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Theorem fsumcvg3 13633
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 3510 . . . 4  |-  ( A  =  (/)  ->  ( A 
C_  ( M ... n )  <->  (/)  C_  ( M ... n ) ) )
21rexbidv 2965 . . 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 463 . . . . . 6  |-  ( (
ph  /\  A  =/=  (/) )  ->  A  C_  Z
)
5 fsumcvg3.1 . . . . . 6  |-  Z  =  ( ZZ>= `  M )
64, 5syl6sseq 3535 . . . . 5  |-  ( (
ph  /\  A  =/=  (/) )  ->  A  C_  ( ZZ>=
`  M ) )
7 ltso 9654 . . . . . 6  |-  <  Or  RR
8 fsumcvg3.3 . . . . . . . 8  |-  ( ph  ->  A  e.  Fin )
98adantr 463 . . . . . . 7  |-  ( (
ph  /\  A  =/=  (/) )  ->  A  e.  Fin )
10 simpr 459 . . . . . . 7  |-  ( (
ph  /\  A  =/=  (/) )  ->  A  =/=  (/) )
11 uzssz 11101 . . . . . . . . . 10  |-  ( ZZ>= `  M )  C_  ZZ
12 zssre 10867 . . . . . . . . . 10  |-  ZZ  C_  RR
1311, 12sstri 3498 . . . . . . . . 9  |-  ( ZZ>= `  M )  C_  RR
145, 13eqsstri 3519 . . . . . . . 8  |-  Z  C_  RR
154, 14syl6ss 3501 . . . . . . 7  |-  ( (
ph  /\  A  =/=  (/) )  ->  A  C_  RR )
169, 10, 153jca 1174 . . . . . 6  |-  ( (
ph  /\  A  =/=  (/) )  ->  ( A  e.  Fin  /\  A  =/=  (/)  /\  A  C_  RR ) )
17 fisupcl 7919 . . . . . 6  |-  ( (  <  Or  RR  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A  C_  RR ) )  ->  sup ( A ,  RR ,  <  )  e.  A
)
187, 16, 17sylancr 661 . . . . 5  |-  ( (
ph  /\  A  =/=  (/) )  ->  sup ( A ,  RR ,  <  )  e.  A )
196, 18sseldd 3490 . . . 4  |-  ( (
ph  /\  A  =/=  (/) )  ->  sup ( A ,  RR ,  <  )  e.  ( ZZ>= `  M ) )
20 fimaxre2 10486 . . . . . . . . . 10  |-  ( ( A  C_  RR  /\  A  e.  Fin )  ->  E. k  e.  RR  A. n  e.  A  n  <_  k
)
2115, 9, 20syl2anc 659 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  (/) )  ->  E. k  e.  RR  A. n  e.  A  n  <_  k
)
2215, 10, 213jca 1174 . . . . . . . 8  |-  ( (
ph  /\  A  =/=  (/) )  ->  ( A  C_  RR  /\  A  =/=  (/)  /\  E. k  e.  RR  A. n  e.  A  n  <_  k
) )
23 suprub 10499 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. k  e.  RR  A. n  e.  A  n  <_  k )  /\  k  e.  A )  ->  k  <_  sup ( A ,  RR ,  <  ) )
2422, 23sylan 469 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  (/) )  /\  k  e.  A )  ->  k  <_  sup ( A ,  RR ,  <  ) )
256sselda 3489 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  (/) )  /\  k  e.  A )  ->  k  e.  ( ZZ>= `  M )
)
2611, 19sseldi 3487 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  (/) )  ->  sup ( A ,  RR ,  <  )  e.  ZZ )
2726adantr 463 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  (/) )  /\  k  e.  A )  ->  sup ( A ,  RR ,  <  )  e.  ZZ )
28 elfz5 11683 . . . . . . . 8  |-  ( ( k  e.  ( ZZ>= `  M )  /\  sup ( A ,  RR ,  <  )  e.  ZZ )  ->  ( k  e.  ( M ... sup ( A ,  RR ,  <  ) )  <->  k  <_  sup ( A ,  RR ,  <  ) ) )
2925, 27, 28syl2anc 659 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  (/) )  /\  k  e.  A )  ->  (
k  e.  ( M ... sup ( A ,  RR ,  <  ) )  <->  k  <_  sup ( A ,  RR ,  <  ) ) )
3024, 29mpbird 232 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  (/) )  /\  k  e.  A )  ->  k  e.  ( M ... sup ( A ,  RR ,  <  ) ) )
