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Theorem dvdsflsumcom 20428
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 11035 . . 3  |-  ( ph  ->  ( 1 ... ( |_ `  A ) )  e.  Fin )
2 fzfid 11035 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1 ... n )  e. 
Fin )
3 elfznn 10819 . . . . . 6  |-  ( n  e.  ( 1 ... ( |_ `  A
) )  ->  n  e.  NN )
43adantl 452 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  n  e.  NN )
5 sgmss 20344 . . . . 5  |-  ( n  e.  NN  ->  { x  e.  NN  |  x  ||  n }  C_  ( 1 ... n ) )
64, 5syl 15 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  { x  e.  NN  |  x  ||  n }  C_  ( 1 ... n ) )
7 ssfi 7083 . . . 4  |-  ( ( ( 1 ... n
)  e.  Fin  /\  { x  e.  NN  |  x  ||  n }  C_  ( 1 ... n
) )  ->  { x  e.  NN  |  x  ||  n }  e.  Fin )
82, 6, 7syl2anc 642 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  { x  e.  NN  |  x  ||  n }  e.  Fin )
9 nnre 9753 . . . . . . . . . . . 12  |-  ( d  e.  NN  ->  d  e.  RR )
109ad2antrl 708 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  -> 
d  e.  RR )
114adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  ->  n  e.  NN )
1211nnred 9761 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  ->  n  e.  RR )
13 dvdsflsumcom.2 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  RR )
1413ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  ->  A  e.  RR )
15 nnz 10045 . . . . . . . . . . . . 13  |-  ( d  e.  NN  ->  d  e.  ZZ )
16 dvdsle 12574 . . . . . . . . . . . . 13  |-  ( ( d  e.  ZZ  /\  n  e.  NN )  ->  ( d  ||  n  ->  d  <_  n )
)
1715, 4, 16syl2anr 464 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  d  e.  NN )  ->  (
d  ||  n  ->  d  <_  n ) )
1817impr 602 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  -> 
d  <_  n )
19 fznnfl 10966 . . . . . . . . . . . . . 14  |-  ( A  e.  RR  ->  (
n  e.  ( 1 ... ( |_ `  A ) )  <->  ( n  e.  NN  /\  n  <_  A ) ) )
2013, 19syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  ( n  e.  ( 1 ... ( |_
`  A ) )  <-> 
( n  e.  NN  /\  n  <_  A )
) )
2120simplbda 607 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  n  <_  A )
2221adantr 451 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  ->  n  <_  A )
2310, 12, 14, 18, 22letrd 8973 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  -> 
d  <_  A )
2423ex 423 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (
d  e.  NN  /\  d  ||  n )  -> 
d  <_  A )
)
2524pm4.71rd 616 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (
d  e.  NN  /\  d  ||  n )  <->  ( d  <_  A  /\  ( d  e.  NN  /\  d  ||  n ) ) ) )
26 ancom 437 . . . . . . . . 9  |-  ( ( d  <_  A  /\  ( d  e.  NN  /\  d  ||  n ) )  <->  ( ( d  e.  NN  /\  d  ||  n )  /\  d  <_  A ) )
27 an32 773 . . . . . . . . 9  |-  ( ( ( d  e.  NN  /\  d  ||  n )  /\  d  <_  A
)  <->  ( ( d  e.  NN  /\  d  <_  A )  /\  d  ||  n ) )
2826, 27bitri 240 . . . . . . . 8  |-  ( ( d  <_  A  /\  ( d  e.  NN  /\  d  ||  n ) )  <->  ( ( d  e.  NN  /\  d  <_  A )  /\  d  ||  n ) )
2925, 28syl6bb 252 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (
d  e.  NN  /\  d  ||  n )  <->  ( (
d  e.  NN  /\  d  <_  A )  /\  d  ||  n ) ) )
30 fznnfl 10966 . . . . . . . . . 10  |-  ( A  e.  RR  ->  (
d  e.  ( 1 ... ( |_ `  A ) )  <->  ( d  e.  NN  /\  d  <_  A ) ) )
3113, 30syl 15 . . . . . . . . 9  |-  ( ph  ->  ( d  e.  ( 1 ... ( |_
`  A ) )  <-> 
( d  e.  NN  /\  d  <_  A )
) )
3231adantr 451 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( d  e.  ( 1 ... ( |_ `  A ) )  <-> 
( d  e.  NN  /\  d  <_  A )
) )
3332anbi1d 685 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (
d  e.  ( 1 ... ( |_ `  A ) )  /\  d  ||  n )  <->  ( (
d  e.  NN  /\  d  <_  A )  /\  d  ||  n ) ) )
3429, 33bitr4d 247 . . . . . 6  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (
