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Theorem fsumcvg4 27568
Description: A serie with finite support is a finite sum, and therefore converges. (Contributed by Thierry Arnoux, 6-Sep-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
Hypotheses
Ref Expression
fsumcvg4.s  |-  S  =  ( ZZ>= `  M )
fsumcvg4.m  |-  ( ph  ->  M  e.  ZZ )
fsumcvg4.c  |-  ( ph  ->  F : S --> CC )
fsumcvg4.f  |-  ( ph  ->  ( `' F "
( CC  \  {
0 } ) )  e.  Fin )
Assertion
Ref Expression
fsumcvg4  |-  ( ph  ->  seq M (  +  ,  F )  e. 
dom 
~~>  )

Proof of Theorem fsumcvg4
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 fsumcvg4.s . 2  |-  S  =  ( ZZ>= `  M )
2 fsumcvg4.m . 2  |-  ( ph  ->  M  e.  ZZ )
3 fsumcvg4.f . 2  |-  ( ph  ->  ( `' F "
( CC  \  {
0 } ) )  e.  Fin )
4 fsumcvg4.c . . . . 5  |-  ( ph  ->  F : S --> CC )
5 ffun 5731 . . . . 5  |-  ( F : S --> CC  ->  Fun 
F )
6 difpreima 6007 . . . . 5  |-  ( Fun 
F  ->  ( `' F " ( CC  \  { 0 } ) )  =  ( ( `' F " CC ) 
\  ( `' F " { 0 } ) ) )
74, 5, 63syl 20 . . . 4  |-  ( ph  ->  ( `' F "
( CC  \  {
0 } ) )  =  ( ( `' F " CC ) 
\  ( `' F " { 0 } ) ) )
8 difss 3631 . . . 4  |-  ( ( `' F " CC ) 
\  ( `' F " { 0 } ) )  C_  ( `' F " CC )
97, 8syl6eqss 3554 . . 3  |-  ( ph  ->  ( `' F "
( CC  \  {
0 } ) ) 
C_  ( `' F " CC ) )
10 fimacnv 6011 . . . 4  |-  ( F : S --> CC  ->  ( `' F " CC )  =  S )
114, 10syl 16 . . 3  |-  ( ph  ->  ( `' F " CC )  =  S
)
129, 11sseqtrd 3540 . 2  |-  ( ph  ->  ( `' F "
( CC  \  {
0 } ) ) 
C_  S )
13 exmidd 416 . . . 4  |-  ( (
ph  /\  k  e.  S )  ->  (
k  e.  ( `' F " ( CC 
\  { 0 } ) )  \/  -.  k  e.  ( `' F " ( CC  \  { 0 } ) ) ) )
14 eqid 2467 . . . . . . 7  |-  ( F `
 k )  =  ( F `  k
)
1514biantru 505 . . . . . 6  |-  ( k  e.  ( `' F " ( CC  \  {
0 } ) )  <-> 
( k  e.  ( `' F " ( CC 
\  { 0 } ) )  /\  ( F `  k )  =  ( F `  k ) ) )
1615a1i 11 . . . . 5  |-  ( (
ph  /\  k  e.  S )  ->  (
k  e.  ( `' F " ( CC 
\  { 0 } ) )  <->  ( k  e.  ( `' F "
( CC  \  {
0 } ) )  /\  ( F `  k )  =  ( F `  k ) ) ) )
17 fvex 5874 . . . . . . . . . . . . . . 15  |-  ( ZZ>= `  M )  e.  _V
181, 17eqeltri 2551 . . . . . . . . . . . . . 14  |-  S  e. 
