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Theorem dvdsflsumcom 20930
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 11271 . . 3  |-  ( ph  ->  ( 1 ... ( |_ `  A ) )  e.  Fin )
2 fzfid 11271 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1 ... n )  e. 
Fin )
3 elfznn 11040 . . . . . 6  |-  ( n  e.  ( 1 ... ( |_ `  A
) )  ->  n  e.  NN )
43adantl 453 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  n  e.  NN )
5 sgmss 20846 . . . . 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 7292 . . . 4  |-  ( ( ( 1 ... n
)  e.  Fin  /\  { x  e.  NN  |  x  ||  n }  C_  ( 1 ... n
) )  ->  { x  e.  NN  |  x  ||  n }  e.  Fin )
82, 6, 7syl2anc 643 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  { x  e.  NN  |  x  ||  n }  e.  Fin )
9 nnre 9967 . . . . . . . . . . . 12  |-  ( d  e.  NN  ->  d  e.  RR )
109ad2antrl 709 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  -> 
d  e.  RR )
114adantr 452 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  ->  n  e.  NN )
1211nnred 9975 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  ->  n  e.  RR )
13 dvdsflsumcom.2 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  RR )
1413ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  ->  A  e.  RR )
15 nnz 10263 . . . . . . . . . . . . 13  |-  ( d  e.  NN  ->  d  e.  ZZ )
16 dvdsle 12854 . . . . . . . . . . . . 13  |-  ( ( d  e.  ZZ  /\  n  e.  NN )  ->  ( d  ||  n  ->  d  <_  n )
)
1715, 4, 16syl2anr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  d  e.  NN )  ->  (
d  ||  n  ->  d  <_  n ) )
1817impr 603 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  -> 
d  <_  n )
19 fznnfl 11202 . . . . . . . . . . . . . 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 608 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  n  <_  A )
2221adantr 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  ->  n  <_  A )
2310, 12, 14, 18, 22letrd 9187 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  -> 
d  <_  A )
2423ex 424 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (
d  e.  NN  /\  d  ||  n )  -> 
d  <_  A )
)
2524pm4.71rd 617 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (
d  e.  NN  /\  d  ||  n )  <->  ( d  <_  A  /\  ( d  e.  NN  /\  d  ||  n ) ) ) )
26 ancom 438 . . . . . . . . 9  |-  ( ( d  <_  A  /\  ( d  e.  NN  /\  d  ||  n ) )  <->  ( ( d  e.  NN  /\  d  ||  n )  /\  d  <_  A ) )
27 an32 774 . . . . . . . . 9  |-  ( ( ( d  e.  NN  /\  d  ||  n )  /\  d  <_  A
)  <->  ( ( d  e.  NN  /\  d  <_  A )  /\  d  ||  n ) )
2826, 27bitri 241 . . . . . . . 8  |-  ( ( d  <_  A  /\  ( d  e.  NN  /\  d  ||  n ) )  <->  ( ( d  e.  NN  /\  d  <_  A )  /\  d  ||  n ) )
2925, 28syl6bb 253 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (
d  e.  NN  /\  d  ||  n )  <->  ( (
d  e.  NN  /\  d  <_  A )  /\  d  ||  n ) ) )
30 fznnfl 11202 . . . . . . . . . 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 452 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( d  e.  ( 1 ... ( |_ `  A ) )  <-> 
( d  e.  NN  /\  d  <_  A )
) )
3332anbi1d 686 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (
d  e.  ( 1 ... ( |_ `  A ) )  /\  d  ||  n )  <->  ( (
d  e.  NN  /\  d  <_  A )  /\  d  ||  n ) ) )
3429, 33bitr4d 248 . . . . . 6  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (
d  e.  NN  /\  d  ||  n )  <->  ( d  e.  ( 1 ... ( |_ `  A ) )  /\  d  ||  n
) ) )
3534pm5.32da 623 . . . . 5  |-  ( ph  ->  ( ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  <->  ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( d  e.  ( 1 ... ( |_ `  A ) )  /\  d  ||  n
) ) ) )
36 an12 773 . . . . 5  |-  ( ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( d  e.  ( 1 ... ( |_
`  A ) )  /\  d  ||  n
) )  <->  ( d  e.  ( 1 ... ( |_ `  A ) )  /\  ( n  e.  ( 1 ... ( |_ `  A ) )  /\  d  ||  n
) ) )
3735, 36syl6bb 253 . . . 4  |-  ( ph  ->  ( ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  <->  ( d  e.  ( 1 ... ( |_ `  A ) )  /\  ( n  e.  ( 1 ... ( |_ `  A ) )  /\  d  ||  n
) ) ) )
38 breq1 4179 . . . . . 6  |-  ( x  =  d  ->  (
x  ||  n  <->  d  ||  n ) )
3938elrab 3056 . . . . 5  |-  ( d  e.  { x  e.  NN  |  x  ||  n }  <->  ( d  e.  NN  /\  d  ||  n ) )
4039anbi2i 676 . . . 4  |-  ( ( n  e.  ( 1 ... ( |_ `  A ) )  /\  d  e.  { x  e.  NN  |  x  ||  n } )  <->  ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( d  e.  NN  /\  d  ||  n ) ) )
41 breq2 4180 . . . . . 6  |-  ( x  =  n  ->  (
d  ||  x  <->  d  ||  n ) )
4241elrab 3056 . . . . 5  |-  ( n  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  d  ||  x } 
<->  ( n  e.  ( 1 ... ( |_
`  A ) )  /\  d  ||  n
) )
4342anbi2i 676 . . . 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 280 . . 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 12517 . 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 11271 . . . 4  |-  ( (
ph  /\  d  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1 ... ( |_ `  ( A  /  d
) ) )  e. 
