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Theorem isumclim3 12238
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 12044 . . 3  |-  ( F  e.  dom  ~~>  <->  F  ~~>  (  ~~>  `  F
) )
31, 2sylib 188 . 2  |-  ( ph  ->  F  ~~>  (  ~~>  `  F
) )
4 sumfc 12198 . . . 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 2297 . . . . 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 2296 . . . . . . 7  |-  ( k  e.  Z  |->  A )  =  ( k  e.  Z  |->  A )
108, 9fmptd 5700 . . . . . 6  |-  ( ph  ->  ( k  e.  Z  |->  A ) : Z --> CC )
11 ffvelrn 5679 . . . . . 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 12208 . . . 4  |-  ( ph  -> 
sum_ m  e.  Z  ( ( k  e.  Z  |->  A ) `  m )  =  (  ~~>  `
 seq  M (  +  ,  ( k  e.  Z  |->  A ) ) ) )
144, 13syl5eqr 2342 . . 3  |-  ( ph  -> 
sum_ k  e.  Z  A  =  (  ~~>  `  seq  M (  +  ,  ( k  e.  Z  |->  A ) ) ) )
15 seqex 11064 . . . . . . 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 10848 . . . . . . . . . . . . . 14  |-  ( M ... j )  C_  ( ZZ>= `  M )
1918, 5sseqtr4i 3224 . . . . . . . . . . . . 13  |-  ( M ... j )  C_  Z
20 resmpt 5016 . . . . . . . . . . . . 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 5542 . . . . . . . . . . 11  |-  ( ( ( k  e.  Z  |->  A )  |`  ( M ... j ) ) `
 m )  =  ( ( k  e.  ( M ... j
)  |->  A ) `  m )
23 fvres 5558 . . . . . . . . . . 11  |-  ( m  e.  ( M ... j )  ->  (
( ( k  e.  Z  |->  A )  |`  ( M ... j ) ) `  m )  =  ( ( k  e.  Z  |->  A ) `
 m ) )
2422, 23syl5reqr 2343 . . . . . . . . . 10  |-  ( m  e.  ( M ... j )  ->  (
( k  e.  Z  |->  A ) `  m
)  =  ( ( k  e.  ( M ... j )  |->  A ) `  m ) )
2524sumeq2i 12188 . . . . . . . . 9  |-  sum_ m  e.  ( M ... j
) ( ( k  e.  Z  |->  A ) `
 m )  = 
sum_ m  e.  ( M ... j ) ( ( k  e.  ( M ... j ) 
|->  A ) `  m
)
26 sumfc 12198 . . . . . . . . 9  |-  sum_ m  e.  ( M ... j
) ( ( k  e.  ( M ... j )  |->  A ) `
 m )  = 
sum_ k  e.  ( M ... j ) A
2725, 26eqtri 2316 . . . . . . . 8  |-  sum_ m  e.  ( M ... j
) ( ( k  e.  Z  |->  A ) `
 m )  = 
sum_ k  e.  ( M ... j ) A
28 eqidd 2297 . . . . . . . . 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 2386 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  ( ZZ>= `  M )
)
31 simpl 443 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  Z )  ->  ph )
32 elfzuz 10810 . . . . . . . . . . 11  |-  ( m  e.  ( M ... j )  ->  m  e.  ( ZZ>= `  M )
)
3332, 5syl6eleqr 2387 . . . . . . . . . 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 12219 . . . . . . . 8  |-  ( (
ph  /\  j  e.  Z )  ->  sum_ m  e.  ( M ... j
) ( ( k  e.  Z  |->  A ) `
 m )  =  (  seq  M (  +  ,  ( k  e.  Z  |->  A ) ) `  j ) )
3627, 35syl5eqr 2342 . . . . . . 7  |-  ( (
ph  /\  j  e.  Z )  ->  sum_ k  e.  ( M ... j
) A  =  (  seq  M (  +  ,  ( k  e.  Z  |->  A ) ) `
 j ) )
3717, 36eqtr2d 2329 . . . . . 6  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq  M (  +  , 
( k  e.  Z  |->  A ) ) `  j )  =  ( F `  j ) )
385, 16, 1, 6, 37climeq 12057 . . . . 5  |-  ( ph  ->  (  seq  M (  +  ,  ( k  e.  Z  |->  A ) )  ~~>  x  <->  F  ~~>  x ) )
3938iotabidv 5256 . . . 4  |-  ( ph  ->  ( iota x  seq  M (  +  ,  ( k  e.  Z  |->  A ) )  ~~>  x )  =  ( iota x F 
~~>  x ) )
40 df-fv 5279 . . . 4  |-  (  ~~>  `  seq  M (  +  ,  ( k  e.  Z  |->  A ) ) )  =  ( iota x  seq  M (  +  ,  ( k  e.  Z  |->  A ) )  ~~>  x )
41 df-fv 5279 . . . 4  |-  (  ~~>  `  F
)  =  ( iota
x F  ~~>  x )
4239, 40, 413eqtr4g 2353 . . 3  |-  ( ph  ->  (  ~~>  `  seq  M (  +  ,  ( k  e.  Z  |->  A ) ) )  =  (  ~~>  `
 F ) )
4314, 42eqtrd 2328 . 2  |-  ( ph  -> 
sum_ k  e.  Z  A  =  (  ~~>  `  F
) )
443, 43breqtrrd 4065 1  |-  ( ph  ->  F  ~~>  sum_ k  e.  Z  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   class class class wbr 4039    e. cmpt 4093   dom cdm 4705    |` cres 4707   iotacio 5233   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751    + caddc 8756   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798    seq cseq 11062    ~~> cli 11974   sum_csu 12174
This theorem is referenced by:  esumcvg  23469
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175
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