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Theorem sumz 12516
Description: Any sum of zero over a summable set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 20-Apr-2014.)
Assertion
Ref Expression
sumz  |-  ( ( A  C_  ( ZZ>= `  M )  \/  A  e.  Fin )  ->  sum_ k  e.  A  0  = 
0 )
Distinct variable groups:    A, k    k, M

Proof of Theorem sumz
Dummy variables  f  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2436 . . . . 5  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2 simpr 448 . . . . 5  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  M  e.  ZZ )
3 simpl 444 . . . . 5  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  A  C_  ( ZZ>= `  M )
)
4 c0ex 9085 . . . . . . . 8  |-  0  e.  _V
54fvconst2 5947 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( (
( ZZ>= `  M )  X.  { 0 } ) `
 k )  =  0 )
6 ifid 3771 . . . . . . 7  |-  if ( k  e.  A , 
0 ,  0 )  =  0
75, 6syl6eqr 2486 . . . . . 6  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( (
( ZZ>= `  M )  X.  { 0 } ) `
 k )  =  if ( k  e.  A ,  0 ,  0 ) )
87adantl 453 . . . . 5  |-  ( ( ( A  C_  ( ZZ>=
`  M )  /\  M  e.  ZZ )  /\  k  e.  ( ZZ>=
`  M ) )  ->  ( ( (
ZZ>= `  M )  X. 
{ 0 } ) `
 k )  =  if ( k  e.  A ,  0 ,  0 ) )
9 0cn 9084 . . . . . 6  |-  0  e.  CC
109a1i 11 . . . . 5  |-  ( ( ( A  C_  ( ZZ>=
`  M )  /\  M  e.  ZZ )  /\  k  e.  A
)  ->  0  e.  CC )
111, 2, 3, 8, 10zsum 12512 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  sum_ k  e.  A  0  =  ( 
~~>  `  seq  M (  +  ,  ( (
ZZ>= `  M )  X. 
{ 0 } ) ) ) )
12 fclim 12347 . . . . . 6  |-  ~~>  : dom  ~~>  --> CC
13 ffun 5593 . . . . . 6  |-  (  ~~>  : dom  ~~>  --> CC 
->  Fun  ~~>  )
1412, 13ax-mp 8 . . . . 5  |-  Fun  ~~>
15 serclim0 12371 . . . . . 6  |-  ( M  e.  ZZ  ->  seq  M (  +  ,  ( ( ZZ>= `  M )  X.  { 0 } ) )  ~~>  0 )
1615adantl 453 . . . . 5  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  seq  M (  +  ,  ( ( ZZ>= `  M )  X.  { 0 } ) )  ~~>  0 )
17 funbrfv 5765 . . . . 5  |-  ( Fun  ~~>  ->  (  seq  M (  +  ,  ( (
ZZ>= `  M )  X. 
{ 0 } ) )  ~~>  0  ->  (  ~~>  ` 
seq  M (  +  ,  ( ( ZZ>= `  M )  X.  {
0 } ) ) )  =  0 ) )
1814, 16, 17mpsyl 61 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  (  ~~>  ` 
seq  M (  +  ,  ( ( ZZ>= `  M )  X.  {
0 } ) ) )  =  0 )
1911, 18eqtrd 2468 . . 3  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  sum_ k  e.  A  0  = 
0 )
20 uzf 10491 . . . . . . . . 9  |-  ZZ>= : ZZ --> ~P ZZ
2120fdmi 5596 . . . . . . . 8  |-  dom  ZZ>=  =  ZZ
2221eleq2i 2500 . . . . . . 7  |-  ( M  e.  dom  ZZ>=  <->  M  e.  ZZ )
23 ndmfv 5755 . . . . . . 7  |-  ( -.  M  e.  dom  ZZ>=  -> 
( ZZ>= `  M )  =  (/) )
2422, 23sylnbir 299 . . . . . 6  |-  ( -.  M  e.  ZZ  ->  (
ZZ>= `  M )  =  (/) )
2524sseq2d 3376 . . . . 5  |-  ( -.  M  e.  ZZ  ->  ( A  C_  ( ZZ>= `  M )  <->  A  C_  (/) ) )
2625biimpac 473 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  -.  M  e.  ZZ )  ->  A  C_  (/) )
27 ss0 3658 . . . 4  |-  ( A 
C_  (/)  ->  A  =  (/) )
28 sumeq1 12483 . . . . 5  |-  ( A  =  (/)  ->  sum_ k  e.  A  0  =  sum_ k  e.  (/)  0 )
29 sum0 12515 . . . . 5  |-  sum_ k  e.  (/)  0  =  0
3028, 29syl6eq 2484 . . . 4  |-  ( A  =  (/)  ->  sum_ k  e.  A  0  = 
0 )
3126, 27, 303syl 19 . . 3  |-  ( ( A  C_  ( ZZ>= `  M )  /\  -.  M  e.  ZZ )  -> 
sum_ k  e.  A 
0  =  0 )
3219, 31pm2.61dan 767 . 2  |-  ( A 
C_  ( ZZ>= `  M
