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Theorem dvdsflsumcom 20978
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 11317 . . 3  |-  ( ph  ->  ( 1 ... ( |_ `  A ) )  e.  Fin )
2 fzfid 11317 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1 ... n )  e. 
Fin )
3 elfznn 11085 . . . . . 6  |-  ( n  e.  ( 1 ... ( |_ `  A
) )  ->  n  e.  NN )
43adantl 454 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  n  e.  NN )
5 sgmss 20894 . . . . 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 7332 . . . 4  |-  ( ( ( 1 ... n
)  e.  Fin  /\  { x  e.  NN  |  x  ||  n }  C_  ( 1 ... n
) )  ->  { x  e.  NN  |  x  ||  n }  e.  Fin )
82, 6, 7syl2anc 644 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  { x  e.  NN  |  x  ||  n }  e.  Fin )
9 nnre 10012 . . . . . . . . . . . 12  |-  ( d  e.  NN  ->  d  e.  RR )
109ad2antrl 710 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  -> 
d  e.  RR )
114adantr 453 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  ->  n  e.  NN )
1211nnred 10020 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  ->  n  e.  RR )
13 dvdsflsumcom.2 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  RR )
1413ad2antrr 708 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  ->  A  e.  RR )
15 nnz 10308 . . . . . . . . . . . . 13  |-  ( d  e.  NN  ->  d  e.  ZZ )
16 dvdsle 12900 . . . . . . . . . . . . 13  |-  ( ( d  e.  ZZ  /\  n  e.  NN )  ->  ( d  ||  n  ->  d  <_  n )
)
1715, 4, 16syl2anr 466 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  d  e.  NN )  ->  (
d  ||  n  ->  d  <_  n ) )
1817impr 604 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  -> 
d  <_  n )
19 fznnfl 11248 . . . . . . . . . . . . . 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 609 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  n  <_  A )
2221adantr 453 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  ->  n  <_  A )
2310, 12, 14, 18, 22letrd 9232 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  -> 
d  <_  A )
2423ex 425 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (
d  e.  NN  /\  d  ||  n )  -> 
d  <_  A )
)
2524pm4.71rd 618 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (
d  e.  NN  /\  d  ||  n )  <->  ( d  <_  A  /\  ( d  e.  NN  /\  d  ||  n ) ) ) )
26 ancom 439 . . . . . . . . 9  |-  ( ( d  <_  A  /\  ( d  e.  NN  /\  d  ||  n ) )  <->  ( ( d  e.  NN  /\  d  ||  n )  /\  d  <_  A ) )
27 an32 775 . . . . . . . . 9  |-  ( ( ( d  e.  NN  /\  d  ||  n )  /\  d  <_  A
)  <->  ( ( d  e.  NN  /\  d  <_  A )  /\  d  ||  n ) )
2826, 27bitri 242 . . . . . . . 8  |-  ( ( d  <_  A  /\  ( d  e.  NN  /\  d  ||  n ) )  <->  ( ( d  e.  NN  /\  d  <_  A )  /\  d  ||  n ) )
2925, 28syl6bb 254 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (
d  e.  NN  /\  d  ||  n )  <->  ( (
d  e.  NN  /\  d  <_  A )  /\  d  ||  n ) ) )
30 fznnfl 11248 . . . . . . . . . 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 453 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( d  e.  ( 1 ... ( |_ `  A ) )  <-> 
( d  e.  NN  /\  d  <_  A )
) )
3332anbi1d 687 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (
d  e.  ( 1 ... ( |_ `  A ) )  /\  d  ||  n )  <->  ( (
d  e.  NN  /\  d  <_  A )  /\  d  ||  n ) ) )
3429, 33bitr4d 249 . . . . . 6  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (
d  e.  NN  /\  d  ||  n )  <->  ( d  e.  ( 1 ... ( |_ `  A ) )  /\  d  ||  n
) ) )
3534pm5.32da 624 . . . . 5  |-  ( ph  ->  ( ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  <->  ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( d  e.  ( 1 ... ( |_ `  A ) )  /\  d  ||  n
) ) ) )
