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Theorem isumclim3 12222
Description: The sequence of partial finite sums of a converging infinite series converge to the infinite sum of the series. Note that  j must not occur in  A. (Contributed by NM, 9-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)
Hypotheses
Ref Expression
isumclim3.1  |-  Z  =  ( ZZ>= `  M )
isumclim3.2  |-  ( ph  ->  M  e.  ZZ )
isumclim3.3  |-  ( ph  ->  F  e.  dom  ~~>  )
isumclim3.4  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
isumclim3.5  |-  ( (
ph  /\  j  e.  Z )  ->  ( F `  j )  =  sum_ k  e.  ( M ... j ) A )
Assertion
Ref Expression
isumclim3  |-  ( ph  ->  F  ~~>  sum_ k  e.  Z  A )
Distinct variable groups:    A, j    j, k, M    ph, j, k   
j, Z, k    j, F
Allowed substitution hints:    A( k)    F( k)

Proof of Theorem isumclim3
Dummy variables  m  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isumclim3.3 . . 3  |-  ( ph  ->  F  e.  dom  ~~>  )
2 climdm 12028 . . 3  |-  ( F  e.  dom  ~~>  <->  F  ~~>  (  ~~>  `  F
) )
31, 2sylib 188 . 2  |-  ( ph  ->  F  ~~>  (  ~~>  `  F
) )
4 sumfc 12182 . . . 4  |-  sum_ m  e.  Z  ( (
k  e.  Z  |->  A ) `  m )  =  sum_ k  e.  Z  A
5 isumclim3.1 . . . . 5  |-  Z  =  ( ZZ>= `  M )
6 isumclim3.2 . . . . 5  |-  ( ph  ->  M  e.  ZZ )
7 eqidd 2284 . . . . 5  |-  ( (
ph  /\  m  e.  Z )  ->  (
( k  e.  Z  |->  A ) `  m
)  =  ( ( k  e.  Z  |->  A ) `  m ) )
8 isumclim3.4 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
9 eqid 2283 . . . . . . 7  |-  ( k  e.  Z  |->  A )  =  ( k  e.  Z  |->  A )
108, 9fmptd 5684 . . . . . 6  |-  ( ph  ->  ( k  e.  Z  |->  A ) : Z --> CC )
11 ffvelrn 5663 . . . . . 6  |-  ( ( ( k  e.  Z  |->  A ) : Z --> CC  /\  m  e.  Z
)  ->  ( (
k  e.  Z  |->  A ) `  m )  e.  CC )
1210, 11sylan 457 . . . . 5  |-  ( (
ph  /\  m  e.  Z )  ->  (
( k  e.  Z  |->  A ) `  m
)  e.  CC )
135, 6, 7, 12isum 12192 . . . 4  |-  ( ph  -> 
sum_ m  e.  Z  ( ( k  e.  Z  |->  A ) `  m )  =  (  ~~>  `
 seq  M (  +  ,  ( k  e.  Z  |->  A ) ) ) )
144, 13syl5eqr 2329 . . 3  |-  ( ph  -> 
sum_ k  e.  Z  A  =  (  ~~>  `  seq  M (  +  ,  ( k  e.  Z  |->  A ) ) ) )
15 seqex 11048 . . . . . . 7  |-  seq  M
(  +  ,  ( k  e.  Z  |->  A ) )  e.  _V
1615a1i 10 . . . . . 6  |-  ( ph  ->  seq  M (  +  ,  ( k  e.  Z  |->  A ) )  e.  _V )
17 isumclim3.5 . . . . . . 7  |-  ( (
ph  /\  j  e.  Z )  ->  ( F `  j )  =  sum_ k  e.  ( M ... j ) A )
18 fzssuz 10832 . . . . . . . . . . . . . 14  |-  ( M ... j )  C_  ( ZZ>= `  M )
1918, 5sseqtr4i 3211 . . . . . . . . . . . . 13  |-  ( M ... j )  C_  Z
20 resmpt 5000 . . . . . . . . . . . . 13  |-  ( ( M ... j ) 
C_  Z  ->  (
( k  e.  Z  |->  A )  |`  ( M ... j ) )  =  ( k  e.  ( M ... j
)  |->  A ) )
2119, 20ax-mp 8 . . . . . . . . . . . 12  |-  ( ( k  e.  Z  |->  A )  |`  ( M ... j ) )  =  ( k  e.  ( M ... j ) 
|->  A )
2221fveq1i 5526 . . . . . . . . . . 11  |-  ( ( ( k  e.  Z  |->  A )  |`  ( M ... j ) ) `
 m )  =  ( ( k  e.  ( M ... j
)  |->  A ) `  m )
23 fvres 5542 . . . . . . . . . . 11  |-  ( m  e.  ( M ... j )  ->  (
( ( k  e.  Z  |->  A )  |`  ( M ... j ) ) `  m )  =  ( ( k  e.  Z  |->  A ) `
 m ) )
2422, 23syl5reqr 2330 . . . . . . . . . 10  |-  ( m  e.  ( M ... j )  ->  (
