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Theorem fsumcvg3 12202
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 3199 . . . 4  |-  ( A  =  (/)  ->  ( A 
C_  ( M ... n )  <->  (/)  C_  ( M ... n ) ) )
21rexbidv 2564 . . 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 451 . . . . . 6  |-  ( (
ph  /\  A  =/=  (/) )  ->  A  C_  Z
)
5 fsumcvg3.1 . . . . . 6  |-  Z  =  ( ZZ>= `  M )
64, 5syl6sseq 3224 . . . . 5  |-  ( (
ph  /\  A  =/=  (/) )  ->  A  C_  ( ZZ>=
`  M ) )
7 ltso 8903 . . . . . 6  |-  <  Or  RR
8 fsumcvg3.3 . . . . . . . 8  |-  ( ph  ->  A  e.  Fin )
98adantr 451 . . . . . . 7  |-  ( (
ph  /\  A  =/=  (/) )  ->  A  e.  Fin )
10 simpr 447 . . . . . . 7  |-  ( (
ph  /\  A  =/=  (/) )  ->  A  =/=  (/) )
11 uzssz 10247 . . . . . . . . . 10  |-  ( ZZ>= `  M )  C_  ZZ
12 zssre 10031 . . . . . . . . . 10  |-  ZZ  C_  RR
1311, 12sstri 3188 . . . . . . . . 9  |-  ( ZZ>= `  M )  C_  RR
145, 13eqsstri 3208 . . . . . . . 8  |-  Z  C_  RR
154, 14syl6ss 3191 . . . . . . 7  |-  ( (
ph  /\  A  =/=  (/) )  ->  A  C_  RR )
169, 10, 153jca 1132 . . . . . 6  |-  ( (
ph  /\  A  =/=  (/) )  ->  ( A  e.  Fin  /\  A  =/=  (/)  /\  A  C_  RR ) )
17 fisupcl 7218 . . . . . 6  |-  ( (  <  Or  RR  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A  C_  RR ) )  ->  sup ( A ,  RR ,  <  )  e.  A
)
187, 16, 17sylancr 644 . . . . 5  |-  ( (
ph  /\  A  =/=  (/) )  ->  sup ( A ,  RR ,  <  )  e.  A )
196, 18sseldd 3181 . . . 4  |-  ( (
ph  /\  A  =/=  (/) )  ->  sup ( A ,  RR ,  <  )  e.  ( ZZ>= `  M ) )
20 fimaxre2 9702 . . . . . . . . . 10  |-  ( ( A  C_  RR  /\  A  e.  Fin )  ->  E. k  e.  RR  A. n  e.  A  n  <_  k
)
2115, 9, 20syl2anc 642 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  (/) )  ->  E. k  e.  RR  A. n  e.  A  n  <_  k
)
2215, 10, 213jca 1132 . . . . . . . 8  |-  ( (
ph  /\  A  =/=  (/) )  ->  ( A  C_  RR  /\  A  =/=  (/)  /\  E. k  e.  RR  A. n  e.  A  n  <_  k
) )
23 suprub 9715 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. k  e.  RR  A. n  e.  A  n  <_  k )  /\  k  e.  A )  ->  k  <_  sup ( A ,  RR ,  <  ) )
2422, 23sylan 457 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  (/) )  /\  k  e.  A )  ->  k  <_  sup ( A ,  RR ,  <  ) )
256sselda 3180 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  (/) )  /\  k  e.  A )  ->  k  e.  ( ZZ>= `  M )
)
2611, 19sseldi 3178 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  (/) )  ->  sup ( A ,  RR ,  <  )  e.  ZZ )
2726adantr 451 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  (/) )  /\  k  e.  A )  ->  sup ( A ,  RR ,  <  )  e.  ZZ )
28 elfz5 10790 . . . . . . . 8  |-  ( ( k  e.  ( ZZ>= `  M )  /\  sup ( A ,  RR ,  <  )  e.  ZZ )  ->  ( k  e.  ( M ... sup ( A ,  RR ,  <  ) )  <->  k  <_  sup ( A ,  RR ,  <  ) ) )
2925, 27, 28syl2anc 642 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  (/) )  /\  k  e.  A )  ->  (
k  e.  ( M ... sup ( A ,  RR ,  <  ) )  <->  k  <_  sup ( A ,  RR ,  <  ) ) )
3024, 29mpbird 223 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  (/) )  /\  k  e.  A )  ->  k  e.  ( M ... sup ( A ,  RR ,  <  ) ) )
3130ex 423 . . . . 5  |-  ( (
ph  /\  A  =/=  (/) )  ->  ( k  e.  A  ->  k  e.  ( M ... sup ( A ,  RR ,  <  ) ) ) )
3231ssrdv 3185 . . . 4  |-  ( (
ph  /\  A  =/=  (/) )  ->  A  C_  ( M ... sup ( A ,  RR ,  <  ) ) )
33 oveq2 5866 . . . . . 6  |-  ( n  =  sup ( A ,  RR ,  <  )  ->  ( M ... n )  =  ( M ... sup ( A ,  RR ,  <  ) ) )
