ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  isumclim3 Unicode version

Theorem isumclim3 11315
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 11187 . . 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 2158 . . . 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 5621 . . . . 5  |-  ( ph  ->  ( k  e.  Z  |->  A ) : Z --> CC )
98ffvelrnda 5601 . . . 4  |-  ( (
ph  /\  m  e.  Z )  ->  (
( k  e.  Z  |->  A ) `  m
)  e.  CC )
104, 5, 6, 9isum 11277 . . 3  |-  ( ph  -> 
sum_ m  e.  Z  ( ( k  e.  Z  |->  A ) `  m )  =  (  ~~>  `
 seq M (  +  ,  ( k  e.  Z  |->  A ) ) ) )
117ralrimiva 2530 . . . 4  |-  ( ph  ->  A. k  e.  Z  A  e.  CC )
12 sumfct 11266 . . . 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 10341 . . . . . . 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 fzssuz 9962 . . . . . . . . . . . . . 14  |-  ( M ... j )  C_  ( ZZ>= `  M )
1918, 4sseqtrri 3163 . . . . . . . . . . . . 13  |-  ( M ... j )  C_  Z
20 resmpt 4913 . . . . . . . . . . . . 13  |-  ( ( M ... j ) 
C_  Z  ->  (
( k  e.  Z  |->  A )  |`  ( M ... j ) )  =  ( k  e.  ( M ... j
)  |->  A ) )
2119, 20ax-mp 5 . . . . . . . . . . . 12  |-  ( ( k  e.  Z  |->  A )  |`  ( M ... j ) )  =  ( k  e.  ( M ... j ) 
|->  A )
2221fveq1i 5468 . . . . . . . . . . 11  |-  ( ( ( k  e.  Z  |->  A )  |`  ( M ... j ) ) `
 m )  =  ( ( k  e.  ( M ... j
)  |->  A ) `  m )
23 fvres 5491 . . . . . . . . . . 11  |-  ( m  e.  ( M ... j )  ->  (
( ( k  e.  Z  |->  A )  |`  ( M ... j ) ) `  m )  =  ( ( k  e.  Z  |->  A ) `
 m ) )
2422, 23syl5reqr 2205 . . . . . . . . . 10  |-  ( m  e.  ( M ... j )  ->  (
( k  e.  Z  |->  A ) `  m
)  =  ( ( k  e.  ( M ... j )  |->  A ) `  m ) )
2524sumeq2i 11256 . . . . . . . . 9  |-  sum_ m  e.  ( M ... j
) ( ( k  e.  Z  |->  A ) `
 m )  = 
sum_ m  e.  ( M ... j ) ( ( k  e.  ( M ... j ) 
|->  A ) `  m
)
26 ssralv 3192 . . . . . . . . . . 11  |-  ( ( M ... j ) 
C_  Z  ->  ( A. k  e.  Z  A  e.  CC  ->  A. k  e.  ( M ... j ) A  e.  CC ) )
2719, 11, 26mpsyl 65 . . . . . . . . . 10  |-  ( ph  ->  A. k  e.  ( M ... j ) A  e.  CC )
28 sumfct 11266 . . . . . . . . . 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 2202 . . . . . . . 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 2158 . . . . . . . 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 2250 . . . . . . . 8  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  ( ZZ>= `  M )
)
354eleq2i 2224 . . . . . . . . . 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 11289 . . . . . . 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 2197 . . . . . 6  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  , 
( k  e.  Z  |->  A ) ) `  j )  =  ( F `  j ) )
404, 15, 1, 5, 39climeq 11191 . . . . 5  |-  ( ph  ->  (  seq M (  +  ,  ( k  e.  Z  |->  A ) )  ~~>  x  <->  F  ~~>  x ) )
4140iotabidv 5155 . . . 4  |-  ( ph  ->  ( iota x  seq M (  +  , 
( k  e.  Z  |->  A ) )  ~~>  x )  =  ( iota x F 
~~>  x ) )
42 df-fv 5177 . . . 4  |-  (  ~~>  `  seq M (  +  , 
( k  e.  Z  |->  A ) ) )  =  ( iota x  seq M (  +  , 
( k  e.  Z  |->  A ) )  ~~>  x )
43 df-fv 5177 . . . 4  |-  (  ~~>  `  F
)  =  ( iota
x F  ~~>  x )
4441, 42, 433eqtr4g 2215 . . 3  |-  ( ph  ->  (  ~~>  `  seq M (  +  ,  ( k  e.  Z  |->  A ) ) )  =  (  ~~>  `
 F ) )
4510, 13, 443eqtr3d 2198 . 2  |-  ( ph  -> 
sum_ k  e.  Z  A  =  (  ~~>  `  F
) )
463, 45breqtrrd 3992 1  |-  ( ph  ->  F  ~~>  sum_ k  e.  Z  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1335    e. wcel 2128   A.wral 2435   _Vcvv 2712    C_ wss 3102   class class class wbr 3965    |-> cmpt 4025   dom cdm 4585    |` cres 4587   iotacio 5132   ` cfv 5169  (class class class)co 5821   CCcc 7725    + caddc 7730   ZZcz 9162   ZZ>=cuz 9434   ...cfz 9907    seqcseq 10339    ~~> cli 11170   sum_csu 11245
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546  ax-cnex 7818  ax-resscn 7819  ax-1cn 7820  ax-1re 7821  ax-icn 7822  ax-addcl 7823  ax-addrcl 7824  ax-mulcl 7825  ax-mulrcl 7826  ax-addcom 7827  ax-mulcom 7828  ax-addass 7829  ax-mulass 7830  ax-distr 7831  ax-i2m1 7832  ax-0lt1 7833  ax-1rid 7834  ax-0id 7835  ax-rnegex 7836  ax-precex 7837  ax-cnre 7838  ax-pre-ltirr 7839  ax-pre-ltwlin 7840  ax-pre-lttrn 7841  ax-pre-apti 7842  ax-pre-ltadd 7843  ax-pre-mulgt0 7844  ax-pre-mulext 7845  ax-arch 7846  ax-caucvg 7847
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-isom 5178  df-riota 5777  df-ov 5824  df-oprab 5825  df-mpo 5826  df-1st 6085  df-2nd 6086  df-recs 6249  df-irdg 6314  df-frec 6335  df-1o 6360  df-oadd 6364  df-er 6477  df-en 6683  df-dom 6684  df-fin 6685  df-pnf 7909  df-mnf 7910  df-xr 7911  df-ltxr 7912  df-le 7913  df-sub 8043  df-neg 8044  df-reap 8445  df-ap 8452  df-div 8541  df-inn 8829  df-2 8887  df-3 8888  df-4 8889  df-n0 9086  df-z 9163  df-uz 9435  df-q 9524  df-rp 9556  df-fz 9908  df-fzo 10037  df-seqfrec 10340  df-exp 10414  df-ihash 10645  df-cj 10737  df-re 10738  df-im 10739  df-rsqrt 10893  df-abs 10894  df-clim 11171  df-sumdc 11246
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator