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Theorem isumz 11390
Description: Any sum of zero over a summable set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Jim Kingdon, 9-Apr-2023.)
Assertion
Ref Expression
isumz  |-  ( ( ( M  e.  ZZ  /\  A  C_  ( ZZ>= `  M )  /\  A. j  e.  ( ZZ>= `  M )DECID  j  e.  A )  \/  A  e.  Fin )  ->  sum_ k  e.  A 
0  =  0 )
Distinct variable groups:    A, j, k   
j, M, k

Proof of Theorem isumz
Dummy variables  a  f  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2177 . . . 4  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2 simp1 997 . . . 4  |-  ( ( M  e.  ZZ  /\  A  C_  ( ZZ>= `  M
)  /\  A. j  e.  ( ZZ>= `  M )DECID  j  e.  A )  ->  M  e.  ZZ )
3 simp2 998 . . . 4  |-  ( ( M  e.  ZZ  /\  A  C_  ( ZZ>= `  M
)  /\  A. j  e.  ( ZZ>= `  M )DECID  j  e.  A )  ->  A  C_  ( ZZ>= `  M )
)
4 c0ex 7948 . . . . . . 7  |-  0  e.  _V
54fvconst2 5731 . . . . . 6  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( (
( ZZ>= `  M )  X.  { 0 } ) `
 k )  =  0 )
65adantl 277 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  A  C_  ( ZZ>= `  M )  /\  A. j  e.  ( ZZ>= `  M )DECID  j  e.  A )  /\  k  e.  (
ZZ>= `  M ) )  ->  ( ( (
ZZ>= `  M )  X. 
{ 0 } ) `
 k )  =  0 )
7 eleq1w 2238 . . . . . . . 8  |-  ( j  =  k  ->  (
j  e.  A  <->  k  e.  A ) )
87dcbid 838 . . . . . . 7  |-  ( j  =  k  ->  (DECID  j  e.  A  <-> DECID  k  e.  A )
)
9 simpl3 1002 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  A  C_  ( ZZ>= `  M )  /\  A. j  e.  ( ZZ>= `  M )DECID  j  e.  A )  /\  k  e.  (
ZZ>= `  M ) )  ->  A. j  e.  (
ZZ>= `  M )DECID  j  e.  A )
10 simpr 110 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  A  C_  ( ZZ>= `  M )  /\  A. j  e.  ( ZZ>= `  M )DECID  j  e.  A )  /\  k  e.  (
ZZ>= `  M ) )  ->  k  e.  (
ZZ>= `  M ) )
118, 9, 10rspcdva 2846 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  A  C_  ( ZZ>= `  M )  /\  A. j  e.  ( ZZ>= `  M )DECID  j  e.  A )  /\  k  e.  (
ZZ>= `  M ) )  -> DECID 
k  e.  A )
12 ifiddc 3568 . . . . . 6  |-  (DECID  k  e.  A  ->  if (
k  e.  A , 
0 ,  0 )  =  0 )
1311, 12syl 14 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  A  C_  ( ZZ>= `  M )  /\  A. j  e.  ( ZZ>= `  M )DECID  j  e.  A )  /\  k  e.  (
ZZ>= `  M ) )  ->  if ( k  e.  A ,  0 ,  0 )  =  0 )
146, 13eqtr4d 2213 . . . 4  |-  ( ( ( M  e.  ZZ  /\  A  C_  ( ZZ>= `  M )  /\  A. j  e.  ( ZZ>= `  M )DECID  j  e.  A )  /\  k  e.  (
ZZ>= `  M ) )  ->  ( ( (
ZZ>= `  M )  X. 