d  e.  NN  /\  d  ||  n )  <->  ( d  e.  ( 1 ... ( |_ `  A ) )  /\  d  ||  n
) ) )
3534pm5.32da 622 . . . . 5  |-  ( ph  ->  ( ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  <->  ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( d  e.  ( 1 ... ( |_ `  A ) )  /\  d  ||  n
) ) ) )
36 an12 772 . . . . 5  |-  ( ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( d  e.  ( 1 ... ( |_
`  A ) )  /\  d  ||  n
) )  <->  ( d  e.  ( 1 ... ( |_ `  A ) )  /\  ( n  e.  ( 1 ... ( |_ `  A ) )  /\  d  ||  n
) ) )
3735, 36syl6bb 252 . . . 4  |-  ( ph  ->  ( ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  <->  ( d  e.  ( 1 ... ( |_ `  A ) )  /\  ( n  e.  ( 1 ... ( |_ `  A ) )  /\  d  ||  n
) ) ) )
38 breq1 4026 . . . . . 6  |-  ( x  =  d  ->  (
x  ||  n  <->  d  ||  n ) )
3938elrab 2923 . . . . 5  |-  ( d  e.  { x  e.  NN  |  x  ||  n }  <->  ( d  e.  NN  /\  d  ||  n ) )
4039anbi2i 675 . . . 4  |-  ( ( n  e.  ( 1 ... ( |_ `  A ) )  /\  d  e.  { x  e.  NN  |  x  ||  n } )  <->  ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( d  e.  NN  /\  d  ||  n ) ) )
41 breq2 4027 . . . . . 6  |-  ( x  =  n  ->  (
d  ||  x  <->  d  ||  n ) )
4241elrab 2923 . . . . 5  |-  ( n  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  d  ||  x } 
<->  ( n  e.  ( 1 ... ( |_
`  A ) )  /\  d  ||  n
) )
4342anbi2i 675 . . . 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 279 . . 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 12237 . 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 11035 . . . 4  |-  ( (
ph  /\  d  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1 ... ( |_ `  ( A  /  d
) ) )  e. 
Fin )
4913adantr 451 . . . . 5  |-  ( (
ph  /\  d  e.  ( 1 ... ( |_ `  A ) ) )  ->  A  e.  RR )
5031simprbda 606 . . . . 5  |-  ( (
ph  /\  d  e.  ( 1 ... ( |_ `  A ) ) )  ->  d  e.  NN )
51 eqid 2283 . . . . 5  |-  ( y  e.  ( 1 ... ( |_ `  ( A  /  d ) ) )  |->  ( d  x.  y ) )  =  ( y  e.  ( 1 ... ( |_
`  ( A  / 
d ) ) ) 
|->  ( d  x.  y
) )
5249, 50, 51dvdsflf1o 20427 . . . 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 5866 . . . . . 6  |-  ( y  =  m  ->  (
d  x.  y )  =  ( d  x.  m ) )
54 ovex 5883 . . . . . 6  |-  ( d  x.  m )  e. 
_V
5553, 51, 54fvmpt 5602 . . . . 5  |-  ( m  e.  ( 1 ... ( |_ `  ( A  /  d ) ) )  ->  ( (
y  e.  ( 1 ... ( |_ `  ( A  /  d
) ) )  |->  ( d  x.  y ) ) `  m )  =  ( d  x.  m ) )
5655adantl 452 . . . 4  |-  ( ( ( ph  /\  d  e.  ( 1 ... ( |_ `  A ) ) )  /\  m  e.  ( 1 ... ( |_ `  ( A  / 
d ) ) ) )  ->  ( (
y  e.  ( 1 ... ( |_ `  ( A  /  d
) ) )  |->  ( d  x.  y ) ) `  m )  =  ( d  x.  m ) )
5744biimpar 471 . . . . . 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 456 . . . . 5  |-  ( (
ph  /\  ( d  e.  ( 1 ... ( |_ `  A ) )  /\  n  e.  {
x  e.  ( 1 ... ( |_ `  A ) )  |  d  ||  x }
) )  ->  B  e.  CC )
5958anassrs 629 . . . 4  |-  ( ( ( ph  /\  d  e.  ( 1 ... ( |_ `  A ) ) )  /\  n  e. 
{ x  e.  ( 1 ... ( |_
`  A ) )  |  d  ||  x } )  ->  B  e.  CC )
6047, 48, 52, 56, 59fsumf1o 12196 . . 3  |-  ( (
ph  /\  d  e.  ( 1 ... ( |_ `  A ) ) )  ->  sum_ n  e. 
{ x  e.  ( 1 ... ( |_
`  A ) )  |  d  ||  x } B  =  sum_ m  e.  ( 1 ... ( |_ `  ( A  /  d ) ) ) C )
6160sumeq2dv 12176 . 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 2315 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   Fincfn 6863   CCcc 8735   RRcr 8736   1c1 8738    x. cmul 8742    <_ cle 8868    / cdiv 9423   NNcn 9746   ZZcz 10024   ...cfz 10782   |_cfl 10924   sum_csu 12158    || cdivides 12531
This theorem is referenced by:  dchrmusum2  20643  dchrvmasumlem1  20644  dchrvmasum2lem  20645  dchrisum0  20669  mudivsum  20679  mulogsum  20681  mulog2sumlem2  20684  vmalogdivsum2  20687  selberglem3  20696  selberg  20697  selberg34r  20720  pntsval2  20725
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-dvds 12532
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