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Theorem isumclim3 12506
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 12311 . . 3  |-  ( F  e.  dom  ~~>  <->  F  ~~>  (  ~~>  `  F
) )
31, 2sylib 189 . 2  |-  ( ph  ->  F  ~~>  (  ~~>  `  F
) )
4 sumfc 12466 . . . 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 2413 . . . . 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 2412 . . . . . . 7  |-  ( k  e.  Z  |->  A )  =  ( k  e.  Z  |->  A )
108, 9fmptd 5860 . . . . . 6  |-  ( ph  ->  ( k  e.  Z  |->  A ) : Z --> CC )
1110ffvelrnda 5837 . . . . 5  |-  ( (
ph  /\  m  e.  Z )  ->  (
( k  e.  Z  |->  A ) `  m
)  e.  CC )
125, 6, 7, 11isum 12476 . . . 4  |-  ( ph  -> 
sum_ m  e.  Z  ( ( k  e.  Z  |->  A ) `  m )  =  (  ~~>  `
 seq  M (  +  ,  ( k  e.  Z  |->  A ) ) ) )
134, 12syl5eqr 2458 . . 3  |-  ( ph  -> 
sum_ k  e.  Z  A  =  (  ~~>  `  seq  M (  +  ,  ( k  e.  Z  |->  A ) ) ) )
14 seqex 11288 . . . . . . 7  |-  seq  M
(  +  ,  ( k  e.  Z  |->  A ) )  e.  _V
1514a1i 11 . . . . . 6  |-  ( ph  ->  seq  M (  +  ,  ( k  e.  Z  |->  A ) )  e.  _V )
16 isumclim3.5 . . . . . . 7  |-  ( (
ph  /\  j  e.  Z )  ->  ( F `  j )  =  sum_ k  e.  ( M ... j ) A )
17 fzssuz 11057 . . . . . . . . . . . . . 14  |-  ( M ... j )  C_  ( ZZ>= `  M )
1817, 5sseqtr4i 3349 . . . . . . . . . . . . 13  |-  ( M ... j )  C_  Z
19 resmpt 5158 . . . . . . . . . . . . 13  |-  ( ( M ... j ) 
C_  Z  ->  (
( k  e.  Z  |->  A )  |`  ( M ... j ) )  =  ( k  e.  ( M ... j
)  |->  A ) )
2018, 19ax-mp 8 . . . . . . . . . . . 12  |-  ( ( k  e.  Z  |->  A )  |`  ( M ... j ) )  =  ( k  e.  ( M ... j ) 
|->  A )
2120fveq1i 5696 . . . . . . . . . . 11  |-  ( ( ( k  e.  Z  |->  A )  |`  ( M ... j ) ) `
 m )  =  ( ( k  e.  ( M ... j
)  |->  A ) `  m )
22 fvres 5712 . . . . . . . . . . 11  |-  ( m  e.  ( M ... j )  ->  (
( ( k  e.  Z  |->  A )  |`  ( M ... j ) ) `  m )  =  ( ( k  e.  Z  |->  A ) `
 m ) )
2321, 22syl5reqr 2459 . . . . . . . . . 10  |-  ( m  e.  ( M ... j )  ->  (
( k  e.  Z  |->  A ) `  m
)  =  ( ( k  e.  ( M ... j )  |->  A ) `  m ) )
2423sumeq2i 12456 . . . . . . . . 9  |-  sum_ m  e.  ( M ... j
) ( ( k  e.  Z  |->  A ) `
 m )  = 
sum_ m  e.  ( M ... j ) ( ( k  e.  ( M ... j ) 
|->  A ) `  m
)
25 sumfc 12466 . . . . . . . . 9  |-  sum_ m  e.  ( M ... j
) ( ( k  e.  ( M ... j )  |->  A ) `
 m )  = 
sum_ k  e.  ( M ... j ) A
2624, 25eqtri 2432 . . . . . . . 8  |-  sum_ m  e.  ( M ... j
) ( ( k  e.  Z  |->  A ) `
 m )  = 
sum_ k  e.  ( M ... j ) A
27 eqidd 2413 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  Z )  /\  m  e.  ( M ... j
) )  ->  (
