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Theorem sumz 13204
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 2443 . . . . 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 9385 . . . . . . . 8  |-  0  e.  _V
54fvconst2 5938 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( (
( ZZ>= `  M )  X.  { 0 } ) `
 k )  =  0 )
6 ifid 3831 . . . . . . 7  |-  if ( k  e.  A , 
0 ,  0 )  =  0
75, 6syl6eqr 2493 . . . . . 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 9384 . . . . 5  |-  ( ( ( A  C_  ( ZZ>=
`  M )  /\  M  e.  ZZ )  /\  k  e.  A
)  ->  0  e.  CC )
101, 2, 3, 8, 9zsum 13200 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  sum_ k  e.  A  0  =  ( 
~~>  `  seq M (  +  ,  ( (
ZZ>= `  M )  X. 
{ 0 } ) ) ) )
11 fclim 13036 . . . . . 6  |-  ~~>  : dom  ~~>  --> CC
12 ffun 5566 . . . . . 6  |-  (  ~~>  : dom  ~~>  --> CC 
->  Fun  ~~>  )
1311, 12ax-mp 5 . . . . 5  |-  Fun  ~~>
14 serclim0 13060 . . . . . 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 5735 . . . . 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 2475 . . 3  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  sum_ k  e.  A  0  = 
0 )
19 uzf 10869 . . . . . . . . 9  |-  ZZ>= : ZZ --> ~P ZZ
2019fdmi 5569 . . . . . . . 8  |-  dom  ZZ>=  =  ZZ
2120eleq2i 2507 . . . . . . 7  |-  ( M  e.  dom  ZZ>=  <->  M  e.  ZZ )
22 ndmfv 5719 . . . . . . 7  |-  ( -.  M  e.  dom  ZZ>=  -> 
( ZZ>= `  M )  =  (/) )
2321, 22sylnbir 307 . . . . . 6  |-  ( -.  M  e.  ZZ  ->  (
ZZ>= `  M )  =  (/) )
2423sseq2d 3389 . . . . 5  |-  ( -.  M  e.  ZZ  ->  ( A  C_  ( ZZ>= `  M )  <->  A  C_  (/) ) )
2524biimpac 486 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  -.  M  e.  ZZ )  ->  A  C_  (/) )
26 ss0 3673 . . . 4  |-  ( A 
C_  (/)  ->  A  =  (/) )
27 sumeq1 13171 . . . . 5  |-  ( A  =  (/)  ->  sum_ k  e.  A  0  =  sum_ k  e.  (/)  0 )
28 sum0 13203 . . . . 5  |-  sum_ k  e.  (/)  0  =  0
2927, 28syl6eq 2491 . . . 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 13192 . . 3  |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  (
( # `  A )  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) ) )
33 eqidd 2444 . . . . . . . . 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 9384 . . . . . . . . 9  |-  ( ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  /\  k  e.  A )  ->  0  e.  CC )
37 elfznn 11483 . . . . . . . . . . 11  |-  ( n  e.  ( 1 ... ( # `  A
) )  ->  n  e.  NN )
384fvconst2 5938 . . . . . . . . . . 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 13202 . . . . . . . 8  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  sum_ k  e.  A  0  =  (  seq 1 (  +  ,  ( NN  X.  { 0 } ) ) `  ( # `  A ) ) )
42 nnuz 10901 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
4342ser0 11863 . . . . . . . . 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 2475 . . . . . . 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 1690 . . . . 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 1369   E.wex 1586    e. wcel 1756    C_ wss 3333   (/)c0 3642   ifcif 3796   ~Pcpw 3865   {csn 3882   class class class wbr 4297    X. cxp 4843   dom cdm 4845   Fun wfun 5417   -->wf 5419   -1-1-onto->wf1o 5422   ` cfv 5423  (class class class)co 6096   Fincfn 7315   CCcc 9285   0cc0 9287   1c1 9288    + caddc 9290   NNcn 10327   ZZcz 10651   ZZ>=cuz 10866   ...cfz 11442    seqcseq 11811   #chash 12108    ~~> cli 12967   sum_csu 13168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-oi 7729  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-fz 11443  df-fzo 11554  df-seq 11812  df-exp 11871  df-hash 12109  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-clim 12971  df-sum 13169
This theorem is referenced by:  fsum00  13266  fsumdvds  13581  pcfac  13966  ovoliunnul  20995  vitalilem5  21097  itg1addlem5  21183  itg10a  21193  itg0  21262  itgz  21263  plymullem1  21687  coemullem  21722  logtayl  22110  ftalem5  22419  chp1  22510  logexprlim  22569  bposlem2  22629  rpvmasumlem  22741  axcgrid  23167  axlowdimlem16  23208  plymulx0  26953  signsplypnf  26956  volsupnfl  28441  stoweidlem37  29837
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