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Theorem fsumcvg4 28167
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 5715 . . . . 5  |-  ( F : S --> CC  ->  Fun 
F )
6 difpreima 5991 . . . . 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 3617 . . . 4  |-  ( ( `' F " CC ) 
\  ( `' F " { 0 } ) )  C_  ( `' F " CC )
97, 8syl6eqss 3539 . . 3  |-  ( ph  ->  ( `' F "
( CC  \  {
0 } ) ) 
C_  ( `' F " CC ) )
10 fimacnv 5995 . . . 4  |-  ( F : S --> CC  ->  ( `' F " CC )  =  S )
114, 10syl 16 . . 3  |-  ( ph  ->  ( `' F " CC )  =  S
)
129, 11sseqtrd 3525 . 2  |-  ( ph  ->  ( `' F "
( CC  \  {
0 } ) ) 
C_  S )
13 exmidd 414 . . . 4  |-  ( (
ph  /\  k  e.  S )  ->  (
k  e.  ( `' F " ( CC 
\  { 0 } ) )  \/  -.  k  e.  ( `' F " ( CC  \  { 0 } ) ) ) )
14 eqid 2454 . . . . . . 7  |-  ( F `
 k )  =  ( F `  k
)
1514biantru 503 . . . . . 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 5858 . . . . . . . . . . . . . . 15  |-  ( ZZ>= `  M )  e.  _V
181, 17eqeltri 2538 . . . . . . . . . . . . . 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 2454 . . . . . . . . . . . . . 14  |-  ( CC 
\  { 0 } )  =  ( CC 
\  { 0 } )
2322ffs2 27782 . . . . . . . . . . . . 13  |-  ( ( S  e.  _V  /\  0  e.  NN0  /\  F : S --> CC )  -> 
( F supp  0 )  =  ( `' F " ( CC  \  {
0 } ) ) )
2419, 21, 4, 23syl3anc 1226 . . . . . . . . . . . 12  |-  ( ph  ->  ( F supp  0 )  =  ( `' F " ( CC  \  {
0 } ) ) )
25 ffn 5713 . . . . . . . . . . . . . 14  |-  ( F : S --> CC  ->  F  Fn  S )
264, 25syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  F  Fn  S )
27 suppvalfn 6898 . . . . . . . . . . . . 13  |-  ( ( F  Fn  S  /\  S  e.  _V  /\  0  e.  NN0 )  ->  ( F supp  0 )  =  {
k  e.  S  | 
( F `  k
)  =/=  0 } )
2826, 19, 21, 27syl3anc 1226 . . . . . . . . . . . 12  |-  ( ph  ->  ( F supp  0 )  =  { k  e.  S  |  ( F `
 k )  =/=  0 } )
2924, 28eqtr3d 2497 . . . . . . . . . . 11  |-  ( ph  ->  ( `' F "
( CC  \  {
0 } ) )  =  { k  e.  S  |  ( F `
 k )  =/=  0 } )
3029eleq2d 2524 . . . . . . . . . 10  |-  ( ph  ->  ( k  e.  ( `' F " ( CC 
\  { 0 } ) )  <->  k  e.  { k  e.  S  | 
( F `  k
)  =/=  0 } ) )
31 rabid 3031 . . . . . . . . . 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 2710 . . . . . . 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 632 . . . . 5  |-  ( (
ph  /\  k  e.  S )  ->  ( -.  k  e.  ( `' F " ( CC 
\  { 0 } ) )  <->  ( -.  k  e.  ( `' F " ( CC  \  { 0 } ) )  /\  ( F `
 k )  =  0 ) ) )
3716, 36orbi12d 707 . . . 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 3967 . . 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 3489 . . 3  |-  ( (
ph  /\  k  e.  ( `' F " ( CC 
\  { 0 } ) ) )  -> 
k  e.  S )
424ffvelrnda 6007 . . 3  |-  ( (
ph  /\  k  e.  S )  ->  ( F `  k )  e.  CC )
4341, 42syldan 468 . 2  |-  ( (
ph  /\  k  e.  ( `' F " ( CC 
\  { 0 } ) ) )  -> 
( F `  k
)  e.  CC )
441, 2, 3, 12, 40, 43fsumcvg3 13633 1  |-  ( ph  ->  seq M (  +  ,  F )  e. 
dom 
~~>  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   {crab 2808   _Vcvv 3106    \ cdif 3458   ifcif 3929   {csn 4016   `'ccnv 4987   dom cdm 4988   "cima 4991   Fun wfun 5564    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   supp csupp 6891   Fincfn 7509   CCcc 9479   0cc0 9481    + caddc 9484   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11082    seqcseq 12089    ~~> cli 13389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fz 11676  df-seq 12090  df-exp 12149  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393
This theorem is referenced by:  eulerpartlems  28563
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