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Theorem isumclim3 10817
Description: The sequence of partial finite sums of a converging infinite series converges 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 10683 . . 3  |-  ( F  e.  dom  ~~>  <->  F  ~~>  (  ~~>  `  F
) )
31, 2sylib 120 . 2  |-  ( ph  ->  F  ~~>  (  ~~>  `  F
) )
4 isumclim3.1 . . . 4  |-  Z  =  ( ZZ>= `  M )
5 isumclim3.2 . . . 4  |-  ( ph  ->  M  e.  ZZ )
6 eqidd 2089 . . . 4  |-  ( (
ph  /\  m  e.  Z )  ->  (
( k  e.  Z  |->  A ) `  m
)  =  ( ( k  e.  Z  |->  A ) `  m ) )
7 isumclim3.4 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
87fmpttd 5453 . . . . 5  |-  ( ph  ->  ( k  e.  Z  |->  A ) : Z --> CC )
98ffvelrnda 5434 . . . 4  |-  ( (
ph  /\  m  e.  Z )  ->  (
( k  e.  Z  |->  A ) `  m
)  e.  CC )
104, 5, 6, 9iisum 10775 . . 3  |-  ( ph  -> 
sum_ m  e.  Z  ( ( k  e.  Z  |->  A ) `  m )  =  (  ~~>  `
 seq M (  +  ,  ( k  e.  Z  |->  A ) ,  CC ) ) )
117ralrimiva 2446 . . . 4  |-  ( ph  ->  A. k  e.  Z  A  e.  CC )
12 sumfct 10763 . . . 4  |-  ( A. k  e.  Z  A  e.  CC  ->  sum_ m  e.  Z  ( ( k  e.  Z  |->  A ) `
 m )  = 
sum_ k  e.  Z  A )
1311, 12syl 14 . . 3  |-  ( ph  -> 
sum_ m  e.  Z  ( ( k  e.  Z  |->  A ) `  m )  =  sum_ k  e.  Z  A
)
14 iseqex 9856 . . . . . . 7  |-  seq M
(  +  ,  ( k  e.  Z  |->  A ) ,  CC )  e.  _V
1514a1i 9 . . . . . 6  |-  ( ph  ->  seq M (  +  ,  ( k  e.  Z  |->  A ) ,  CC )  e.  _V )
16 isumclim3.5 . . . . . . 7  |-  ( (
ph  /\  j  e.  Z )  ->  ( F `  j )  =  sum_ k  e.  ( M ... j ) A )
17 simpl 107 . . . . . . . 8  |-  ( (
ph  /\  j  e.  Z )  ->  ph )
18 fzssuz 9479 . . . . . . . . . . . . . 14  |-  ( M ... j )  C_  ( ZZ>= `  M )
1918, 4sseqtr4i 3059 . . . . . . . . . . . . 13  |-  ( M ... j )  C_  Z
20 resmpt 4760 . . . . . . . . . . . . 13  |-  ( ( M ... j ) 
C_  Z  ->  (
( k  e.  Z  |->  A )  |`  ( M ... j ) )  =  ( k  e.  ( M ... j
)  |->  A ) )
2119, 20ax-mp 7 . . . . . . . . . . . 12  |-  ( ( k  e.  Z  |->  A )  |`  ( M ... j ) )  =  ( k  e.  ( M ... j ) 
|->  A )
2221fveq1i 5306 . . . . . . . . . . 11  |-  ( ( ( k  e.  Z  |->  A )  |`  ( M ... j ) ) `
 m )  =  ( ( k  e.  ( M ... j
)  |->  A ) `  m )
23 fvres 5329 . . . . . . . . . . 11  |-  ( m  e.  ( M ... j )  ->  (
( ( k  e.  Z  |->  A )  |`  ( M ... j ) ) `  m )  =  ( ( k  e.  Z  |->  A ) `
 m ) )
2422, 23syl5reqr 2135 . . . . . . . . . 10  |-  ( m  e.  ( M ... j )  ->  (
( k  e.  Z  |->  A ) `  m
)  =  ( ( k  e.  ( M ... j )  |->  A ) `  m ) )
2524sumeq2i 10753 . . . . . . . . 9  |-  sum_ m  e.  ( M ... j
) ( ( k  e.  Z  |->  A ) `
 m )  = 
sum_ m  e.  ( M ... j ) ( ( k  e.  ( M ... j ) 
|->  A ) `  m
)
26 ssralv 3085 . . . . . . . . . . 11  |-  ( ( M ... j ) 
C_  Z  ->  ( A. k  e.  Z  A  e.  CC  ->  A. k  e.  ( M ... j ) A  e.  CC ) )
2719, 11, 26mpsyl 64 . . . . . . . . . 10  |-  ( ph  ->  A. k  e.  ( M ... j ) A  e.  CC )
28 sumfct 10763 . . . . . . . . . 10  |-  ( A. k  e.  ( M ... j ) A  e.  CC  ->  sum_ m  e.  ( M ... j
) ( ( k  e.  ( M ... j )  |->  A ) `
 m )  = 
sum_ k  e.  ( M ... j ) A )
2927, 28syl 14 . . . . . . . . 9  |-  ( ph  -> 
sum_ m  e.  ( M ... j ) ( ( k  e.  ( M ... j ) 
|->  A ) `  m
)  =  sum_ k  e.  ( M ... j
) A )
3025, 29syl5eq 2132 . . . . . . . 8  |-  ( ph  -> 
sum_ m  e.  ( M ... j ) ( ( k  e.  Z  |->  A ) `  m