( k  e.  Z  |->  A ) `  m
)  =  ( ( k  e.  ( M ... j )  |->  A ) `  m ) )
2524sumeq2i 12172 . . . . . . . . 9  |-  sum_ m  e.  ( M ... j
) ( ( k  e.  Z  |->  A ) `
 m )  = 
sum_ m  e.  ( M ... j ) ( ( k  e.  ( M ... j ) 
|->  A ) `  m
)
26 sumfc 12182 . . . . . . . . 9  |-  sum_ m  e.  ( M ... j
) ( ( k  e.  ( M ... j )  |->  A ) `
 m )  = 
sum_ k  e.  ( M ... j ) A
2725, 26eqtri 2303 . . . . . . . 8  |-  sum_ m  e.  ( M ... j
) ( ( k  e.  Z  |->  A ) `
 m )  = 
sum_ k  e.  ( M ... j ) A
28 eqidd 2284 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  Z )  /\  m  e.  ( M ... j
) )  ->  (
( k  e.  Z  |->  A ) `  m
)  =  ( ( k  e.  Z  |->  A ) `  m ) )
29 simpr 447 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  Z )
3029, 5syl6eleq 2373 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  ( ZZ>= `  M )
)
31 simpl 443 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  Z )  ->  ph )
32 elfzuz 10794 . . . . . . . . . . 11  |-  ( m  e.  ( M ... j )  ->  m  e.  ( ZZ>= `  M )
)
3332, 5syl6eleqr 2374 . . . . . . . . . 10  |-  ( m  e.  ( M ... j )  ->  m  e.  Z )
3431, 33, 12syl2an 463 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  Z )  /\  m  e.  ( M ... j
) )  ->  (
( k  e.  Z  |->  A ) `  m
)  e.  CC )
3528, 30, 34fsumser 12203 . . . . . . . 8  |-  ( (
ph  /\  j  e.  Z )  ->  sum_ m  e.  ( M ... j
) ( ( k  e.  Z  |->  A ) `
 m )  =  (  seq  M (  +  ,  ( k  e.  Z  |->  A ) ) `  j ) )
3627, 35syl5eqr 2329 . . . . . . 7  |-  ( (
ph  /\  j  e.  Z )  ->  sum_ k  e.  ( M ... j
) A  =  (  seq  M (  +  ,  ( k  e.  Z  |->  A ) ) `
 j ) )
3717, 36eqtr2d 2316 . . . . . 6  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq  M (  +  , 
( k  e.  Z  |->  A ) ) `  j )  =  ( F `  j ) )
385, 16, 1, 6, 37climeq 12041 . . . . 5  |-  ( ph  ->  (  seq  M (  +  ,  ( k  e.  Z  |->  A ) )  ~~>  x  <->  F  ~~>  x ) )
3938iotabidv 5240 . . . 4  |-  ( ph  ->  ( iota x  seq  M (  +  ,  ( k  e.  Z  |->  A ) )  ~~>  x )  =  ( iota x F 
~~>  x ) )
40 df-fv 5263 . . . 4  |-  (  ~~>  `  seq  M (  +  ,  ( k  e.  Z  |->  A ) ) )  =  ( iota x  seq  M (  +  ,  ( k  e.  Z  |->  A ) )  ~~>  x )
41 df-fv 5263 . . . 4  |-  (  ~~>  `  F
)  =  ( iota
x F  ~~>  x )
4239, 40, 413eqtr4g 2340 . . 3  |-  ( ph  ->  (  ~~>  `  seq  M (  +  ,  ( k  e.  Z  |->  A ) ) )  =  (  ~~>  `
 F ) )
4314, 42eqtrd 2315 . 2  |-  ( ph  -> 
sum_ k  e.  Z  A  =  (  ~~>  `  F
) )
443, 43breqtrrd 4049 1  |-  ( ph  ->  F  ~~>  sum_ k  e.  Z  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   dom cdm 4689    |` cres 4691   iotacio 5217   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735    + caddc 8740   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046    ~~> cli 11958   sum_csu 12158
This theorem is referenced by:  esumcvg  23454
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-int 3863  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-se 4353  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-isom 5264  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-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  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-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159
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