_V
1918a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  S  e.  _V )
20 0nn0 10806 . . . . . . . . . . . . . 14  |-  0  e.  NN0
2120a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  0  e.  NN0 )
22 eqid 2467 . . . . . . . . . . . . . 14  |-  ( CC 
\  { 0 } )  =  ( CC 
\  { 0 } )
2322ffs2 27223 . . . . . . . . . . . . 13  |-  ( ( S  e.  _V  /\  0  e.  NN0  /\  F : S --> CC )  -> 
( F supp  0 )  =  ( `' F " ( CC  \  {
0 } ) ) )
2419, 21, 4, 23syl3anc 1228 . . . . . . . . . . . 12  |-  ( ph  ->  ( F supp  0 )  =  ( `' F " ( CC  \  {
0 } ) ) )
25 ffn 5729 . . . . . . . . . . . . . 14  |-  ( F : S --> CC  ->  F  Fn  S )
264, 25syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  F  Fn  S )
27 suppvalfn 6905 . . . . . . . . . . . . 13  |-  ( ( F  Fn  S  /\  S  e.  _V  /\  0  e.  NN0 )  ->  ( F supp  0 )  =  {
k  e.  S  | 
( F `  k
)  =/=  0 } )
2826, 19, 21, 27syl3anc 1228 . . . . . . . . . . . 12  |-  ( ph  ->  ( F supp  0 )  =  { k  e.  S  |  ( F `
 k )  =/=  0 } )
2924, 28eqtr3d 2510 . . . . . . . . . . 11  |-  ( ph  ->  ( `' F "
( CC  \  {
0 } ) )  =  { k  e.  S  |  ( F `
 k )  =/=  0 } )
3029eleq2d 2537 . . . . . . . . . 10  |-  ( ph  ->  ( k  e.  ( `' F " ( CC 
\  { 0 } ) )  <->  k  e.  { k  e.  S  | 
( F `  k
)  =/=  0 } ) )
31 rabid 3038 . . . . . . . . . 10  |-  ( k  e.  { k  e.  S  |  ( F `
 k )  =/=  0 }  <->  ( k  e.  S  /\  ( F `  k )  =/=  0 ) )
3230, 31syl6bb 261 . . . . . . . . 9  |-  ( ph  ->  ( k  e.  ( `' F " ( CC 
\  { 0 } ) )  <->  ( k  e.  S  /\  ( F `  k )  =/=  0 ) ) )
3332baibd 907 . . . . . . . 8  |-  ( (
ph  /\  k  e.  S )  ->  (
k  e.  ( `' F " ( CC 
\  { 0 } ) )  <->  ( F `  k )  =/=  0
) )
3433necon2bbid 2723 . . . . . . 7  |-  ( (
ph  /\  k  e.  S )  ->  (
( F `  k
)  =  0  <->  -.  k  e.  ( `' F " ( CC  \  { 0 } ) ) ) )
3534biimprd 223 . . . . . 6  |-  ( (
ph  /\  k  e.  S )  ->  ( -.  k  e.  ( `' F " ( CC 
\  { 0 } ) )  ->  ( F `  k )  =  0 ) )
3635pm4.71d 634 . . . . 5  |-  ( (
ph  /\  k  e.  S )  ->  ( -.  k  e.  ( `' F " ( CC 
\  { 0 } ) )  <->  ( -.  k  e.  ( `' F " ( CC  \  { 0 } ) )  /\  ( F `
 k )  =  0 ) ) )
3716, 36orbi12d 709 . . . 4  |-  ( (
ph  /\  k  e.  S )  ->  (
( k  e.  ( `' F " ( CC 
\  { 0 } ) )  \/  -.  k  e.  ( `' F " ( CC  \  { 0 } ) ) )  <->  ( (
k  e.  ( `' F " ( CC 
\  { 0 } ) )  /\  ( F `  k )  =  ( F `  k ) )  \/  ( -.  k  e.  ( `' F "
( CC  \  {
0 } ) )  /\  ( F `  k )  =  0 ) ) ) )
3813, 37mpbid 210 . . 3  |-  ( (
ph  /\  k  e.  S )  ->  (
( k  e.  ( `' F " ( CC 
\  { 0 } ) )  /\  ( F `  k )  =  ( F `  k ) )  \/  ( -.  k  e.  ( `' F "
( CC  \  {
0 } ) )  /\  ( F `  k )  =  0 ) ) )
39 eqif 3977 . . 3  |-  ( ( F `  k )  =  if ( k  e.  ( `' F " ( CC  \  {
0 } ) ) ,  ( F `  k ) ,  0 )  <->  ( ( k  e.  ( `' F " ( CC  \  {
0 } ) )  /\  ( F `  k )  =  ( F `  k ) )  \/  ( -.  k  e.  ( `' F " ( CC 
\  { 0 } ) )  /\  ( F `  k )  =  0 ) ) )
4038, 39sylibr 212 . 2  |-  ( (
ph  /\  k  e.  S )  ->  ( F `  k )  =  if ( k  e.  ( `' F "
( CC  \  {
0 } ) ) ,  ( F `  k ) ,  0 ) )
4112sselda 3504 . . 3  |-  ( (
ph  /\  k  e.  ( `' F " ( CC 
\  { 0 } ) ) )  -> 
k  e.  S )
424ffvelrnda 6019 . . 3  |-  ( (
ph  /\  k  e.  S )  ->  ( F `  k )  e.  CC )
4341, 42syldan 470 . 2  |-  ( (
ph  /\  k  e.  ( `' F " ( CC 
\  { 0 } ) ) )  -> 
( F `  k
)  e.  CC )
441, 2, 3, 12, 40, 43fsumcvg3 13510 1  |-  ( ph  ->  seq M (  +  ,  F )  e. 
dom 
~~>  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   {crab 2818   _Vcvv 3113    \ cdif 3473   ifcif 3939   {csn 4027   `'ccnv 4998   dom cdm 4999   "cima 5002   Fun wfun 5580    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   supp csupp 6898   Fincfn 7513   CCcc 9486   0cc0 9488    + caddc 9491   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11078    seqcseq 12071    ~~> cli 13266
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-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-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-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  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-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fz 11669  df-seq 12072  df-exp 12131  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270
This theorem is referenced by:  eulerpartlems  27939
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