{ 0 } ) `
 k )  =  if ( k  e.  A ,  0 ,  0 ) )
15 simp3 999 . . . . 5  |-  ( ( M  e.  ZZ  /\  A  C_  ( ZZ>= `  M
)  /\  A. j  e.  ( ZZ>= `  M )DECID  j  e.  A )  ->  A. j  e.  ( ZZ>= `  M )DECID  j  e.  A )
16 eleq1w 2238 . . . . . . 7  |-  ( j  =  a  ->  (
j  e.  A  <->  a  e.  A ) )
1716dcbid 838 . . . . . 6  |-  ( j  =  a  ->  (DECID  j  e.  A  <-> DECID  a  e.  A )
)
1817cbvralv 2703 . . . . 5  |-  ( A. j  e.  ( ZZ>= `  M )DECID  j  e.  A  <->  A. a  e.  ( ZZ>= `  M )DECID  a  e.  A )
1915, 18sylib 122 . . . 4  |-  ( ( M  e.  ZZ  /\  A  C_  ( ZZ>= `  M
)  /\  A. j  e.  ( ZZ>= `  M )DECID  j  e.  A )  ->  A. a  e.  ( ZZ>= `  M )DECID  a  e.  A )
20 0cnd 7947 . . . 4  |-  ( ( ( M  e.  ZZ  /\  A  C_  ( ZZ>= `  M )  /\  A. j  e.  ( ZZ>= `  M )DECID  j  e.  A )  /\  k  e.  A
)  ->  0  e.  CC )
211, 2, 3, 14, 19, 20zsumdc 11385 . . 3  |-  ( ( M  e.  ZZ  /\  A  C_  ( ZZ>= `  M
)  /\  A. j  e.  ( ZZ>= `  M )DECID  j  e.  A )  ->  sum_ k  e.  A  0  =  ( 
~~>  `  seq M (  +  ,  ( (
ZZ>= `  M )  X. 
{ 0 } ) ) ) )
22 fclim 11295 . . . . 5  |-  ~~>  : dom  ~~>  --> CC
23 ffun 5367 . . . . 5  |-  (  ~~>  : dom  ~~>  --> CC 
->  Fun  ~~>  )
2422, 23ax-mp 5 . . . 4  |-  Fun  ~~>
25 serclim0 11306 . . . . 5  |-  ( M  e.  ZZ  ->  seq M (  +  , 
( ( ZZ>= `  M
)  X.  { 0 } ) )  ~~>  0 )
262, 25syl 14 . . . 4  |-  ( ( M  e.  ZZ  /\  A  C_  ( ZZ>= `  M
)  /\  A. j  e.  ( ZZ>= `  M )DECID  j  e.  A )  ->  seq M (  +  , 
( ( ZZ>= `  M
)  X.  { 0 } ) )  ~~>  0 )
27 funbrfv 5553 . . . 4  |-  ( Fun  ~~>  ->  (  seq M (  +  ,  ( (
ZZ>= `  M )  X. 
{ 0 } ) )  ~~>  0  ->  (  ~~>  ` 
seq M (  +  ,  ( ( ZZ>= `  M )  X.  {
0 } ) ) )  =  0 ) )
2824, 26, 27mpsyl 65 . . 3  |-  ( ( M  e.  ZZ  /\  A  C_  ( ZZ>= `  M
)  /\  A. j  e.  ( ZZ>= `  M )DECID  j  e.  A )  ->  (  ~~>  ` 
seq M (  +  ,  ( ( ZZ>= `  M )  X.  {
0 } ) ) )  =  0 )
2921, 28eqtrd 2210 . 2  |-  ( ( M  e.  ZZ  /\  A  C_  ( ZZ>= `  M
)  /\  A. j  e.  ( ZZ>= `  M )DECID  j  e.  A )  ->  sum_ k  e.  A  0  = 
0 )
30 fz1f1o 11376 . . 3  |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  (
( `  A )  e.  NN  /\  E. f 
f : ( 1 ... ( `  A
) ) -1-1-onto-> A ) ) )
31 sumeq1 11356 . . . . 5  |-  ( A  =  (/)  ->  sum_ k  e.  A  0  =  sum_ k  e.  (/)  0 )
32 sum0 11389 . . . . 5  |-  sum_ k  e.  (/)  0  =  0
3331, 32eqtrdi 2226 . . . 4  |-  ( A  =  (/)  ->  sum_ k  e.  A  0  = 
0 )
34 eqidd 2178 . . . . . . . . 9  |-  ( k  =  ( f `  n )  ->  0  =  0 )
35 simpl 109 . . . . . . . . 9  |-  ( ( ( `  A )  e.  NN  /\  f : ( 1 ... ( `  A ) ) -1-1-onto-> A )  ->  ( `  A )  e.  NN )
