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Theorem isumclim3 11386
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 11258 . . 3  |-  ( F  e.  dom  ~~>  <->  F  ~~>  (  ~~>  `  F
) )
31, 2sylib 121 . 2  |-  ( ph  ->  F  ~~>  (  ~~>  `  F
) )
4 isumclim3.1 . . . 4  |-  Z  =  ( ZZ>= `  M )
5 isumclim3.2 . . . 4  |-  ( ph  ->  M  e.  ZZ )
6 eqidd 2171 . . . 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 5651 . . . . 5  |-  ( ph  ->  ( k  e.  Z  |->  A ) : Z --> CC )
98ffvelrnda 5631 . . . 4  |-  ( (
ph  /\  m  e.  Z )  ->  (
( k  e.  Z  |->  A ) `  m
)  e.  CC )
104, 5, 6, 9isum 11348 . . 3  |-  ( ph  -> 
sum_ m  e.  Z  ( ( k  e.  Z  |->  A ) `  m )  =  (  ~~>  `
 seq M (  +  ,  ( k  e.  Z  |->  A ) ) ) )
117ralrimiva 2543 . . . 4  |-  ( ph  ->  A. k  e.  Z  A  e.  CC )
12 sumfct 11337 . . . 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 seqex 10403 . . . . . . 7  |-  seq M
(  +  ,  ( k  e.  Z  |->  A ) )  e.  _V
1514a1i 9 . . . . . 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 simpl 108 . . . . . . . 8  |-  ( (
ph  /\  j  e.  Z )  ->  ph )
18 fvres 5520 . . . . . . . . . . 11  |-  ( m  e.  ( M ... j )  ->  (
( ( k  e.  Z  |->  A )  |`  ( M ... j ) ) `  m )  =  ( ( k  e.  Z  |->  A ) `
 m ) )
19 fzssuz 10021 . . . . . . . . . . . . . 14  |-  ( M ... j )  C_  ( ZZ>= `  M )
2019, 4sseqtrri 3182 . . . . . . . . . . . . 13  |-  ( M ... j )  C_  Z
21 resmpt 4939 . . . . . . . . . . . . 13  |-  ( ( M ... j ) 
C_  Z  ->  (
( k  e.  Z  |->  A )  |`  ( M ... j ) )  =  ( k  e.  ( M ... j
)  |->  A ) )
2220, 21ax-mp 5 . . . . . . . . . . . 12  |-  ( ( k  e.  Z  |->  A )  |`  ( M ... j ) )  =  ( k  e.  ( M ... j ) 
|->  A )
2322fveq1i 5497 . . . . . . . . . . 11  |-  ( ( ( k  e.  Z  |->  A )  |`  ( M ... j ) ) `
 m )  =  ( ( k  e.  ( M ... j
)  |->  A ) `  m )
2418, 23eqtr3di 2218 . . . . . . . . . 10  |-  ( m  e.  ( M ... j )  ->  (
( k  e.  Z  |->  A ) `  m
)  =  ( ( k  e.  ( M ... j )  |->  A ) `  m ) )
2524sumeq2i 11327 . . . . . . . . 9  |-  sum_ m  e.  ( M ... j
) ( ( k  e.  Z  |->  A ) `
 m )  = 
sum_ m  e.  ( M ... j ) ( ( k  e.  ( M ... j ) 
|->  A ) `  m
)
26 ssralv 3211 . . . . . . . . . . 11  |-  ( ( M ... j ) 
C_  Z  ->  ( A. k  e.  Z  A  e.  CC  ->  A. k  e.  ( M ... j ) A  e.  CC ) )
2720, 11, 26mpsyl 65 . . . . . . . . . 10  |-  ( ph  ->  A. k  e.  ( M ... j ) A  e.  CC )
28 sumfct 11337 . . . . . . . . . 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, 29eqtrid 2215 . . . . . . . 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 2171 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  Z )  /\  m  e.  ( ZZ>= `  M )
)  ->  ( (
k  e.  Z  |->  A ) `  m )  =  ( ( k  e.  Z  |->  A ) `
 m ) )
33 simpr 109 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  Z )
3433, 4eleqtrdi 2263 . . . . . . . 8  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  ( ZZ>= `  M )
)
354eleq2i 2237 . . . . . . . . . 10  |-  ( m  e.  Z  <->  m  e.  ( ZZ>= `  M )
)
3635biimpri 132 . . . . . . . . 9  |-  ( m  e.  ( ZZ>= `  M
)  ->  m  e.  Z )
3717, 36, 9syl2an 287 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  Z )  /\  m  e.  ( ZZ>= `  M )
)  ->  ( (
k  e.  Z  |->  A ) `  m )  e.  CC )
3832, 34, 37fsum3ser 11360 . . . . . . 7  |-  ( (
ph  /\  j  e.  Z )  ->  sum_ m  e.  ( M ... j
) ( ( k  e.  Z  |->  A ) `
 m )  =  (  seq M (  +  ,  ( k  e.  Z  |->  A ) ) `  j ) )
3916, 31, 383eqtr2rd 2210 . . . . . 6  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  , 
( k  e.  Z  |->  A ) ) `  j )  =  ( F `  j ) )
404, 15, 1, 5, 39climeq 11262 . . . . 5  |-  ( ph  ->  (  seq M (  +  ,  ( k  e.  Z  |->  A ) )  ~~>  x  <->  F  ~~>  x ) )
4140iotabidv 5181 . . . 4  |-  ( ph  ->  ( iota x  seq M (  +  , 
( k  e.  Z  |->  A ) )  ~~>  x )  =  ( iota x F 
~~>  x ) )
42 df-fv 5206 . . . 4  |-  (  ~~>  `  seq M (  +  , 
( k  e.  Z  |->  A ) ) )  =  ( iota x  seq M (  +  , 
( k  e.  Z  |->  A ) )  ~~>  x )
43 df-fv 5206 . . . 4  |-  (  ~~>  `  F
)  =  ( iota
x F  ~~>  x )
4441, 42, 433eqtr4g 2228 . . 3  |-  ( ph  ->  (  ~~>  `  seq M (  +  ,  ( k  e.  Z  |->  A ) ) )  =  (  ~~>  `
 F ) )
4510, 13, 443eqtr3d 2211 . 2  |-  ( ph  -> 
sum_ k  e.  Z  A  =  (  ~~>  `  F
) )
463, 45breqtrrd 4017 1  |-  ( ph  ->  F  ~~>  sum_ k  e.  Z  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    e. wcel 2141   A.wral 2448   _Vcvv 2730    C_ wss 3121   class class class wbr 3989    |-> cmpt 4050   dom cdm 4611    |` cres 4613   iotacio 5158   ` cfv 5198  (class class class)co 5853   CCcc 7772    + caddc 7777   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965    seqcseq 10401    ~~> cli 11241   sum_csu 11316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317
This theorem is referenced by: (None)
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