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Theorem sumz 13503
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 2467 . . . . 5  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2 simpr 461 . . . . 5  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  M  e.  ZZ )
3 simpl 457 . . . . 5  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  A  C_  ( ZZ>= `  M )
)
4 c0ex 9586 . . . . . . . 8  |-  0  e.  _V
54fvconst2 6114 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( (
( ZZ>= `  M )  X.  { 0 } ) `
 k )  =  0 )
6 ifid 3976 . . . . . . 7  |-  if ( k  e.  A , 
0 ,  0 )  =  0
75, 6syl6eqr 2526 . . . . . 6  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( (
( ZZ>= `  M )  X.  { 0 } ) `
 k )  =  if ( k  e.  A ,  0 ,  0 ) )
87adantl 466 . . . . 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 9585 . . . . 5  |-  ( ( ( A  C_  ( ZZ>=
`  M )  /\  M  e.  ZZ )  /\  k  e.  A
)  ->  0  e.  CC )
101, 2, 3, 8, 9zsum 13499 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  sum_ k  e.  A  0  =  ( 
~~>  `  seq M (  +  ,  ( (
ZZ>= `  M )  X. 
{ 0 } ) ) ) )
11 fclim 13335 . . . . . 6  |-  ~~>  : dom  ~~>  --> CC
12 ffun 5731 . . . . . 6  |-  (  ~~>  : dom  ~~>  --> CC 
->  Fun  ~~>  )
1311, 12ax-mp 5 . . . . 5  |-  Fun  ~~>
14 serclim0 13359 . . . . . 6  |-  ( M  e.  ZZ  ->  seq M (  +  , 
( ( ZZ>= `  M
)  X.  { 0 } ) )  ~~>  0 )
1514adantl 466 . . . . 5  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  seq M (  +  , 
( ( ZZ>= `  M
)  X.  { 0 } ) )  ~~>  0 )
16 funbrfv 5904 . . . . 5  |-  ( Fun  ~~>  ->  (  seq M (  +  ,  ( (
ZZ>= `  M )  X. 
{ 0 } ) )  ~~>  0  ->  (  ~~>  ` 
seq M (  +  ,  ( ( ZZ>= `  M )  X.  {
0 } ) ) )  =  0 ) )
1713, 15, 16mpsyl 63 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  (  ~~>  ` 
seq M (  +  ,  ( ( ZZ>= `  M )  X.  {
0 } ) ) )  =  0 )
1810, 17eqtrd 2508 . . 3  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  sum_ k  e.  A  0  = 
0 )
19 uzf 11081 . . . . . . . . 9  |-  ZZ>= : ZZ --> ~P ZZ
2019fdmi 5734 . . . . . . . 8  |-  dom  ZZ>=  =  ZZ
2120eleq2i 2545 . . . . . . 7  |-  ( M  e.  dom  ZZ>=  <->  M  e.  ZZ )
22 ndmfv 5888 . . . . . . 7  |-  ( -.  M  e.  dom  ZZ>=  -> 
( ZZ>= `  M )  =  (/) )
2321, 22sylnbir 307 . . . . . 6  |-  ( -.  M  e.  ZZ  ->  (
ZZ>= `  M )  =  (/) )
2423sseq2d 3532 . . . . 5  |-  ( -.  M  e.  ZZ  ->  ( A  C_  ( ZZ>= `  M )  <->  A  C_  (/) ) )
2524biimpac 486 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  -.  M  e.  ZZ )  ->  A  C_  (/) )
26 ss0 3816 . . . 4  |-  ( A 
C_  (/)  ->  A  =  (/) )
27 sumeq1 13470 . . . . 5  |-  ( A  =  (/)  ->  sum_ k  e.  A  0  =  sum_ k  e.  (/)  0 )
28 sum0 13502 . . . . 5  |-  sum_ k  e.  (/)  0  =  0
2927, 28syl6eq 2524 . . . 4  |-  ( A  =  (/)  ->  sum_ k  e.  A  0  = 
0 )
3025, 26, 293syl 20 . . 3  |-  ( ( A  C_  ( ZZ>= `  M )  /\  -.  M  e.  ZZ )  -> 
sum_ k  e.  A 
0  =  0 )
3118, 30pm2.61dan 789 . 2  |-  ( A 
C_  ( ZZ>= `  M
)  ->  sum_ k  e.  A  0  =  0 )
32 fz1f1o 13491 . . 3  |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  (
( # `  A )  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) ) )
33 eqidd 2468 . . . . . . . . 9  |-  ( k  =  ( f `  n )  ->  0  =  0 )
