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Theorem fsumcvg3 13206
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 3377 . . . 4  |-  ( A  =  (/)  ->  ( A 
C_  ( M ... n )  <->  (/)  C_  ( M ... n ) ) )
21rexbidv 2736 . . 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 3402 . . . . 5  |-  ( (
ph  /\  A  =/=  (/) )  ->  A  C_  ( ZZ>=
`  M ) )
7 ltso 9455 . . . . . 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 10880 . . . . . . . . . 10  |-  ( ZZ>= `  M )  C_  ZZ
12 zssre 10653 . . . . . . . . . 10  |-  ZZ  C_  RR
1311, 12sstri 3365 . . . . . . . . 9  |-  ( ZZ>= `  M )  C_  RR
145, 13eqsstri 3386 . . . . . . . 8  |-  Z  C_  RR
154, 14syl6ss 3368 . . . . . . 7  |-  ( (
ph  /\  A  =/=  (/) )  ->  A  C_  RR )
169, 10, 153jca 1168 . . . . . 6  |-  ( (
ph  /\  A  =/=  (/) )  ->  ( A  e.  Fin  /\  A  =/=  (/)  /\  A  C_  RR ) )
17 fisupcl 7717 . . . . . 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 3357 . . . 4  |-  ( (
ph  /\  A  =/=  (/) )  ->  sup ( A ,  RR ,  <  )  e.  ( ZZ>= `  M ) )
20 fimaxre2 10278 . . . . . . . . . 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 1168 . . . . . . . 8  |-  ( (
ph  /\  A  =/=  (/) )  ->  ( A  C_  RR  /\  A  =/=  (/)  /\  E. k  e.  RR  A. n  e.  A  n  <_  k
) )
23 suprub 10291 . . . . . . . 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 3356 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  (/) )  /\  k  e.  A )  ->  k  e.  ( ZZ>= `  M )
)
2611, 19sseldi 3354 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  (/) )  ->  sup ( A ,  RR ,  <  )  e.  ZZ )
2726adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  (/) )  /\  k  e.  A )  ->  sup ( A ,  RR ,  <  )  e.  ZZ )
28 elfz5 11445 . . . . . . . 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 3362 . . . 4  |-  ( (
ph  /\  A  =/=  (/) )  ->  A  C_  ( M ... sup ( A ,  RR ,  <  ) ) )
33 oveq2 6099 . . . . . 6  |-  ( n  =  sup ( A ,  RR ,  <  )  ->  ( M ... n )  =  ( M ... sup ( A ,  RR ,  <  ) ) )
3433sseq2d 3384 . . . . 5  |-  ( n  =  sup ( A ,  RR ,  <  )  ->  ( A  C_  ( M ... n )  <-> 
A  C_  ( M ... sup ( A ,  RR ,  <  ) ) ) )
3534rspcev 3073 . . . 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 10875 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
3937, 38syl 16 . . . 4  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
40 0ss 3666 . . . 4  |-  (/)  C_  ( M ... M )
41 oveq2 6099 . . . . . 6  |-  ( n  =  M  ->  ( M ... n )  =  ( M ... M
) )
4241sseq2d 3384 . . . . 5  |-  ( n  =  M  ->  ( (/)  C_  ( M ... n
)  <->  (/)  C_  ( M ... M ) ) )
4342rspcev 3073 . . . 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 2686 . 2  |-  ( ph  ->  E. n  e.  (
ZZ>= `  M ) A 
C_  ( M ... n ) )
465eleq2i 2507 . . . . . 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 13204 . . 3  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  M )  /\  A  C_  ( M ... n ) ) )  ->  seq M (  +  ,  F )  ~~>  (  seq M (  +  ,  F ) `
 n ) )
55 climrel 12970 . . . 4  |-  Rel  ~~>
5655releldmi 5076 . . 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 2845 1  |-  ( ph  ->  seq M (  +  ,  F )  e. 
dom 
~~>  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   E.wrex 2716    C_ wss 3328   (/)c0 3637   ifcif 3791   class class class wbr 4292    Or wor 4640   dom cdm 4840   ` cfv 5418  (class class class)co 6091   Fincfn 7310   supcsup 7690   CCcc 9280   RRcr 9281   0cc0 9282    + caddc 9285    < clt 9418    <_ cle 9419   ZZcz 10646   ZZ>=cuz 10861   ...cfz 11437    seqcseq 11806    ~~> cli 12962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fz 11438  df-seq 11807  df-exp 11866  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-clim 12966
This theorem is referenced by:  isumless  13308  radcnv0  21881  fsumcvg4  26380
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