)  ->  sum_ k  e.  A  0  =  0 )
33 fz1f1o 12504 . . 3  |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  (
( # `  A )  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) ) )
34 eqidd 2437 . . . . . . . . 9  |-  ( k  =  ( f `  n )  ->  0  =  0 )
35 simpl 444 . . . . . . . . 9  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  ( # `
 A )  e.  NN )
36 simpr 448 . . . . . . . . 9  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )
379a1i 11 . . . . . . . . 9  |-  ( ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  /\  k  e.  A )  ->  0  e.  CC )
38 elfznn 11080 . . . . . . . . . . 11  |-  ( n  e.  ( 1 ... ( # `  A
) )  ->  n  e.  NN )
394fvconst2 5947 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  (
( NN  X.  {
0 } ) `  n )  =  0 )
4038, 39syl 16 . . . . . . . . . 10  |-  ( n  e.  ( 1 ... ( # `  A
) )  ->  (
( NN  X.  {
0 } ) `  n )  =  0 )
4140adantl 453 . . . . . . . . 9  |-  ( ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  /\  n  e.  ( 1 ... ( # `
 A ) ) )  ->  ( ( NN  X.  { 0 } ) `  n )  =  0 )
4234, 35, 36, 37, 41fsum 12514 . . . . . . . 8  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  sum_ k  e.  A  0  =  (  seq  1 (  +  ,  ( NN  X.  { 0 } ) ) `  ( # `  A ) ) )
43 nnuz 10521 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
4443ser0 11375 . . . . . . . . 9  |-  ( (
# `  A )  e.  NN  ->  (  seq  1 (  +  , 
( NN  X.  {
0 } ) ) `
 ( # `  A
) )  =  0 )
4544adantr 452 . . . . . . . 8  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  (  seq  1 (  +  , 
( NN  X.  {
0 } ) ) `
 ( # `  A
) )  =  0 )
4642, 45eqtrd 2468 . . . . . . 7  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  sum_ k  e.  A  0  = 
0 )
4746ex 424 . . . . . 6  |-  ( (
# `  A )  e.  NN  ->  ( f : ( 1 ... ( # `  A
) ) -1-1-onto-> A  ->  sum_ k  e.  A  0  =  0 ) )
4847exlimdv 1646 . . . . 5  |-  ( (
# `  A )  e.  NN  ->  ( E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A  ->  sum_ k  e.  A  0  =  0 ) )
4948imp 419 . . . 4  |-  ( ( ( # `  A
)  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A )  ->  sum_ k  e.  A 
0  =  0 )
5030, 49jaoi 369 . . 3  |-  ( ( A  =  (/)  \/  (
( # `  A )  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) )  ->  sum_ k  e.  A  0  =  0 )
5133, 50syl 16 . 2  |-  ( A  e.  Fin  ->  sum_ k  e.  A  0  = 
0 )
5232, 51jaoi 369 1  |-  ( ( A  C_  ( ZZ>= `  M )  \/  A  e.  Fin )  ->  sum_ k  e.  A  0  = 
0 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725    C_ wss 3320   (/)c0 3628   ifcif 3739   ~Pcpw 3799   {csn 3814   class class class wbr 4212    X. cxp 4876   dom cdm 4878   Fun wfun 5448   -->wf 5450   -1-1-onto->wf1o 5453   ` cfv 5454  (class class class)co 6081   Fincfn 7109   CCcc 8988   0cc0 8990   1c1 8991    + caddc 8993   NNcn 10000   ZZcz 10282   ZZ>=cuz 10488   ...cfz 11043    seq cseq 11323   #chash 11618    ~~> cli 12278   sum_csu 12479
This theorem is referenced by:  fsum00  12577  fsumdvds  12893  pcfac  13268  ovoliunnul  19403  vitalilem5  19504  itg1addlem5  19592  itg10a  19602  itg0  19671  itgz  19672  plymullem1  20133  coemullem  20168  logtayl  20551  ftalem5  20859  chp1  20950  logexprlim  21009  bposlem2  21069  rpvmasumlem  21181  axcgrid  25855  axlowdimlem16  25896  volsupnfl  26251  stoweidlem37  27762
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-sum 12480
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