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Theorem sumz 13781
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 2423 . . . . 5  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2 simpr 463 . . . . 5  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  M  e.  ZZ )
3 simpl 459 . . . . 5  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  A  C_  ( ZZ>= `  M )
)
4 c0ex 9639 . . . . . . . 8  |-  0  e.  _V
54fvconst2 6133 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( (
( ZZ>= `  M )  X.  { 0 } ) `
 k )  =  0 )
6 ifid 3947 . . . . . . 7  |-  if ( k  e.  A , 
0 ,  0 )  =  0
75, 6syl6eqr 2482 . . . . . 6  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( (
( ZZ>= `  M )  X.  { 0 } ) `
 k )  =  if ( k  e.  A ,  0 ,  0 ) )
87adantl 468 . . . . 5  |-  ( ( ( A  C_  ( ZZ>=
`  M )  /\  M  e.  ZZ )  /\  k  e.  ( ZZ>=
`  M ) )  ->  ( ( (
ZZ>= `  M )  X. 
{ 0 } ) `
 k )  =  if ( k  e.  A ,  0 ,  0 ) )
9 0cnd 9638 . . . . 5  |-  ( ( ( A  C_  ( ZZ>=
`  M )  /\  M  e.  ZZ )  /\  k  e.  A
)  ->  0  e.  CC )
101, 2, 3, 8, 9zsum 13777 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  sum_ k  e.  A  0  =  ( 
~~>  `  seq M (  +  ,  ( (
ZZ>= `  M )  X. 
{ 0 } ) ) ) )
11 fclim 13610 . . . . . 6  |-  ~~>  : dom  ~~>  --> CC
12 ffun 5746 . . . . . 6  |-  (  ~~>  : dom  ~~>  --> CC 
->  Fun  ~~>  )
1311, 12ax-mp 5 . . . . 5  |-  Fun  ~~>
14 serclim0 13634 . . . . . 6  |-  ( M  e.  ZZ  ->  seq M (  +  , 
( ( ZZ>= `  M
)  X.  { 0 } ) )  ~~>  0 )
1514adantl 468 . . . . 5  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  seq M (  +  , 
( ( ZZ>= `  M
)  X.  { 0 } ) )  ~~>  0 )
16 funbrfv 5917 . . . . 5  |-  ( Fun  ~~>  ->  (  seq M (  +  ,  ( (
ZZ>= `  M )  X. 
{ 0 } ) )  ~~>  0  ->  (  ~~>  ` 
seq M (  +  ,  ( ( ZZ>= `  M )  X.  {
0 } ) ) )  =  0 ) )
1713, 15, 16mpsyl 66 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  (  ~~>  ` 
seq M (  +  ,  ( ( ZZ>= `  M )  X.  {
0 } ) ) )  =  0 )
1810, 17eqtrd 2464 . . 3  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  sum_ k  e.  A  0  = 
0 )
19 uzf 11164 . . . . . . . . 9  |-  ZZ>= : ZZ --> ~P ZZ
2019fdmi 5749 . . . . . . . 8  |-  dom  ZZ>=  =  ZZ
2120eleq2i 2501 . . . . . . 7  |-  ( M  e.  dom  ZZ>=  <->  M  e.  ZZ )
22 ndmfv 5903 . . . . . . 7  |-  ( -.  M  e.  dom  ZZ>=  -> 
( ZZ>= `  M )  =  (/) )
2321, 22sylnbir 309 . . . . . 6  |-  ( -.  M  e.  ZZ  ->  (
ZZ>= `  M )  =  (/) )
2423sseq2d 3493 . . . . 5  |-  ( -.  M  e.  ZZ  ->  ( A  C_  ( ZZ>= `  M )  <->  A  C_  (/) ) )
2524biimpac 489 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  -.  M  e.  ZZ )  ->  A  C_  (/) )
26 ss0 3794 . . . 4  |-  ( A 
C_  (/)  ->  A  =  (/) )
27 sumeq1 13748 . . . . 5  |-  ( A  =  (/)  ->  sum_ k  e.  A  0  =  sum_ k  e.  (/)  0 )
28 sum0 13780 . . . . 5  |-  sum_ k  e.  (/)  0  =  0
2927, 28syl6eq 2480 . . . 4  |-  ( A  =  (/)  ->  sum_ k  e.  A  0  = 
0 )
3025, 26, 293syl 18 . . 3  |-  ( ( A  C_  ( ZZ>= `  M )  /\  -.  M  e.  ZZ )  -> 
sum_ k  e.  A 
0  =  0 )
3118, 30pm2.61dan 799 . 2  |-  ( A 
C_  ( ZZ>= `  M
)  ->  sum_ k  e.  A  0  =  0 )
32 fz1f1o 13769 . . 3  |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  (
( # `  A )  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) ) )
33 eqidd 2424 . . . . . . . . 9  |-  ( k  =  ( f `  n )  ->  0  =  0 )
34 simpl 459 . . . . . . . . 9  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  ( # `
 A )  e.  NN )
35 simpr 463 . . . . . . . . 9  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )
36 0cnd 9638 . . . . . . . . 9  |-  ( ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  /\  k  e.  A )  ->  0  e.  CC )
