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Theorem fsumcvg4 26518
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 5662 . . . . 5  |-  ( F : S --> CC  ->  Fun 
F )
6 difpreima 5933 . . . . 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 3584 . . . 4  |-  ( ( `' F " CC ) 
\  ( `' F " { 0 } ) )  C_  ( `' F " CC )
97, 8syl6eqss 3507 . . 3  |-  ( ph  ->  ( `' F "
( CC  \  {
0 } ) ) 
C_  ( `' F " CC ) )
10 fimacnv 5937 . . . 4  |-  ( F : S --> CC  ->  ( `' F " CC )  =  S )
114, 10syl 16 . . 3  |-  ( ph  ->  ( `' F " CC )  =  S
)
129, 11sseqtrd 3493 . 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 2451 . . . . . . 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 5802 . . . . . . . . . . . . . . 15  |-  ( ZZ>= `  M )  e.  _V
181, 17eqeltri 2535 . . . . . . . . . . . . . 14  |-  S  e. 
_V
1918a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  S  e.  _V )
20 0nn0 10698 . . . . . . . . . . . . . 14  |-  0  e.  NN0
2120a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  0  e.  NN0 )
22 eqid 2451 . . . . . . . . . . . . . 14  |-  ( CC 
\  { 0 } )  =  ( CC 
\  { 0 } )
2322ffs2 26172 . . . . . . . . . . . . 13  |-  ( ( S  e.  _V  /\  0  e.  NN0  /\  F : S --> CC )  -> 
( F supp  0 )  =  ( `' F " ( CC  \  {
0 } ) ) )
2419, 21, 4, 23syl3anc 1219 . . . . . . . . . . . 12  |-  ( ph  ->  ( F supp  0 )  =  ( `' F " ( CC  \  {
0 } ) ) )
25 ffn 5660 . . . . . . . . . . . . . 14  |-  ( F : S --> CC  ->  F  Fn  S )
264, 25syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  F  Fn  S )
27 suppvalfn 6800 . . . . . . . . . . . . 13  |-  ( ( F  Fn  S  /\  S  e.  _V  /\  0  e.  NN0 )  ->  ( F supp  0 )  =  {
k  e.  S  | 
( F `  k
)  =/=  0 } )
2826, 19, 21, 27syl3anc 1219 . . . . . . . . . . . 12  |-  ( ph  ->  ( F supp  0 )  =  { k  e.  S  |  ( F `
 k )  =/=  0 } )
2924, 28eqtr3d 2494 . . . . . . . . . . 11  |-  ( ph  ->  ( `' F "
( CC  \  {
0 } ) )  =  { k  e.  S  |  ( F `
 k )  =/=  0 } )
3029eleq2d 2521 . . . . . . . . . 10  |-  ( ph  ->  ( k  e.  ( `' F " ( CC 
\  { 0 } ) )  <->  k  e.  { k  e.  S  | 
( F `  k
)  =/=  0 } ) )
31 rabid 2996 . . . . . . . . . 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 900 . . . . . . . 8  |-  ( (
ph  /\  k  e.  S )  ->  (
k  e.  ( `' F " ( CC 
\  { 0 } ) )  <->  ( F `  k )  =/=  0
) )
3433necon2bbid 2704 . . . . . . 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 3928 . . 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 3457 . . 3  |-  ( (
ph  /\  k  e.  ( `' F " ( CC 
\  { 0 } ) ) )  -> 
k  e.  S )
424ffvelrnda 5945 . . 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 13317 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 1370    e. wcel 1758    =/= wne 2644   {crab 2799   _Vcvv 3071    \ cdif 3426   ifcif 3892   {csn 3978   `'ccnv 4940   dom cdm 4941   "cima 4944   Fun wfun 5513    Fn wfn 5514   -->wf 5515   ` cfv 5519  (class class class)co 6193   supp csupp 6793   Fincfn 7413   CCcc 9384   0cc0 9386    + caddc 9389   NN0cn0 10683   ZZcz 10750   ZZ>=cuz 10965    seqcseq 11916    ~~> cli 13073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-recs 6935  df-rdg 6969  df-1o 7023  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-sup 7795  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-n0 10684  df-z 10751  df-uz 10966  df-rp 11096  df-fz 11548  df-seq 11917  df-exp 11976  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-clim 13077
This theorem is referenced by:  eulerpartlems  26880
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