36 simpr 110 . . . . . . . . 9  |-  ( ( ( `  A )  e.  NN  /\  f : ( 1 ... ( `  A ) ) -1-1-onto-> A )  ->  f : ( 1 ... ( `  A
) ) -1-1-onto-> A )
37 0cnd 7947 . . . . . . . . 9  |-  ( ( ( ( `  A
)  e.  NN  /\  f : ( 1 ... ( `  A )
)
-1-1-onto-> A )  /\  k  e.  A )  ->  0  e.  CC )
38 elfznn 10049 . . . . . . . . . . 11  |-  ( n  e.  ( 1 ... ( `  A )
)  ->  n  e.  NN )
394fvconst2 5731 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  (
( NN  X.  {
0 } ) `  n )  =  0 )
4038, 39syl 14 . . . . . . . . . 10  |-  ( n  e.  ( 1 ... ( `  A )
)  ->  ( ( NN  X.  { 0 } ) `  n )  =  0 )
4140adantl 277 . . . . . . . . 9  |-  ( ( ( ( `  A
)  e.  NN  /\  f : ( 1 ... ( `  A )
)
-1-1-onto-> A )  /\  n  e.  ( 1 ... ( `  A ) ) )  ->  ( ( NN 
X.  { 0 } ) `  n )  =  0 )
4234, 35, 36, 37, 41fsum3 11388 . . . . . . . 8  |-  ( ( ( `  A )  e.  NN  /\  f : ( 1 ... ( `  A ) ) -1-1-onto-> A )  ->  sum_ k  e.  A 
0  =  (  seq 1 (  +  , 
( n  e.  NN  |->  if ( n  <_  ( `  A ) ,  ( ( NN  X.  {
0 } ) `  n ) ,  0 ) ) ) `  ( `  A ) ) )
43 nnuz 9559 . . . . . . . . . . . . 13  |-  NN  =  ( ZZ>= `  1 )
4443fser0const 10511 . . . . . . . . . . . 12  |-  ( ( `  A )  e.  NN  ->  ( n  e.  NN  |->  if ( n  <_  ( `  A ) ,  ( ( NN  X.  {
0 } ) `  n ) ,  0 ) )  =  ( NN  X.  { 0 } ) )
4544seqeq3d 10448 . . . . . . . . . . 11  |-  ( ( `  A )  e.  NN  ->  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  ( `  A ) ,  ( ( NN 
X.  { 0 } ) `  n ) ,  0 ) ) )  =  seq 1
(  +  ,  ( NN  X.  { 0 } ) ) )
4645fveq1d 5516 . . . . . . . . . 10  |-  ( ( `  A )  e.  NN  ->  (  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  ( `  A
) ,  ( ( NN  X.  { 0 } ) `  n
) ,  0 ) ) ) `  ( `  A ) )  =  (  seq 1 (  +  ,  ( NN 
X.  { 0 } ) ) `  ( `  A ) ) )
4743ser0 10509 . . . . . . . . . 10  |-  ( ( `  A )  e.  NN  ->  (  seq 1 (  +  ,  ( NN 
X.  { 0 } ) ) `  ( `  A ) )  =  0 )
4846, 47eqtrd 2210 . . . . . . . . 9  |-  ( ( `  A )  e.  NN  ->  (  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  ( `  A
) ,  ( ( NN  X.  { 0 } ) `  n
) ,  0 ) ) ) `  ( `  A ) )  =  0 )
4935, 48syl 14 . . . . . . . 8  |-  ( ( ( `  A )  e.  NN  /\  f : ( 1 ... ( `  A ) ) -1-1-onto-> A )  ->  (  seq 1
(  +  ,  ( n  e.  NN  |->  if ( n  <_  ( `  A ) ,  ( ( NN  X.  {
0 } ) `  n ) ,  0 ) ) ) `  ( `  A ) )  =  0 )
5042, 49eqtrd 2210 . . . . . . 7  |-  ( ( ( `  A )  e.  NN  /\  f : ( 1 ... ( `  A ) ) -1-1-onto-> A )  ->  sum_ k  e.  A 
0  =  0 )
5150ex 115 . . . . . 6  |-  ( ( `  A )  e.  NN  ->  ( f : ( 1 ... ( `  A
) ) -1-1-onto-> A  ->  sum_ k  e.  A  0  =  0 ) )
5251exlimdv 1819 . . . . 5  |-  ( ( `  A )  e.  NN  ->  ( E. f  f : ( 1 ... ( `  A )