3433sseq2d 3206 . . . . 5  |-  ( n  =  sup ( A ,  RR ,  <  )  ->  ( A  C_  ( M ... n )  <-> 
A  C_  ( M ... sup ( A ,  RR ,  <  ) ) ) )
3534rspcev 2884 . . . 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 642 . . 3  |-  ( (
ph  /\  A  =/=  (/) )  ->  E. n  e.  ( ZZ>= `  M ) A  C_  ( M ... n ) )
37 fsumcvg3.2 . . . . 5  |-  ( ph  ->  M  e.  ZZ )
38 uzid 10242 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
3937, 38syl 15 . . . 4  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
40 0ss 3483 . . . 4  |-  (/)  C_  ( M ... M )
41 oveq2 5866 . . . . . 6  |-  ( n  =  M  ->  ( M ... n )  =  ( M ... M
) )
4241sseq2d 3206 . . . . 5  |-  ( n  =  M  ->  ( (/)  C_  ( M ... n
)  <->  (/)  C_  ( M ... M ) ) )
4342rspcev 2884 . . . 4  |-  ( ( M  e.  ( ZZ>= `  M )  /\  (/)  C_  ( M ... M ) )  ->  E. n  e.  (
ZZ>= `  M ) (/)  C_  ( M ... n
) )
4439, 40, 43sylancl 643 . . 3  |-  ( ph  ->  E. n  e.  (
ZZ>= `  M ) (/)  C_  ( M ... n
) )
452, 36, 44pm2.61ne 2521 . 2  |-  ( ph  ->  E. n  e.  (
ZZ>= `  M ) A 
C_  ( M ... n ) )
465eleq2i 2347 . . . . . . . 8  |-  ( k  e.  Z  <->  k  e.  ( ZZ>= `  M )
)
47 fsumcvg3.5 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  if ( k  e.  A ,  B , 
0 ) )
4846, 47sylan2br 462 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  if ( k  e.  A ,  B ,  0 ) )
4948adantlr 695 . . . . . 6  |-  ( ( ( ph  /\  (
n  e.  ( ZZ>= `  M )  /\  A  C_  ( M ... n
) ) )  /\  k  e.  ( ZZ>= `  M ) )  -> 
( F `  k
)  =  if ( k  e.  A ,  B ,  0 ) )
50 simprl 732 . . . . . 6  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  M )  /\  A  C_  ( M ... n ) ) )  ->  n  e.  ( ZZ>= `  M )
)
51 fsumcvg3.6 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
5251adantlr 695 . . . . . 6  |-  ( ( ( ph  /\  (
n  e.  ( ZZ>= `  M )  /\  A  C_  ( M ... n
) ) )  /\  k  e.  A )  ->  B  e.  CC )
53 simprr 733 . . . . . 6  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  M )  /\  A  C_  ( M ... n ) ) )  ->  A  C_  ( M ... n ) )
5449, 50, 52, 53fsumcvg2 12200 . . . . 5  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  M )  /\  A  C_  ( M ... n ) ) )  ->  seq  M (  +  ,  F )  ~~>  (  seq  M (  +  ,  F ) `
 n ) )
55 climrel 11966 . . . . . 6  |-  Rel  ~~>
5655releldmi 4915 . . . . 5  |-  (  seq 
M (  +  ,  F )  ~~>  (  seq 
M (  +  ,  F ) `  n
)  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )
5754, 56syl 15 . . . 4  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  M )  /\  A  C_  ( M ... n ) ) )  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )
5857expr 598 . . 3  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( A  C_  ( M ... n
)  ->  seq  M (  +  ,  F )  e.  dom  ~~>  ) )
5958rexlimdva 2667 . 2  |-  ( ph  ->  ( E. n  e.  ( ZZ>= `  M ) A  C_  ( M ... n )  ->  seq  M (  +  ,  F
)  e.  dom  ~~>  ) )
6045, 59mpd 14 1  |-  ( ph  ->  seq  M (  +  ,  F )  e. 
dom 
~~>  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    C_ wss 3152   (/)c0 3455   ifcif 3565   class class class wbr 4023    Or wor 4313   dom cdm 4689   ` cfv 5255  (class class class)co 5858   Fincfn 6863   supcsup 7193   CCcc 8735   RRcr 8736   0cc0 8737    + caddc 8740    < clt 8867    <_ cle 8868   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046    ~~> cli 11958
This theorem is referenced by:  isumless  12304  radcnv0  19792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962
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