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Theorem fsumcvg3 13513
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 3525 . . . 4  |-  ( A  =  (/)  ->  ( A 
C_  ( M ... n )  <->  (/)  C_  ( M ... n ) ) )
21rexbidv 2973 . . 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 465 . . . . . 6  |-  ( (
ph  /\  A  =/=  (/) )  ->  A  C_  Z
)
5 fsumcvg3.1 . . . . . 6  |-  Z  =  ( ZZ>= `  M )
64, 5syl6sseq 3550 . . . . 5  |-  ( (
ph  /\  A  =/=  (/) )  ->  A  C_  ( ZZ>=
`  M ) )
7 ltso 9664 . . . . . 6  |-  <  Or  RR
8 fsumcvg3.3 . . . . . . . 8  |-  ( ph  ->  A  e.  Fin )
98adantr 465 . . . . . . 7  |-  ( (
ph  /\  A  =/=  (/) )  ->  A  e.  Fin )
10 simpr 461 . . . . . . 7  |-  ( (
ph  /\  A  =/=  (/) )  ->  A  =/=  (/) )
11 uzssz 11100 . . . . . . . . . 10  |-  ( ZZ>= `  M )  C_  ZZ
12 zssre 10870 . . . . . . . . . 10  |-  ZZ  C_  RR
1311, 12sstri 3513 . . . . . . . . 9  |-  ( ZZ>= `  M )  C_  RR
145, 13eqsstri 3534 . . . . . . . 8  |-  Z  C_  RR
154, 14syl6ss 3516 . . . . . . 7  |-  ( (
ph  /\  A  =/=  (/) )  ->  A  C_  RR )
169, 10, 153jca 1176 . . . . . 6  |-  ( (
ph  /\  A  =/=  (/) )  ->  ( A  e.  Fin  /\  A  =/=  (/)  /\  A  C_  RR ) )
17 fisupcl 7926 . . . . . 6  |-  ( (  <  Or  RR  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A  C_  RR ) )  ->  sup ( A ,  RR ,  <  )  e.  A
)
187, 16, 17sylancr 663 . . . . 5  |-  ( (
ph  /\  A  =/=  (/) )  ->  sup ( A ,  RR ,  <  )  e.  A )
196, 18sseldd 3505 . . . 4  |-  ( (
ph  /\  A  =/=  (/) )  ->  sup ( A ,  RR ,  <  )  e.  ( ZZ>= `  M ) )
20 fimaxre2 10490 . . . . . . . . . 10  |-  ( ( A  C_  RR  /\  A  e.  Fin )  ->  E. k  e.  RR  A. n  e.  A  n  <_  k
)
2115, 9, 20syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  (/) )  ->  E. k  e.  RR  A. n  e.  A  n  <_  k
)
2215, 10, 213jca 1176 . . . . . . . 8  |-  ( (
ph  /\  A  =/=  (/) )  ->  ( A  C_  RR  /\  A  =/=  (/)  /\  E. k  e.  RR  A. n  e.  A  n  <_  k
) )
23 suprub 10503 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. k  e.  RR  A. n  e.  A  n  <_  k )  /\  k  e.  A )  ->  k  <_  sup ( A ,  RR ,  <  ) )
2422, 23sylan 471 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  (/) )  /\  k  e.  A )  ->  k  <_  sup ( A ,  RR ,  <  ) )
256sselda 3504 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  (/) )  /\  k  e.  A )  ->  k  e.  ( ZZ>= `  M )
)
2611, 19sseldi 3502 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  (/) )  ->  sup ( A ,  RR ,  <  )  e.  ZZ )
2726adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  (/) )  /\  k  e.  A )  ->  sup ( A ,  RR ,  <  )  e.  ZZ )
28 elfz5 11679 . . . . . . . 8  |-  ( ( k  e.  ( ZZ>= `  M )  /\  sup ( A ,  RR ,  <  )  e.  ZZ )  ->  ( k  e.  ( M ... sup ( A ,  RR ,  <  ) )  <->  k  <_  sup ( A ,  RR ,  <  ) ) )
2925, 27, 28syl2anc 661 . . . . . . 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 434 . . . . 5  |-  ( (
ph  /\  A  =/=  (/) )  ->  ( k  e.  A  ->  k  e.  ( M ... sup ( A ,  RR ,  <  ) ) ) )
3231ssrdv 3510 . . . 4  |-  ( (
ph  /\  A  =/=  (/) )  ->  A  C_  ( M ... sup ( A ,  RR ,  <  ) ) )