)
-1-1-onto-> A  ->  sum_ k  e.  A 
0  =  0 ) )
5352imp 124 . . . 4  |-  ( ( ( `  A )  e.  NN  /\  E. f 
f : ( 1 ... ( `  A
) ) -1-1-onto-> A )  ->  sum_ k  e.  A  0  = 
0 )
5433, 53jaoi 716 . . 3  |-  ( ( A  =  (/)  \/  (
( `  A )  e.  NN  /\  E. f 
f : ( 1 ... ( `  A
) ) -1-1-onto-> A ) )  ->  sum_ k  e.  A  0  =  0 )
5530, 54syl 14 . 2  |-  ( A  e.  Fin  ->  sum_ k  e.  A  0  = 
0 )
5629, 55jaoi 716 1  |-  ( ( ( M  e.  ZZ  /\  A  C_  ( ZZ>= `  M )  /\  A. j  e.  ( ZZ>= `  M )DECID  j  e.  A )  \/  A  e.  Fin )  ->  sum_ k  e.  A 
0  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    \/ wo 708  DECID wdc 834    /\ w3a 978    = wceq 1353   E.wex 1492    e. wcel 2148   A.wral 2455    C_ wss 3129   (/)c0 3422   ifcif 3534   {csn 3592   class class class wbr 4002    |-> cmpt 4063    X. cxp 4623   dom cdm 4625   Fun wfun 5209   -->wf 5211   -1-1-onto->wf1o 5214   ` cfv 5215  (class class class)co 5872   Fincfn 6737   CCcc 7806   0cc0 7808   1c1 7809    + caddc 7811    <_ cle 7989   NNcn 8915   ZZcz 9249   ZZ>=cuz 9524   ...cfz 10004    seqcseq 10440  ♯chash 10748    ~~> cli 11279   sum_csu 11354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-iinf 4586  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-mulrcl 7907  ax-addcom 7908  ax-mulcom 7909  ax-addass 7910  ax-mulass 7911  ax-distr 7912  ax-i2m1 7913  ax-0lt1 7914  ax-1rid 7915  ax-0id 7916  ax-rnegex 7917  ax-precex 7918  ax-cnre 7919  ax-pre-ltirr 7920  ax-pre-ltwlin 7921  ax-pre-lttrn 7922  ax-pre-apti 7923  ax-pre-ltadd 7924  ax-pre-mulgt0 7925  ax-pre-mulext 7926  ax-arch 7927  ax-caucvg 7928
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-tr 4101  df-id 4292  df-po 4295  df-iso 4296  df-iord 4365  df-on 4367  df-ilim 4368  df-suc 4370  df-iom 4589  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-isom 5224  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-1st 6138  df-2nd 6139  df-recs 6303  df-irdg 6368  df-frec 6389  df-1o 6414  df-oadd 6418  df-er 6532  df-en 6738  df-dom 6739  df-fin 6740  df-pnf 7990  df-mnf 7991  df-xr 7992  df-ltxr 7993  df-le 7994  df-sub 8126  df-neg 8127  df-reap 8528  df-ap 8535  df-div 8626  df-inn 8916  df-2 8974  df-3 8975  df-4 8976  df-n0 9173  df-z 9250  df-uz 9525  df-q 9616  df-rp 9650  df-fz 10005  df-fzo 10138  df-seqfrec 10441  df-exp 10515  df-ihash 10749  df-cj 10844  df-re 10845  df-im 10846  df-rsqrt 11000  df-abs 11001  df-clim 11280  df-sumdc 11355
This theorem is referenced by:  fsum00  11463  pcfac  12340  nconstwlpolem0  14670
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