MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fsumcvg3 Structured version   Unicode version

Theorem fsumcvg3 12515
Description: A finite sum is convergent. (Contributed by Mario Carneiro, 24-Apr-2014.)
Hypotheses
Ref Expression
fsumcvg3.1  |-  Z  =  ( ZZ>= `  M )
fsumcvg3.2  |-  ( ph  ->  M  e.  ZZ )
fsumcvg3.3  |-  ( ph  ->  A  e.  Fin )
fsumcvg3.4  |-  ( ph  ->  A  C_  Z )
fsumcvg3.5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  if ( k  e.  A ,  B , 
0 ) )
fsumcvg3.6  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
Assertion
Ref Expression
fsumcvg3  |-  ( ph  ->  seq  M (  +  ,  F )  e. 
dom 
~~>  )
Distinct variable groups:    A, k    k, F    k, M    ph, k
Allowed substitution hints:    B( k)    Z( k)

Proof of Theorem fsumcvg3
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 sseq1 3361 . . . 4  |-  ( A  =  (/)  ->  ( A 
C_  ( M ... n )  <->  (/)  C_  ( M ... n ) ) )
21rexbidv 2718 . . 3  |-  ( A  =  (/)  ->  ( E. n  e.  ( ZZ>= `  M ) A  C_  ( M ... n )  <->  E. n  e.  ( ZZ>=
`  M ) (/)  C_  ( M ... n
) ) )
3 fsumcvg3.4 . . . . . . 7  |-  ( ph  ->  A  C_  Z )
43adantr 452 . . . . . 6  |-  ( (
ph  /\  A  =/=  (/) )  ->  A  C_  Z
)
5 fsumcvg3.1 . . . . . 6  |-  Z  =  ( ZZ>= `  M )
64, 5syl6sseq 3386 . . . . 5  |-  ( (
ph  /\  A  =/=  (/) )  ->  A  C_  ( ZZ>=
`  M ) )
7 ltso 9148 . . . . . 6  |-  <  Or  RR
8 fsumcvg3.3 . . . . . . . 8  |-  ( ph  ->  A  e.  Fin )
98adantr 452 . . . . . . 7  |-  ( (
ph  /\  A  =/=  (/) )  ->  A  e.  Fin )
10 simpr 448 . . . . . . 7  |-  ( (
ph  /\  A  =/=  (/) )  ->  A  =/=  (/) )
11 uzssz 10497 . . . . . . . . . 10  |-  ( ZZ>= `  M )  C_  ZZ
12 zssre 10281 . . . . . . . . . 10  |-  ZZ  C_  RR
1311, 12sstri 3349 . . . . . . . . 9  |-  ( ZZ>= `  M )  C_  RR
145, 13eqsstri 3370 . . . . . . . 8  |-  Z  C_  RR
154, 14syl6ss 3352 . . . . . . 7  |-  ( (
ph  /\  A  =/=  (/) )  ->  A  C_  RR )
169, 10, 153jca 1134 . . . . . 6  |-  ( (
ph  /\  A  =/=  (/) )  ->  ( A  e.  Fin  /\  A  =/=  (/)  /\  A  C_  RR ) )
17 fisupcl 7464 . . . . . 6  |-  ( (  <  Or  RR  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A  C_  RR ) )  ->  sup ( A ,  RR ,  <  )  e.  A
)
187, 16, 17sylancr 645 . . . . 5  |-  ( (
ph  /\  A  =/=  (/) )  ->  sup ( A ,  RR ,  <  )  e.  A )
196, 18sseldd 3341 . . . 4  |-  ( (
ph  /\  A  =/=  (/) )  ->  sup ( A ,  RR ,  <  )  e.  ( ZZ>= `  M ) )
20 fimaxre2 9948 . . . . . . . . . 10  |-  ( ( A  C_  RR  /\  A  e.  Fin )  ->  E. k  e.  RR  A. n  e.  A  n  <_  k
)
2115, 9, 20syl2anc 643 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  (/) )  ->  E. k  e.  RR  A. n  e.  A  n  <_  k
)
2215, 10, 213jca 1134 . . . . . . . 8  |-  ( (
ph  /\  A  =/=  (/) )  ->  ( A  C_  RR  /\  A  =/=  (/)  /\  E. k  e.  RR  A. n  e.  A  n  <_  k
) )
23 suprub 9961 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. k  e.  RR  A. n  e.  A  n  <_  k )  /\  k  e.  A )  ->  k  <_  sup ( A ,  RR ,  <  ) )
2422, 23sylan 458 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  (/) )  /\  k  e.  A )  ->  k  <_  sup ( A ,  RR ,  <  ) )
256sselda 3340 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  (/) )  /\  k  e.  A )  ->  k  e.  ( ZZ>= `  M )
)
2611, 19sseldi 3338 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  (/) )  ->  sup ( A ,  RR ,  <  )  e.  ZZ )
2726adantr 452 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  (/) )  /\  k  e.  A )  ->  sup ( A ,  RR ,  <  )  e.  ZZ )
28 elfz5 11043 . . . . . . . 8  |-  ( ( k  e.  ( ZZ>= `  M )  /\  sup ( A ,  RR ,  <  )  e.  ZZ )  ->  ( k  e.  ( M ... sup ( A ,  RR ,  <  ) )  <->  k  <_  sup ( A ,  RR ,  <  ) ) )
2925, 27, 28syl2anc 643 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  (/) )  /\  k  e.  A )  ->  (
k  e.  ( M ... sup ( A ,  RR ,  <  ) )  <->  k  <_  sup ( A ,  RR ,  <  ) ) )