)  =  sum_ k  e.  ( M ... j
) A )
3117, 30syl 14 . . . . . . 7  |-  ( (
ph  /\  j  e.  Z )  ->  sum_ m  e.  ( M ... j
) ( ( k  e.  Z  |->  A ) `
 m )  = 
sum_ k  e.  ( M ... j ) A )
32 eqidd 2089 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  Z )  /\  m  e.  ( ZZ>= `  M )
)  ->  ( (
k  e.  Z  |->  A ) `  m )  =  ( ( k  e.  Z  |->  A ) `
 m ) )
33 simpr 108 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  Z )
3433, 4syl6eleq 2180 . . . . . . . 8  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  ( ZZ>= `  M )
)
354eleq2i 2154 . . . . . . . . . 10  |-  ( m  e.  Z  <->  m  e.  ( ZZ>= `  M )
)
3635biimpri 131 . . . . . . . . 9  |-  ( m  e.  ( ZZ>= `  M
)  ->  m  e.  Z )
3717, 36, 9syl2an 283 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  Z )  /\  m  e.  ( ZZ>= `  M )
)  ->  ( (
k  e.  Z  |->  A ) `  m )  e.  CC )
3832, 34, 37fisumser 10790 . . . . . . 7  |-  ( (
ph  /\  j  e.  Z )  ->  sum_ m  e.  ( M ... j
) ( ( k  e.  Z  |->  A ) `
 m )  =  (  seq M (  +  ,  ( k  e.  Z  |->  A ) ,  CC ) `  j ) )
3916, 31, 383eqtr2rd 2127 . . . . . 6  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  , 
( k  e.  Z  |->  A ) ,  CC ) `  j )  =  ( F `  j ) )
404, 15, 1, 5, 39climeq 10687 . . . . 5  |-  ( ph  ->  (  seq M (  +  ,  ( k  e.  Z  |->  A ) ,  CC )  ~~>  x  <->  F  ~~>  x ) )
4140iotabidv 5001 . . . 4  |-  ( ph  ->  ( iota x  seq M (  +  , 
( k  e.  Z  |->  A ) ,  CC ) 
~~>  x )  =  ( iota x F  ~~>  x ) )
42 df-fv 5023 . . . 4  |-  (  ~~>  `  seq M (  +  , 
( k  e.  Z  |->  A ) ,  CC ) )  =  ( iota x  seq M
(  +  ,  ( k  e.  Z  |->  A ) ,  CC )  ~~>  x )
43 df-fv 5023 . . . 4  |-  (  ~~>  `  F
)  =  ( iota
x F  ~~>  x )
4441, 42, 433eqtr4g 2145 . . 3  |-  ( ph  ->  (  ~~>  `  seq M (  +  ,  ( k  e.  Z  |->  A ) ,  CC ) )  =  (  ~~>  `  F
) )
4510, 13, 443eqtr3d 2128 . 2  |-  ( ph  -> 
sum_ k  e.  Z  A  =  (  ~~>  `  F
) )
463, 45breqtrrd 3871 1  |-  ( ph  ->  F  ~~>  sum_ k  e.  Z  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   A.wral 2359   _Vcvv 2619    C_ wss 2999   class class class wbr 3845    |-> cmpt 3899   dom cdm 4438    |` cres 4440   iotacio 4978   ` cfv 5015  (class class class)co 5652   CCcc 7348    + caddc 7353   ZZcz 8750   ZZ>=cuz 9019   ...cfz 9424    seqcseq4 9851    ~~> cli 10666   sum_csu 10742
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-mulrcl 7444  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-precex 7455  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-apti 7460  ax-pre-ltadd 7461  ax-pre-mulgt0 7462  ax-pre-mulext 7463  ax-arch 7464  ax-caucvg 7465
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-isom 5024  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-frec 6156  df-1o 6181  df-oadd 6185  df-er 6292  df-en 6458  df-dom 6459  df-fin 6460  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-reap 8052  df-ap 8059  df-div 8140  df-inn 8423  df-2 8481  df-3 8482  df-4 8483  df-n0 8674  df-z 8751  df-uz 9020  df-q 9105  df-rp 9135  df-fz 9425  df-fzo 9554  df-iseq 9853  df-seq3 9854  df-exp 9955  df-ihash 10184  df-cj 10276  df-re 10277  df-im 10278  df-rsqrt 10431  df-abs 10432  df-clim 10667  df-isum 10743
This theorem is referenced by: (None)
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