( k  e.  Z  |->  A ) `  m
)  =  ( ( k  e.  Z  |->  A ) `  m ) )
28 simpr 448 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  Z )
2928, 5syl6eleq 2502 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  ( ZZ>= `  M )
)
30 simpl 444 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  Z )  ->  ph )
31 elfzuz 11019 . . . . . . . . . . 11  |-  ( m  e.  ( M ... j )  ->  m  e.  ( ZZ>= `  M )
)
3231, 5syl6eleqr 2503 . . . . . . . . . 10  |-  ( m  e.  ( M ... j )  ->  m  e.  Z )
3330, 32, 11syl2an 464 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  Z )  /\  m  e.  ( M ... j
) )  ->  (
( k  e.  Z  |->  A ) `  m
)  e.  CC )
3427, 29, 33fsumser 12487 . . . . . . . 8  |-  ( (
ph  /\  j  e.  Z )  ->  sum_ m  e.  ( M ... j
) ( ( k  e.  Z  |->  A ) `
 m )  =  (  seq  M (  +  ,  ( k  e.  Z  |->  A ) ) `  j ) )
3526, 34syl5eqr 2458 . . . . . . 7  |-  ( (
ph  /\  j  e.  Z )  ->  sum_ k  e.  ( M ... j
) A  =  (  seq  M (  +  ,  ( k  e.  Z  |->  A ) ) `
 j ) )
3616, 35eqtr2d 2445 . . . . . 6  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq  M (  +  , 
( k  e.  Z  |->  A ) ) `  j )  =  ( F `  j ) )
375, 15, 1, 6, 36climeq 12324 . . . . 5  |-  ( ph  ->  (  seq  M (  +  ,  ( k  e.  Z  |->  A ) )  ~~>  x  <->  F  ~~>  x ) )
3837iotabidv 5406 . . . 4  |-  ( ph  ->  ( iota x  seq  M (  +  ,  ( k  e.  Z  |->  A ) )  ~~>  x )  =  ( iota x F 
~~>  x ) )
39 df-fv 5429 . . . 4  |-  (  ~~>  `  seq  M (  +  ,  ( k  e.  Z  |->  A ) ) )  =  ( iota x  seq  M (  +  ,  ( k  e.  Z  |->  A ) )  ~~>  x )
40 df-fv 5429 . . . 4  |-  (  ~~>  `  F
)  =  ( iota
x F  ~~>  x )
4138, 39, 403eqtr4g 2469 . . 3  |-  ( ph  ->  (  ~~>  `  seq  M (  +  ,  ( k  e.  Z  |->  A ) ) )  =  (  ~~>  `
 F ) )
4213, 41eqtrd 2444 . 2  |-  ( ph  -> 
sum_ k  e.  Z  A  =  (  ~~>  `  F
) )
433, 42breqtrrd 4206 1  |-  ( ph  ->  F  ~~>  sum_ k  e.  Z  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2924    C_ wss 3288   class class class wbr 4180    e. cmpt 4234   dom cdm 4845    |` cres 4847   iotacio 5383   ` cfv 5421  (class class class)co 6048   CCcc 8952    + caddc 8957   ZZcz 10246   ZZ>=cuz 10452   ...cfz 11007    seq cseq 11286    ~~> cli 12241   sum_csu 12442
This theorem is referenced by:  esumcvg  24437
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-oi 7443  df-card 7790  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-n0 10186  df-z 10247  df-uz 10453  df-rp 10577  df-fz 11008  df-fzo 11099  df-seq 11287  df-exp 11346  df-hash 11582  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-clim 12245  df-sum 12443
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