3024, 29mpbird 224 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  (/) )  /\  k  e.  A )  ->  k  e.  ( M ... sup ( A ,  RR ,  <  ) ) )
3130ex 424 . . . . 5  |-  ( (
ph  /\  A  =/=  (/) )  ->  ( k  e.  A  ->  k  e.  ( M ... sup ( A ,  RR ,  <  ) ) ) )
3231ssrdv 3346 . . . 4  |-  ( (
ph  /\  A  =/=  (/) )  ->  A  C_  ( M ... sup ( A ,  RR ,  <  ) ) )
33 oveq2 6081 . . . . . 6  |-  ( n  =  sup ( A ,  RR ,  <  )  ->  ( M ... n )  =  ( M ... sup ( A ,  RR ,  <  ) ) )
3433sseq2d 3368 . . . . 5  |-  ( n  =  sup ( A ,  RR ,  <  )  ->  ( A  C_  ( M ... n )  <-> 
A  C_  ( M ... sup ( A ,  RR ,  <  ) ) ) )
3534rspcev 3044 . . . 4  |-  ( ( sup ( A ,  RR ,  <  )  e.  ( ZZ>= `  M )  /\  A  C_  ( M ... sup ( A ,  RR ,  <  ) ) )  ->  E. n  e.  ( ZZ>= `  M ) A  C_  ( M ... n ) )
3619, 32, 35syl2anc 643 . . 3  |-  ( (
ph  /\  A  =/=  (/) )  ->  E. n  e.  ( ZZ>= `  M ) A  C_  ( M ... n ) )
37 fsumcvg3.2 . . . . 5  |-  ( ph  ->  M  e.  ZZ )
38 uzid 10492 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
3937, 38syl 16 . . . 4  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
40 0ss 3648 . . . 4  |-  (/)  C_  ( M ... M )
41 oveq2 6081 . . . . . 6  |-  ( n  =  M  ->  ( M ... n )  =  ( M ... M
) )
4241sseq2d 3368 . . . . 5  |-  ( n  =  M  ->  ( (/)  C_  ( M ... n
)  <->  (/)  C_  ( M ... M ) ) )
4342rspcev 3044 . . . 4  |-  ( ( M  e.  ( ZZ>= `  M )  /\  (/)  C_  ( M ... M ) )  ->  E. n  e.  (
ZZ>= `  M ) (/)  C_  ( M ... n
) )
4439, 40, 43sylancl 644 . . 3  |-  ( ph  ->  E. n  e.  (
ZZ>= `  M ) (/)  C_  ( M ... n
) )
452, 36, 44pm2.61ne 2673 . 2  |-  ( ph  ->  E. n  e.  (
ZZ>= `  M ) A 
C_  ( M ... n ) )
465eleq2i 2499 . . . . . 6  |-  ( k  e.  Z  <->  k  e.  ( ZZ>= `  M )
)
47 fsumcvg3.5 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  if ( k  e.  A ,  B , 
0 ) )
4846, 47sylan2br 463 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  if ( k  e.  A ,  B ,  0 ) )
4948adantlr 696 . . . 4  |-  ( ( ( ph  /\  (
n  e.  ( ZZ>= `  M )  /\  A  C_  ( M ... n
) ) )  /\  k  e.  ( ZZ>= `  M ) )  -> 
( F `  k
)  =  if ( k  e.  A ,  B ,  0 ) )
50 simprl 733 . . . 4  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  M )  /\  A  C_  ( M ... n ) ) )  ->  n  e.  ( ZZ>= `  M )
)
51 fsumcvg3.6 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
5251adantlr 696 . . . 4  |-  ( ( ( ph  /\  (
n  e.  ( ZZ>= `  M )  /\  A  C_  ( M ... n
) ) )  /\  k  e.  A )  ->  B  e.  CC )
53 simprr 734 . . . 4  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  M )  /\  A  C_  ( M ... n ) ) )  ->  A  C_  ( M ... n ) )
5449, 50, 52, 53fsumcvg2 12513 . . 3  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  M )  /\  A  C_  ( M ... n ) ) )  ->  seq  M (  +  ,  F )  ~~>  (  seq  M (  +  ,  F ) `
 n ) )
55 climrel 12278 . . . 4  |-  Rel  ~~>
5655releldmi 5098 . . 3  |-  (  seq 
M (  +  ,  F )  ~~>  (  seq 
M (  +  ,  F ) `  n
)  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )
5754, 56syl 16 . 2  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  M )  /\  A  C_  ( M ... n ) ) )  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )
5845, 57rexlimddv 2826 1  |-  ( ph  ->  seq  M (  +  ,  F )  e. 
dom 
~~>  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698    C_ wss 3312   (/)c0 3620   ifcif 3731   class class class wbr 4204    Or wor 4494   dom cdm 4870   ` cfv 5446  (class class class)co 6073   Fincfn 7101   supcsup 7437   CCcc 8980   RRcr 8981   0cc0 8982    + caddc 8985    < clt 9112    <_ cle 9113   ZZcz 10274   ZZ>=cuz 10480   ...cfz 11035    seq cseq 11315    ~~> cli 12270
This theorem is referenced by:  isumless  12617  radcnv0  20324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274
  Copyright terms: Public domain W3C validator