33 oveq2 6291 . . . . . 6  |-  ( n  =  sup ( A ,  RR ,  <  )  ->  ( M ... n )  =  ( M ... sup ( A ,  RR ,  <  ) ) )
3433sseq2d 3532 . . . . 5  |-  ( n  =  sup ( A ,  RR ,  <  )  ->  ( A  C_  ( M ... n )  <-> 
A  C_  ( M ... sup ( A ,  RR ,  <  ) ) ) )
3534rspcev 3214 . . . 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 661 . . 3  |-  ( (
ph  /\  A  =/=  (/) )  ->  E. n  e.  ( ZZ>= `  M ) A  C_  ( M ... n ) )
37 fsumcvg3.2 . . . . 5  |-  ( ph  ->  M  e.  ZZ )
38 uzid 11095 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
3937, 38syl 16 . . . 4  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
40 0ss 3814 . . . 4  |-  (/)  C_  ( M ... M )
41 oveq2 6291 . . . . . 6  |-  ( n  =  M  ->  ( M ... n )  =  ( M ... M
) )
4241sseq2d 3532 . . . . 5  |-  ( n  =  M  ->  ( (/)  C_  ( M ... n
)  <->  (/)  C_  ( M ... M ) ) )
4342rspcev 3214 . . . 4  |-  ( ( M  e.  ( ZZ>= `  M )  /\  (/)  C_  ( M ... M ) )  ->  E. n  e.  (
ZZ>= `  M ) (/)  C_  ( M ... n
) )
4439, 40, 43sylancl 662 . . 3  |-  ( ph  ->  E. n  e.  (
ZZ>= `  M ) (/)  C_  ( M ... n
) )
452, 36, 44pm2.61ne 2782 . 2  |-  ( ph  ->  E. n  e.  (
ZZ>= `  M ) A 
C_  ( M ... n ) )
465eleq2i 2545 . . . . . 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 476 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  if ( k  e.  A ,  B ,  0 ) )
4948adantlr 714 . . . 4  |-  ( ( ( ph  /\  (
n  e.  ( ZZ>= `  M )  /\  A  C_  ( M ... n
) ) )  /\  k  e.  ( ZZ>= `  M ) )  -> 
( F `  k
)  =  if ( k  e.  A ,  B ,  0 ) )
50 simprl 755 . . . 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 714 . . . 4  |-  ( ( ( ph  /\  (
n  e.  ( ZZ>= `  M )  /\  A  C_  ( M ... n
) ) )  /\  k  e.  A )  ->  B  e.  CC )
53 simprr 756 . . . 4  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  M )  /\  A  C_  ( M ... n ) ) )  ->  A  C_  ( M ... n ) )
5449, 50, 52, 53fsumcvg2 13511 . . 3  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  M )  /\  A  C_  ( M ... n ) ) )  ->  seq M (  +  ,  F )  ~~>  (  seq M (  +  ,  F ) `
 n ) )
55 climrel 13277 . . . 4  |-  Rel  ~~>
5655releldmi 5238 . . 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 2959 1  |-  ( ph  ->  seq M (  +  ,  F )  e. 
dom 
~~>  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815    C_ wss 3476   (/)c0 3785   ifcif 3939   class class class wbr 4447    Or wor 4799   dom cdm 4999   ` cfv 5587  (class class class)co 6283   Fincfn 7516   supcsup 7899   CCcc 9489   RRcr 9490   0cc0 9491    + caddc 9494    < clt 9627    <_ cle 9628   ZZcz 10863   ZZ>=cuz 11081   ...cfz 11671    seqcseq 12074    ~~> cli 13269
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 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569
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 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7900  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-n0 10795  df-z 10864  df-uz 11082  df-rp 11220  df-fz 11672  df-seq 12075  df-exp 12134  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-clim 13273
This theorem is referenced by:  isumless  13619  radcnv0  22561  fsumcvg4  27584
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