34 simpl 457 . . . . . . . . 9  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  ( # `
 A )  e.  NN )
35 simpr 461 . . . . . . . . 9  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )
36 0cnd 9585 . . . . . . . . 9  |-  ( ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  /\  k  e.  A )  ->  0  e.  CC )
37 elfznn 11710 . . . . . . . . . . 11  |-  ( n  e.  ( 1 ... ( # `  A
) )  ->  n  e.  NN )
384fvconst2 6114 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  (
( NN  X.  {
0 } ) `  n )  =  0 )
3937, 38syl 16 . . . . . . . . . 10  |-  ( n  e.  ( 1 ... ( # `  A
) )  ->  (
( NN  X.  {
0 } ) `  n )  =  0 )
4039adantl 466 . . . . . . . . 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 13501 . . . . . . . 8  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  sum_ k  e.  A  0  =  (  seq 1 (  +  ,  ( NN  X.  { 0 } ) ) `  ( # `  A ) ) )
42 nnuz 11113 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
4342ser0 12123 . . . . . . . . 9  |-  ( (
# `  A )  e.  NN  ->  (  seq 1 (  +  , 
( NN  X.  {
0 } ) ) `
 ( # `  A
) )  =  0 )
4443adantr 465 . . . . . . . 8  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  (  seq 1 (  +  , 
( NN  X.  {
0 } ) ) `
 ( # `  A
) )  =  0 )
4541, 44eqtrd 2508 . . . . . . 7  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  sum_ k  e.  A  0  = 
0 )
4645ex 434 . . . . . 6  |-  ( (
# `  A )  e.  NN  ->  ( f : ( 1 ... ( # `  A
) ) -1-1-onto-> A  ->  sum_ k  e.  A  0  =  0 ) )
4746exlimdv 1700 . . . . 5  |-  ( (
# `  A )  e.  NN  ->  ( E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A  ->  sum_ k  e.  A  0  =  0 ) )
4847imp 429 . . . 4  |-  ( ( ( # `  A
)  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A )  ->  sum_ k  e.  A 
0  =  0 )
4929, 48jaoi 379 . . 3  |-  ( ( A  =  (/)  \/  (
( # `  A )  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) )  ->  sum_ k  e.  A  0  =  0 )
5032, 49syl 16 . 2  |-  ( A  e.  Fin  ->  sum_ k  e.  A  0  = 
0 )
5131, 50jaoi 379 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 368    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767    C_ wss 3476   (/)c0 3785   ifcif 3939   ~Pcpw 4010   {csn 4027   class class class wbr 4447    X. cxp 4997   dom cdm 4999   Fun wfun 5580   -->wf 5582   -1-1-onto->wf1o 5585   ` cfv 5586  (class class class)co 6282   Fincfn 7513   CCcc 9486   0cc0 9488   1c1 9489    + caddc 9491   NNcn 10532   ZZcz 10860   ZZ>=cuz 11078   ...cfz 11668    seqcseq 12071   #chash 12369    ~~> cli 13266   sum_csu 13467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fz 11669  df-fzo 11789  df-seq 12072  df-exp 12131  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-sum 13468
This theorem is referenced by:  fsum00  13571  fsumdvds  13884  pcfac  14273  ovoliunnul  21653  vitalilem5  21756  itg1addlem5  21842  itg10a  21852  itg0  21921  itgz  21922  plymullem1  22346  coemullem  22381  logtayl  22769  ftalem5  23078  chp1  23169  logexprlim  23228  bposlem2  23288  rpvmasumlem  23400  axcgrid  23895  axlowdimlem16  23936  plymulx0  28144  signsplypnf  28147  volsupnfl  29636  sumnnodd  31172  stoweidlem37  31337  fourierdlem103  31510  fourierdlem104  31511
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