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Theorem isumz 10745
Description: Any sum of zero over a summable set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Jim Kingdon, 16-Sep-2022.)
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 2088 . . . 4  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2 simp1 943 . . . 4  |-  ( ( M  e.  ZZ  /\  A  C_  ( ZZ>= `  M
)  /\  A. j  e.  ( ZZ>= `  M )DECID  j  e.  A )  ->  M  e.  ZZ )
3 simp2 944 . . . 4  |-  ( ( M  e.  ZZ  /\  A  C_  ( ZZ>= `  M
)  /\  A. j  e.  ( ZZ>= `  M )DECID  j  e.  A )  ->  A  C_  ( ZZ>= `  M )
)
4 c0ex 7461 . . . . . . 7  |-  0  e.  _V
54fvconst2 5495 . . . . . 6  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( (
( ZZ>= `  M )  X.  { 0 } ) `
 k )  =  0 )
65adantl 271 . . . . 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 2148 . . . . . . . 8  |-  ( j  =  k  ->  (
j  e.  A  <->  k  e.  A ) )
87dcbid 786 . . . . . . 7  |-  ( j  =  k  ->  (DECID  j  e.  A  <-> DECID  k  e.  A )
)
9 simpl3 948 . . . . . . 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 108 . . . . . . 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 2727 . . . . . 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 3420 . . . . . 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 2123 . . . 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 945 . . . . 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 2148 . . . . . . 7  |-  ( j  =  a  ->  (
j  e.  A  <->  a  e.  A ) )
1716dcbid 786 . . . . . 6  |-  ( j  =  a  ->  (DECID  j  e.  A  <-> DECID  a  e.  A )
)
1817cbvralv 2590 . . . . 5  |-  ( A. j  e.  ( ZZ>= `  M )DECID  j  e.  A  <->  A. a  e.  ( ZZ>= `  M )DECID  a  e.  A )
1915, 18sylib 120 . . . 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 7460 . . . 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, 20zisum 10738 . . 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 } ) ,  CC ) ) )
22 fclim 10646 . . . . 5  |-  ~~>  : dom  ~~>  --> CC
23 ffun 5150 . . . . 5  |-  (  ~~>  : dom  ~~>  --> CC 
->  Fun  ~~>  )
2422, 23ax-mp 7 . . . 4  |-  Fun  ~~>
25 iserclim0 10658 . . . . 5  |-  ( M  e.  ZZ  ->  seq M (  +  , 
( ( ZZ>= `  M
)  X.  { 0 } ) ,  CC ) 
~~>  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 } ) ,  CC ) 
~~>  0 )
27 funbrfv 5327 . . . 4  |-  ( Fun  ~~>  ->  (  seq M (  +  ,  ( (
ZZ>= `  M )  X. 
{ 0 } ) ,  CC )  ~~>  0  -> 
(  ~~>  `  seq M (  +  ,  ( (
ZZ>= `  M )  X. 
{ 0 } ) ,  CC ) )  =  0 ) )
2824, 26, 27mpsyl 64 . . 3  |-  ( ( M  e.  ZZ  /\  A  C_  ( ZZ>= `  M
)  /\  A. j  e.  ( ZZ>= `  M )DECID  j  e.  A )  ->  (  ~~>  ` 
seq M (  +  ,  ( ( ZZ>= `  M )  X.  {
0 } ) ,  CC ) )  =  0 )
2921, 28eqtrd 2120 . 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 10728 . . 3  |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  (
( `  A )  e.  NN  /\  E. f 
f : ( 1 ... ( `  A
) ) -1-1-onto-> A ) ) )
31 sumeq1 10708 . . . . 5  |-  ( A  =  (/)  ->  sum_ k  e.  A  0  =  sum_ k  e.  (/)  0 )
32 sum0 10744 . . . . 5  |-  sum_ k  e.  (/)  0  =  0
3331, 32syl6eq 2136 . . . 4  |-  ( A  =  (/)  ->  sum_ k  e.  A  0  = 
0 )
34 eqidd 2089 . . . . . . . . 9  |-  ( k  =  ( f `  n )  ->  0  =  0 )
35 simpl 107 . . . . . . . . 9  |-  ( ( ( `  A )  e.  NN  /\  f : ( 1 ... ( `  A ) ) -1-1-onto-> A )  ->  ( `  A )  e.  NN )
36 simpr 108 . . . . . . . . 9  |-  ( ( ( `  A )  e.  NN  /\  f : ( 1 ... ( `  A ) ) -1-1-onto-> A )  ->  f : ( 1 ... ( `  A
) ) -1-1-onto-> A )
37 0cnd 7460 . . . . . . . . 9  |-  ( ( ( ( `  A
)  e.  NN  /\  f : ( 1 ... ( `  A )
)
-1-1-onto-> A )  /\  k  e.  A )  ->  0  e.  CC )
38 elfznn 9437 . . . . . . . . . . 11  |-  ( n  e.  ( 1 ... ( `  A )
)  ->  n  e.  NN )
394fvconst2 5495 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  (
( NN  X.  {
0 } ) `  n )  =  0 )
4038, 39syl 14 . . . . . . . . . 10  |-  ( n  e.  ( 1 ... ( `  A )