Fin )
4913adantr 452 . . . . 5  |-  ( (
ph  /\  d  e.  ( 1 ... ( |_ `  A ) ) )  ->  A  e.  RR )
5031simprbda 607 . . . . 5  |-  ( (
ph  /\  d  e.  ( 1 ... ( |_ `  A ) ) )  ->  d  e.  NN )
51 eqid 2408 . . . . 5  |-  ( y  e.  ( 1 ... ( |_ `  ( A  /  d ) ) )  |->  ( d  x.  y ) )  =  ( y  e.  ( 1 ... ( |_
`  ( A  / 
d ) ) ) 
|->  ( d  x.  y
) )
5249, 50, 51dvdsflf1o 20929 . . . 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 6052 . . . . . 6  |-  ( y  =  m  ->  (
d  x.  y )  =  ( d  x.  m ) )
54 ovex 6069 . . . . . 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 453 . . . 4  |-  ( ( ( ph  /\  d  e.  ( 1 ... ( |_ `  A ) ) )  /\  m  e.  ( 1 ... ( |_ `  ( A  / 
d ) ) ) )  ->  ( (
y  e.  ( 1 ... ( |_ `  ( A  /  d
) ) )  |->  ( d  x.  y ) ) `  m )  =  ( d  x.  m ) )
5744biimpar 472 . . . . . 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 457 . . . . 5  |-  ( (
ph  /\  ( d  e.  ( 1 ... ( |_ `  A ) )  /\  n  e.  {
x  e.  ( 1 ... ( |_ `  A ) )  |  d  ||  x }
) )  ->  B  e.  CC )
5958anassrs 630 . . . 4  |-  ( ( ( ph  /\  d  e.  ( 1 ... ( |_ `  A ) ) )  /\  n  e. 
{ x  e.  ( 1 ... ( |_
`  A ) )  |  d  ||  x } )  ->  B  e.  CC )
6047, 48, 52, 56, 59fsumf1o 12476 . . 3  |-  ( (
ph  /\  d  e.  ( 1 ... ( |_ `  A ) ) )  ->  sum_ n  e. 
{ x  e.  ( 1 ... ( |_
`  A ) )  |  d  ||  x } B  =  sum_ m  e.  ( 1 ... ( |_ `  ( A  /  d ) ) ) C )
6160sumeq2dv 12456 . 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 2440 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   {crab 2674    C_ wss 3284   class class class wbr 4176    e. cmpt 4230   ` cfv 5417  (class class class)co 6044   Fincfn 7072   CCcc 8948   RRcr 8949   1c1 8951    x. cmul 8955    <_ cle 9081    / cdiv 9637   NNcn 9960   ZZcz 10242   ...cfz 11003   |_cfl 11160   sum_csu 12438    || cdivides 12811
This theorem is referenced by:  dchrmusum2  21145  dchrvmasumlem1  21146  dchrvmasum2lem  21147  dchrisum0  21171  mudivsum  21181  mulogsum  21183  mulog2sumlem2  21186  vmalogdivsum2  21189  selberglem3  21198  selberg  21199  selberg34r  21222  pntsval2  21227
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-sup 7408  df-oi 7439  df-card 7786  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-n0 10182  df-z 10243  df-uz 10449  df-rp 10573  df-fz 11004  df-fzo 11095  df-fl 11161  df-seq 11283  df-exp 11342  df-hash 11578  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-clim 12241  df-sum 12439  df-dvds 12812
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