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Theorem sumz 12471
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 2404 . . . . 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 9041 . . . . . . . 8  |-  0  e.  _V
54fvconst2 5906 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( (
( ZZ>= `  M )  X.  { 0 } ) `
 k )  =  0 )
6 ifid 3731 . . . . . . 7  |-  if ( k  e.  A , 
0 ,  0 )  =  0
75, 6syl6eqr 2454 . . . . . 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 9040 . . . . . 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 12467 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  sum_ k  e.  A  0  =  ( 
~~>  `  seq  M (  +  ,  ( (
ZZ>= `  M )  X. 
{ 0 } ) ) ) )
12 fclim 12302 . . . . . 6  |-  ~~>  : dom  ~~>  --> CC
13 ffun 5552 . . . . . 6  |-  (  ~~>  : dom  ~~>  --> CC 
->  Fun  ~~>  )
1412, 13ax-mp 8 . . . . 5  |-  Fun  ~~>
15 serclim0 12326 . . . . . 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 5724 . . . . 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 2436 . . 3  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  sum_ k  e.  A  0  = 
0 )
20 uzf 10447 . . . . . . . . 9  |-  ZZ>= : ZZ --> ~P ZZ
2120fdmi 5555 . . . . . . . 8  |-  dom  ZZ>=  =  ZZ
2221eleq2i 2468 . . . . . . 7  |-  ( M  e.  dom  ZZ>=  <->  M  e.  ZZ )
23 ndmfv 5714 . . . . . . 7  |-  ( -.  M  e.  dom  ZZ>=  -> 
( ZZ>= `  M )  =  (/) )
2422, 23sylnbir 299 . . . . . 6  |-  ( -.  M  e.  ZZ  ->  (
ZZ>= `  M )  =  (/) )
2524sseq2d 3336 . . . . 5  |-  ( -.  M  e.  ZZ  ->  ( A  C_  ( ZZ>= `  M )  <->  A  C_  (/) ) )
2625biimpac 473 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  -.  M  e.  ZZ )  ->  A  C_  (/) )
27 ss0 3618 . . . 4  |-  ( A 
C_  (/)  ->  A  =  (/) )
28 sumeq1 12438 . . . . 5  |-  ( A  =  (/)  ->  sum_ k  e.  A  0  =  sum_ k  e.  (/)  0 )
29 sum0 12470 . . . . 5  |-  sum_ k  e.  (/)  0  =  0
3028, 29syl6eq 2452 . . . 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 12459 . . 3  |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  (
( # `  A )  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) ) )
34 eqidd 2405 . . . . . . . . 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 11036 . . . . . . . . . . 11  |-  ( n  e.  ( 1 ... ( # `  A
) )  ->  n  e.  NN )
394fvconst2 5906 . . . . . . . . . . 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 12469 . . . . . . . 8  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  sum_ k  e.  A  0  =  (  seq  1 (  +  ,  ( NN  X.  { 0 } ) ) `  ( # `  A ) ) )
43 nnuz 10477 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
4443ser0 11330 . . . . . . . . 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 2436 . . . . . . 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 1643 . . . . 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 1547    = wceq 1649    e. wcel 1721    C_ wss 3280   (/)c0 3588   ifcif 3699   ~Pcpw 3759   {csn 3774   class class class wbr 4172    X. cxp 4835   dom cdm 4837   Fun wfun 5407   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   Fincfn 7068   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949   NNcn 9956   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999    seq cseq 11278   #chash 11573    ~~> cli 12233   sum_csu 12434
This theorem is referenced by:  fsum00  12532  fsumdvds  12848  pcfac  13223  ovoliunnul  19356  vitalilem5  19457  itg1addlem5  19545  itg10a  19555  itg0  19624  itgz  19625  plymullem1  20086  coemullem  20121  logtayl  20504  ftalem5  20812  chp1  20903  logexprlim  20962  bposlem2  21022  rpvmasumlem  21134  axcgrid  25759  axlowdimlem16  25800  volsupnfl  26150  stoweidlem37  27653
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435
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