)  ->  ( ( NN  X.  { 0 } ) `  n )  =  0 )
4140adantl 271 . . . . . . . . 9  |-  ( ( ( ( `  A
)  e.  NN  /\  f : ( 1 ... ( `  A )
)
-1-1-onto-> A )  /\  n  e.  ( 1 ... ( `  A ) ) )  ->  ( ( NN 
X.  { 0 } ) `  n )  =  0 )
4234, 35, 36, 37, 41fisum 10742 . . . . . . . 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 ) ) ,  CC ) `  ( `  A
) ) )
43 nnuz 9023 . . . . . . . . . . . . 13  |-  NN  =  ( ZZ>= `  1 )
4443fser0const 9916 . . . . . . . . . . . 12  |-  ( ( `  A )  e.  NN  ->  ( n  e.  NN  |->  if ( n  <_  ( `  A ) ,  ( ( NN  X.  {
0 } ) `  n ) ,  0 ) )  =  ( NN  X.  { 0 } ) )
45 iseqeq3 9825 . . . . . . . . . . . 12  |-  ( ( n  e.  NN  |->  if ( n  <_  ( `  A ) ,  ( ( NN  X.  {
0 } ) `  n ) ,  0 ) )  =  ( NN  X.  { 0 } )  ->  seq 1 (  +  , 
( n  e.  NN  |->  if ( n  <_  ( `  A ) ,  ( ( NN  X.  {
0 } ) `  n ) ,  0 ) ) ,  CC )  =  seq 1
(  +  ,  ( NN  X.  { 0 } ) ,  CC ) )
4644, 45syl 14 . . . . . . . . . . 11  |-  ( ( `  A )  e.  NN  ->  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  ( `  A ) ,  ( ( NN 
X.  { 0 } ) `  n ) ,  0 ) ) ,  CC )  =  seq 1 (  +  ,  ( NN  X.  { 0 } ) ,  CC ) )
4746fveq1d 5291 . . . . . . . . . 10  |-  ( ( `  A )  e.  NN  ->  (  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  ( `  A
) ,  ( ( NN  X.  { 0 } ) `  n
) ,  0 ) ) ,  CC ) `
 ( `  A
) )  =  (  seq 1 (  +  ,  ( NN  X.  { 0 } ) ,  CC ) `  ( `  A ) ) )
4843iser0 9912 . . . . . . . . . 10  |-  ( ( `  A )  e.  NN  ->  (  seq 1 (  +  ,  ( NN 
X.  { 0 } ) ,  CC ) `
 ( `  A
) )  =  0 )
4947, 48eqtrd 2120 . . . . . . . . 9  |-  ( ( `  A )  e.  NN  ->  (  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  ( `  A
) ,  ( ( NN  X.  { 0 } ) `  n
) ,  0 ) ) ,  CC ) `
 ( `  A
) )  =  0 )
5035, 49syl 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 ) ) ,  CC ) `  ( `  A
) )  =  0 )
5142, 50eqtrd 2120 . . . . . . 7  |-  ( ( ( `  A )  e.  NN  /\  f : ( 1 ... ( `  A ) ) -1-1-onto-> A )  ->  sum_ k  e.  A 
0  =  0 )
5251ex 113 . . . . . 6  |-  ( ( `  A )  e.  NN  ->  ( f : ( 1 ... ( `  A
) ) -1-1-onto-> A  ->  sum_ k  e.  A  0  =  0 ) )
5352exlimdv 1747 . . . . 5  |-  ( ( `  A )  e.  NN  ->  ( E. f  f : ( 1 ... ( `  A )
)
-1-1-onto-> A  ->  sum_ k  e.  A 
0  =  0 ) )
5453imp 122 . . . 4  |-  ( ( ( `  A )  e.  NN  /\  E. f 
f : ( 1 ... ( `  A
) ) -1-1-onto-> A )  ->  sum_ k  e.  A  0  = 
0 )
5533, 54jaoi 671 . . 3  |-  ( ( A  =  (/)  \/  (
( `  A )  e.  NN  /\  E. f 
f : ( 1 ... ( `  A
) ) -1-1-onto-> A ) )  ->  sum_ k  e.  A  0  =  0 )
5630, 55syl 14 . 2  |-  ( A  e.  Fin  ->  sum_ k  e.  A  0  = 
0 )
5729, 56jaoi 671 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 102    \/ wo 664  DECID wdc 780    /\ w3a 924    = wceq 1289   E.wex 1426    e. wcel 1438   A.wral 2359    C_ wss 2997   (/)c0 3284   ifcif 3389   {csn 3441   class class class wbr 3837    |-> cmpt 3891    X. cxp 4426   dom cdm 4428   Fun wfun 4996   -->wf 4998   -1-1-onto->wf1o 5001   ` cfv 5002  (class class class)co 5634   Fincfn 6437   CCcc 7327   0cc0 7329   1c1 7330    + caddc 7332    <_ cle 7502   NNcn 8394   ZZcz 8720   ZZ>=cuz 8988   ...cfz 9393    seqcseq4 9816  ♯chash 10148    ~~> cli 10630   sum_csu 10706
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442  ax-arch 7443  ax-caucvg 7444
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-isom 5011  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-frec 6138  df-1o 6163  df-oadd 6167  df-er 6272  df-en 6438  df-dom 6439  df-fin 6440  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-3 8453  df-4 8454  df-n0 8644  df-z 8721  df-uz 8989  df-q 9074  df-rp 9104  df-fz 9394  df-fzo 9519  df-iseq 9818  df-seq3 9819  df-exp 9920  df-ihash 10149  df-cj 10241  df-re 10242  df-im 10243  df-rsqrt 10396  df-abs 10397  df-clim 10631  df-isum 10707
This theorem is referenced by:  fsum00  10819
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