.  ( 1 ... ( |_ `  A ) )  /\  n  e.  {
x  e.  ( 1 ... ( |_ `  A ) )  |  d  ||  x }
) )  ->  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  d  e.  { x  e.  NN  |  x  ||  n } ) )
5857, 45syldan 470 . . . . 5  |-  ( (
ph  /\  ( d  e.  ( 1 ... ( |_ `  A ) )  /\  n  e.  {
x  e.  ( 1 ... ( |_ `  A ) )  |  d  ||  x }
) )  ->  B  e.  CC )
5958anassrs 648 . . . 4  |-  ( ( ( ph  /\  d  e.  ( 1 ... ( |_ `  A ) ) )  /\  n  e. 
{ x  e.  ( 1 ... ( |_
`  A ) )  |  d  ||  x } )  ->  B  e.  CC )
6047, 48, 52, 56, 59fsumf1o 13192 . . 3  |-  ( (
ph  /\  d  e.  ( 1 ... ( |_ `  A ) ) )  ->  sum_ n  e. 
{ x  e.  ( 1 ... ( |_
`  A ) )  |  d  ||  x } B  =  sum_ m  e.  ( 1 ... ( |_ `  ( A  /  d ) ) ) C )
6160sumeq2dv 13172 . 2  |-  ( ph  -> 
sum_ d  e.  ( 1 ... ( |_
`  A ) )
sum_ n  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  d  ||  x } B  =  sum_ d  e.  ( 1 ... ( |_ `  A ) ) sum_ m  e.  ( 1 ... ( |_ `  ( A  /  d ) ) ) C )
6246, 61eqtrd 2470 1  |-  ( ph  -> 
sum_ n  e.  (
1 ... ( |_ `  A ) ) sum_ d  e.  { x  e.  NN  |  x  ||  n } B  =  sum_ d  e.  ( 1 ... ( |_ `  A ) ) sum_ m  e.  ( 1 ... ( |_ `  ( A  /  d ) ) ) C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2714    C_ wss 3323   class class class wbr 4287    e. cmpt 4345   ` cfv 5413  (class class class)co 6086   Fincfn 7302   CCcc 9272   RRcr 9273   1c1 9275    x. cmul 9279    <_ cle 9411    / cdiv 9985   NNcn 10314   ZZcz 10638   ...cfz 11429   |_cfl 11632   sum_csu 13155    || cdivides 13527
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-sum 13156  df-dvds 13528
This theorem is referenced by:  dchrmusum2  22718  dchrvmasumlem1  22719  dchrvmasum2lem  22720  dchrisum0  22744  mudivsum  22754  mulogsum  22756  mulog2sumlem2  22759  vmalogdivsum2  22762  selberglem3  22771  selberg  22772  selberg34r  22795  pntsval2  22800
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