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Theorem isumclim3 12548
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 12353 . . 3  |-  ( F  e.  dom  ~~>  <->  F  ~~>  (  ~~>  `  F
) )
31, 2sylib 190 . 2  |-  ( ph  ->  F  ~~>  (  ~~>  `  F
) )
4 sumfc 12508 . . . 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 2439 . . . . 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 2438 . . . . . . 7  |-  ( k  e.  Z  |->  A )  =  ( k  e.  Z  |->  A )
108, 9fmptd 5896 . . . . . 6  |-  ( ph  ->  ( k  e.  Z  |->  A ) : Z --> CC )
1110ffvelrnda 5873 . . . . 5  |-  ( (
ph  /\  m  e.  Z )  ->  (
( k  e.  Z  |->  A ) `  m
)  e.  CC )
125, 6, 7, 11isum 12518 . . . 4  |-  ( ph  -> 
sum_ m  e.  Z  ( ( k  e.  Z  |->  A ) `  m )  =  (  ~~>  `
 seq  M (  +  ,  ( k  e.  Z  |->  A ) ) ) )
134, 12syl5eqr 2484 . . 3  |-  ( ph  -> 
sum_ k  e.  Z  A  =  (  ~~>  `  seq  M (  +  ,  ( k  e.  Z  |->  A ) ) ) )
14 seqex 11330 . . . . . . 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 11098 . . . . . . . . . . . . . 14  |-  ( M ... j )  C_  ( ZZ>= `  M )
1817, 5sseqtr4i 3383 . . . . . . . . . . . . 13  |-  ( M ... j )  C_  Z
19 resmpt 5194 . . . . . . . . . . . . 13  |-  ( ( M ... j ) 
C_  Z  ->  (
( k  e.  Z  |->  A )  |`  ( M ... j ) )  =  ( k  e.  ( M ... j
)  |->  A ) )
2018, 19ax-mp 5 . . . . . . . . . . . 12  |-  ( ( k  e.  Z  |->  A )  |`  ( M ... j ) )  =  ( k  e.  ( M ... j ) 
|->  A )
2120fveq1i 5732 . . . . . . . . . . 11  |-  ( ( ( k  e.  Z  |->  A )  |`  ( M ... j ) ) `
 m )  =  ( ( k  e.  ( M ... j
)  |->  A ) `  m )
22 fvres 5748 . . . . . . . . . . 11  |-  ( m  e.  ( M ... j )  ->  (
( ( k  e.  Z  |->  A )  |`  ( M ... j ) ) `  m )  =  ( ( k  e.  Z  |->  A ) `
 m ) )
2321, 22syl5reqr 2485 . . . . . . . . . 10  |-  ( m  e.  ( M ... j )  ->  (
( k  e.  Z  |->  A ) `  m
)  =  ( ( k  e.  ( M ... j )  |->  A ) `  m ) )
2423sumeq2i 12498 . . . . . . . . 9  |-  sum_ m  e.  ( M ... j
) ( ( k  e.  Z  |->  A ) `
 m )  = 
sum_ m  e.  ( M ... j ) ( ( k  e.  ( M ... j ) 
|->  A ) `  m
)
25 sumfc 12508 . . . . . . . . 9  |-  sum_ m  e.  ( M ... j
) ( ( k  e.  ( M ... j )  |->  A ) `
 m )  = 
sum_ k  e.  ( M ... j ) A
2624, 25eqtri 2458 . . . . . . . 8  |-  sum_ m  e.  ( M ... j
) ( ( k  e.  Z  |->  A ) `
 m )  = 
sum_ k  e.  ( M ... j ) A
27 eqidd 2439 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  Z )  /\  m  e.  ( M ... j
) )  ->  (
( k  e.  Z  |->  A ) `  m
)  =  ( ( k  e.  Z  |->  A ) `  m ) )
28 simpr 449 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  Z )
2928, 5syl6eleq 2528 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  ( ZZ>= `  M )
)
30 simpl 445 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  Z )  ->  ph )
31 elfzuz 11060 . . . . . . . . . . 11  |-  ( m  e.  ( M ... j )  ->  m  e.  ( ZZ>= `  M )
)
3231, 5syl6eleqr 2529 . . . . . . . . . 10  |-  ( m  e.  ( M ... j )  ->  m  e.  Z )
3330, 32, 11syl2an 465 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  Z )  /\  m  e.  ( M ... j
) )  ->  (
( k  e.  Z  |->  A ) `  m
)  e.  CC )
3427, 29, 33fsumser 12529 . . . . . . . 8  |-  ( (
ph  /\  j  e.  Z )  ->  sum_ m  e.  ( M ... j
) ( ( k  e.  Z  |->  A ) `
 m )  =  (  seq  M (  +  ,  ( k  e.  Z  |->  A ) ) `  j ) )
3526, 34syl5eqr 2484 . . . . . . 7  |-  ( (
ph  /\  j  e.  Z )  ->  sum_ k  e.  ( M ... j
) A  =  (  seq  M (  +  ,  ( k  e.  Z  |->  A ) ) `
 j ) )
3616, 35eqtr2d 2471 . . . . . 6  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq  M (  +  , 
( k  e.  Z  |->  A ) ) `  j )  =  ( F `  j ) )
375, 15, 1, 6, 36climeq 12366 . . . . 5  |-  ( ph  ->  (  seq  M (  +  ,  ( k  e.  Z  |->  A ) )  ~~>  x  <->  F  ~~>  x ) )
3837iotabidv 5442 . . . 4  |-  ( ph  ->  ( iota x  seq  M (  +  ,  ( k  e.  Z  |->  A ) )  ~~>  x )  =  ( iota x F 
~~>  x ) )
39 df-fv 5465 . . . 4  |-  (  ~~>  `  seq  M (  +  ,  ( k  e.  Z  |->  A ) ) )  =  ( iota x  seq  M (  +  ,  ( k  e.  Z  |->  A ) )  ~~>  x )
40 df-fv 5465 . . . 4  |-  (  ~~>  `  F
)  =  ( iota
x F  ~~>  x )
4138, 39, 403eqtr4g 2495 . . 3  |-  ( ph  ->  (  ~~>  `  seq  M (  +  ,  ( k  e.  Z  |->  A ) ) )  =  (  ~~>  `
 F ) )
4213, 41eqtrd 2470 . 2  |-  ( ph  -> 
sum_ k  e.  Z  A  =  (  ~~>  `  F
) )
433, 42breqtrrd 4241 1  |-  ( ph  ->  F  ~~>  sum_ k  e.  Z  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    C_ wss 3322   class class class wbr 4215    e. cmpt 4269   dom cdm 4881    |` cres 4883   iotacio 5419   ` cfv 5457  (class class class)co 6084   CCcc 8993    + caddc 8998   ZZcz 10287   ZZ>=cuz 10493   ...cfz 11048    seq cseq 11328    ~~> cli 12283   sum_csu 12484
This theorem is referenced by:  esumcvg  24481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fz 11049  df-fzo 11141  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-sum 12485
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