MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvdsflsumcom Structured version   Unicode version

Theorem dvdsflsumcom 22662
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 11913 . . 3  |-  ( ph  ->  ( 1 ... ( |_ `  A ) )  e.  Fin )
2 fzfid 11913 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1 ... n )  e. 
Fin )
3 elfznn 11596 . . . . . 6  |-  ( n  e.  ( 1 ... ( |_ `  A
) )  ->  n  e.  NN )
43adantl 466 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  n  e.  NN )
5 sgmss 22578 . . . . 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 7645 . . . 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 10441 . . . . . . . . . . . 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 10449 . . . . . . . . . . 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 10780 . . . . . . . . . . . . 13  |-  ( d  e.  NN  ->  d  e.  ZZ )
16 dvdsle 13697 . . . . . . . . . . . . 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 11819 . . . . . . . . . . . . . 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 9640 . . . . . . . . . 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 11819 . . . . . . . . . 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 4404 . . . . . 6  |-  ( x  =  d  ->  (
x  ||  n  <->  d  ||  n ) )
3938elrab 3224 . . . . 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 4405 . . . . . 6  |-  ( x  =  n  ->  (
d  ||  x  <->  d  ||  n ) )
4241elrab 3224 . . . . 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 13360 . 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 11913 . . . 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 2454 . . . . 5  |-  ( y  e.  ( 1 ... ( |_ `  ( A  /  d ) ) )  |->  ( d  x.  y ) )  =  ( y  e.  ( 1 ... ( |_
`  ( A  / 
d ) ) ) 
|->  ( d  x.  y
) )
5249, 50, 51dvdsflf1o 22661 . . . 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 6209 . . . . . 6  |-  ( y  =  m  ->  (
d  x.  y )  =  ( d  x.  m ) )
54 ovex 6226 . . . . . 6  |-  ( d  x.  m )  e. 
_V
5553, 51, 54fvmpt 5884 . . . . 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 13319 . . 3  |-  ( (
ph  /\  d  e.  ( 1 ... ( |_ `  A ) ) )  ->  sum_ n  e. 
{ x  e.  ( 1 ... ( |_
`  A ) )  |  d  ||  x } B  =  sum_ m  e.  ( 1 ... ( |_ `  ( A  /  d ) ) ) C )
6160sumeq2dv 13299 . 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 2495 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 1370    e. wcel 1758   {crab 2803    C_ wss 3437   class class class wbr 4401    |-> cmpt 4459   ` cfv 5527  (class class class)co 6201   Fincfn 7421   CCcc 9392   RRcr 9393   1c1 9395    x. cmul 9399    <_ cle 9531    / cdiv 10105   NNcn 10434   ZZcz 10758   ...cfz 11555   |_cfl 11758   sum_csu 13282    || cdivides 13654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-pre-sup 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-sup 7803  df-oi 7836  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-div 10106  df-nn 10435  df-2 10492  df-3 10493  df-n0 10692  df-z 10759  df-uz 10974  df-rp 11104  df-fz 11556  df-fzo 11667  df-fl 11760  df-seq 11925  df-exp 11984  df-hash 12222  df-cj 12707  df-re 12708  df-im 12709  df-sqr 12843  df-abs 12844  df-clim 13085  df-sum 13283  df-dvds 13655
This theorem is referenced by:  dchrmusum2  22877  dchrvmasumlem1  22878  dchrvmasum2lem  22879  dchrisum0  22903  mudivsum  22913  mulogsum  22915  mulog2sumlem2  22918  vmalogdivsum2  22921  selberglem3  22930  selberg  22931  selberg34r  22954  pntsval2  22959
  Copyright terms: Public domain W3C validator