37 elfznn 11830 . . . . . . . . . . 11  |-  ( n  e.  ( 1 ... ( # `  A
) )  ->  n  e.  NN )
384fvconst2 6133 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  (
( NN  X.  {
0 } ) `  n )  =  0 )
3937, 38syl 17 . . . . . . . . . 10  |-  ( n  e.  ( 1 ... ( # `  A
) )  ->  (
( NN  X.  {
0 } ) `  n )  =  0 )
4039adantl 468 . . . . . . . . 9  |-  ( ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  /\  n  e.  ( 1 ... ( # `
 A ) ) )  ->  ( ( NN  X.  { 0 } ) `  n )  =  0 )
4133, 34, 35, 36, 40fsum 13779 . . . . . . . 8  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  sum_ k  e.  A  0  =  (  seq 1 (  +  ,  ( NN  X.  { 0 } ) ) `  ( # `  A ) ) )
42 nnuz 11196 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
4342ser0 12266 . . . . . . . . 9  |-  ( (
# `  A )  e.  NN  ->  (  seq 1 (  +  , 
( NN  X.  {
0 } ) ) `
 ( # `  A
) )  =  0 )
4443adantr 467 . . . . . . . 8  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  (  seq 1 (  +  , 
( NN  X.  {
0 } ) ) `
 ( # `  A
) )  =  0 )
4541, 44eqtrd 2464 . . . . . . 7  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  sum_ k  e.  A  0  = 
0 )
4645ex 436 . . . . . 6  |-  ( (
# `  A )  e.  NN  ->  ( f : ( 1 ... ( # `  A
) ) -1-1-onto-> A  ->  sum_ k  e.  A  0  =  0 ) )
4746exlimdv 1769 . . . . 5  |-  ( (
# `  A )  e.  NN  ->  ( E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A  ->  sum_ k  e.  A  0  =  0 ) )
4847imp 431 . . . 4  |-  ( ( ( # `  A
)  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A )  ->  sum_ k  e.  A 
0  =  0 )
4929, 48jaoi 381 . . 3  |-  ( ( A  =  (/)  \/  (
( # `  A )  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) )  ->  sum_ k  e.  A  0  =  0 )
5032, 49syl 17 . 2  |-  ( A  e.  Fin  ->  sum_ k  e.  A  0  = 
0 )
5131, 50jaoi 381 1  |-  ( ( A  C_  ( ZZ>= `  M )  \/  A  e.  Fin )  ->  sum_ k  e.  A  0  = 
0 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 370    /\ wa 371    = wceq 1438   E.wex 1660    e. wcel 1869    C_ wss 3437   (/)c0 3762   ifcif 3910   ~Pcpw 3980   {csn 3997   class class class wbr 4421    X. cxp 4849   dom cdm 4851   Fun wfun 5593   -->wf 5595   -1-1-onto->wf1o 5598   ` cfv 5599  (class class class)co 6303   Fincfn 7575   CCcc 9539   0cc0 9541   1c1 9542    + caddc 9544   NNcn 10611   ZZcz 10939   ZZ>=cuz 11161   ...cfz 11786    seqcseq 12214   #chash 12516    ~~> cli 13541   sum_csu 13745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-inf2 8150  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-pre-sup 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-oadd 7192  df-er 7369  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-sup 7960  df-oi 8029  df-card 8376  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-n0 10872  df-z 10940  df-uz 11162  df-rp 11305  df-fz 11787  df-fzo 11918  df-seq 12215  df-exp 12274  df-hash 12517  df-cj 13156  df-re 13157  df-im 13158  df-sqrt 13292  df-abs 13293  df-clim 13545  df-sum 13746
This theorem is referenced by:  fsum00  13851  fsumdvds  14341  pcfac  14837  ovoliunnul  22452  vitalilem5  22562  itg1addlem5  22650  itg10a  22660  itg0  22729  itgz  22730  plymullem1  23160  coemullem  23196  logtayl  23597  ftalem5  23993  ftalem5OLD  23995  chp1  24086  logexprlim  24145  bposlem2  24205  rpvmasumlem  24317  axcgrid  24938  axlowdimlem16  24979  plymulx0  29438  signsplypnf  29441  volsupnfl  31943  binomcxplemnn0  36600  binomcxplemnotnn0  36607  sumnnodd  37574  stoweidlem37  37762  fourierdlem103  37937  fourierdlem104  37938  etransclem24  37987  etransclem32  37995  etransclem35  37998  sge0z  38049  aacllem  39884
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