3130ex 432 . . . . 5  |-  ( (
ph  /\  A  =/=  (/) )  ->  ( k  e.  A  ->  k  e.  ( M ... sup ( A ,  RR ,  <  ) ) ) )
3231ssrdv 3495 . . . 4  |-  ( (
ph  /\  A  =/=  (/) )  ->  A  C_  ( M ... sup ( A ,  RR ,  <  ) ) )
33 oveq2 6278 . . . . . 6  |-  ( n  =  sup ( A ,  RR ,  <  )  ->  ( M ... n )  =  ( M ... sup ( A ,  RR ,  <  ) ) )
3433sseq2d 3517 . . . . 5  |-  ( n  =  sup ( A ,  RR ,  <  )  ->  ( A  C_  ( M ... n )  <-> 
A  C_  ( M ... sup ( A ,  RR ,  <  ) ) ) )
3534rspcev 3207 . . . 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 659 . . 3  |-  ( (
ph  /\  A  =/=  (/) )  ->  E. n  e.  ( ZZ>= `  M ) A  C_  ( M ... n ) )
37 fsumcvg3.2 . . . . 5  |-  ( ph  ->  M  e.  ZZ )
38 uzid 11096 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
3937, 38syl 16 . . . 4  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
40 0ss 3813 . . . 4  |-  (/)  C_  ( M ... M )
41 oveq2 6278 . . . . . 6  |-  ( n  =  M  ->  ( M ... n )  =  ( M ... M
) )
4241sseq2d 3517 . . . . 5  |-  ( n  =  M  ->  ( (/)  C_  ( M ... n
)  <->  (/)  C_  ( M ... M ) ) )
4342rspcev 3207 . . . 4  |-  ( ( M  e.  ( ZZ>= `  M )  /\  (/)  C_  ( M ... M ) )  ->  E. n  e.  (
ZZ>= `  M ) (/)  C_  ( M ... n
) )
4439, 40, 43sylancl 660 . . 3  |-  ( ph  ->  E. n  e.  (
ZZ>= `  M ) (/)  C_  ( M ... n
) )
452, 36, 44pm2.61ne 2769 . 2  |-  ( ph  ->  E. n  e.  (
ZZ>= `  M ) A 
C_  ( M ... n ) )
465eleq2i 2532 . . . . . 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 474 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  if ( k  e.  A ,  B ,  0 ) )
4948adantlr 712 . . . 4  |-  ( ( ( ph  /\  (
n  e.  ( ZZ>= `  M )  /\  A  C_  ( M ... n
) ) )  /\  k  e.  ( ZZ>= `  M ) )  -> 
( F `  k
)  =  if ( k  e.  A ,  B ,  0 ) )
50 simprl 754 . . . 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 712 . . . 4  |-  ( ( ( ph  /\  (
n  e.  ( ZZ>= `  M )  /\  A  C_  ( M ... n
) ) )  /\  k  e.  A )  ->  B  e.  CC )
53 simprr 755 . . . 4  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  M )  /\  A  C_  ( M ... n ) ) )  ->  A  C_  ( M ... n ) )
5449, 50, 52, 53fsumcvg2 13631 . . 3  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  M )  /\  A  C_  ( M ... n ) ) )  ->  seq M (  +  ,  F )  ~~>  (  seq M (  +  ,  F ) `
 n ) )
55 climrel 13397 . . . 4  |-  Rel  ~~>
5655releldmi 5228 . . 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 2950 1  |-  ( ph  ->  seq M (  +  ,  F )  e. 
dom 
~~>  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805    C_ wss 3461   (/)c0 3783   ifcif 3929   class class class wbr 4439    Or wor 4788   dom cdm 4988   ` cfv 5570  (class class class)co 6270   Fincfn 7509   supcsup 7892   CCcc 9479   RRcr 9480   0cc0 9481    + caddc 9484    < clt 9617    <_ cle 9618   ZZcz 10860   ZZ>=cuz 11082   ...cfz 11675    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-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:  isumless  13739  radcnv0  22977  fsumcvg4  28167
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