36 an12 774 . . . . 5  |-  ( ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( d  e.  ( 1 ... ( |_
`  A ) )  /\  d  ||  n
) )  <->  ( d  e.  ( 1 ... ( |_ `  A ) )  /\  ( n  e.  ( 1 ... ( |_ `  A ) )  /\  d  ||  n
) ) )
3735, 36syl6bb 254 . . . 4  |-  ( ph  ->  ( ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( d  e.  NN  /\  d  ||  n ) )  <->  ( d  e.  ( 1 ... ( |_ `  A ) )  /\  ( n  e.  ( 1 ... ( |_ `  A ) )  /\  d  ||  n
) ) ) )
38 breq1 4218 . . . . . 6  |-  ( x  =  d  ->  (
x  ||  n  <->  d  ||  n ) )
3938elrab 3094 . . . . 5  |-  ( d  e.  { x  e.  NN  |  x  ||  n }  <->  ( d  e.  NN  /\  d  ||  n ) )
4039anbi2i 677 . . . 4  |-  ( ( n  e.  ( 1 ... ( |_ `  A ) )  /\  d  e.  { x  e.  NN  |  x  ||  n } )  <->  ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( d  e.  NN  /\  d  ||  n ) ) )
41 breq2 4219 . . . . . 6  |-  ( x  =  n  ->  (
d  ||  x  <->  d  ||  n ) )
4241elrab 3094 . . . . 5  |-  ( n  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  d  ||  x } 
<->  ( n  e.  ( 1 ... ( |_
`  A ) )  /\  d  ||  n
) )
4342anbi2i 677 . . . 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 281 . . 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 12563 . 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 11317 . . . 4  |-  ( (
ph  /\  d  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1 ... ( |_ `  ( A  /  d
) ) )  e. 
Fin )
4913adantr 453 . . . . 5  |-  ( (
ph  /\  d  e.  ( 1 ... ( |_ `  A ) ) )  ->  A  e.  RR )
5031simprbda 608 . . . . 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 20977 . . . 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 6092 . . . . . 6  |-  ( y  =  m  ->  (
d  x.  y )  =  ( d  x.  m ) )
54 ovex 6109 . . . . . 6  |-  ( d  x.  m )  e. 
_V
5553, 51, 54fvmpt 5809 . . . . 5  |-  ( m  e.  ( 1 ... ( |_ `  ( A  /  d ) ) )  ->  ( (
y  e.  ( 1 ... ( |_ `  ( A  /  d
) ) )  |->  ( d  x.  y ) ) `  m )  =  ( d  x.  m ) )
5655adantl 454 . . . 4  |-  ( ( ( ph  /\  d  e.  ( 1 ... ( |_ `  A ) ) )  /\  m  e.  ( 1 ... ( |_ `  ( A  / 
d ) ) ) )  ->  ( (
y  e.  ( 1 ... ( |_ `  ( A  /  d
) ) )  |->  ( d  x.  y ) ) `  m )  =  ( d  x.  m ) )
5744biimpar 473 . . . . . 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 458 . . . . 5  |-  ( (
ph  /\  ( d  e.  ( 1 ... ( |_ `  A ) )  /\  n  e.  {
x  e.  ( 1 ... ( |_ `  A ) )  |  d  ||  x }
) )  ->  B  e.  CC )
5958anassrs 631 . . . 4  |-  ( ( ( ph  /\  d  e.  ( 1 ... ( |_ `  A ) ) )  /\  n  e. 
{ x  e.  ( 1 ... ( |_
`  A ) )  |  d  ||  x } )  ->  B  e.  CC )
6047, 48, 52, 56, 59fsumf1o 12522 . . 3  |-  ( (
ph  /\  d  e.  ( 1 ... ( |_ `  A ) ) )  ->  sum_ n  e. 
{ x  e.  ( 1 ... ( |_
`  A ) )  |  d  ||  x } B  =  sum_ m  e.  ( 1 ... ( |_ `  ( A  /  d ) ) ) C )
6160sumeq2dv 12502 . 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 set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   {crab 2711    C_ wss 3322   class class class wbr 4215    e. cmpt 4269   ` cfv 5457  (class class class)co 6084   Fincfn 7112   CCcc 8993   RRcr 8994   1c1 8996    x. cmul 9000    <_ cle 9126    / cdiv 9682   NNcn 10005   ZZcz 10287   ...cfz 11048   |_cfl 11206   sum_csu 12484    || cdivides 12857
This theorem is referenced by:  dchrmusum2  21193  dchrvmasumlem1  21194  dchrvmasum2lem  21195  dchrisum0  21219  mudivsum  21229  mulogsum  21231  mulog2sumlem2  21234  vmalogdivsum2  21237  selberglem3  21246  selberg  21247  selberg34r  21270  pntsval2  21275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fz 11049  df-fzo 11141  df-fl 11207  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-sum 12485  df-dvds 12858
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