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Theorem isumclim3 11143
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 11015 . . 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 2116 . . . 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 5541 . . . . 5  |-  ( ph  ->  ( k  e.  Z  |->  A ) : Z --> CC )
98ffvelrnda 5521 . . . 4  |-  ( (
ph  /\  m  e.  Z )  ->  (
( k  e.  Z  |->  A ) `  m
)  e.  CC )
104, 5, 6, 9isum 11105 . . 3  |-  ( ph  -> 
sum_ m  e.  Z  ( ( k  e.  Z  |->  A ) `  m )  =  (  ~~>  `
 seq M (  +  ,  ( k  e.  Z  |->  A ) ) ) )
117ralrimiva 2480 . . . 4  |-  ( ph  ->  A. k  e.  Z  A  e.  CC )
12 sumfct 11094 . . . 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 10171 . . . . . . 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 9796 . . . . . . . . . . . . . 14  |-  ( M ... j )  C_  ( ZZ>= `  M )
1918, 4sseqtrri 3100 . . . . . . . . . . . . 13  |-  ( M ... j )  C_  Z
20 resmpt 4835 . . . . . . . . . . . . 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 5388 . . . . . . . . . . 11  |-  ( ( ( k  e.  Z  |->  A )  |`  ( M ... j ) ) `
 m )  =  ( ( k  e.  ( M ... j
)  |->  A ) `  m )
23 fvres 5411 . . . . . . . . . . 11  |-  ( m  e.  ( M ... j )  ->  (
( ( k  e.  Z  |->  A )  |`  ( M ... j ) ) `  m )  =  ( ( k  e.  Z  |->  A ) `
 m ) )
2422, 23syl5reqr 2163 . . . . . . . . . 10  |-  ( m  e.  ( M ... j )  ->  (
( k  e.  Z  |->  A ) `  m
)  =  ( ( k  e.  ( M ... j )  |->  A ) `  m ) )
2524sumeq2i 11084 . . . . . . . . 9  |-  sum_ m  e.  ( M ... j
) ( ( k  e.  Z  |->  A ) `
 m )  = 
sum_ m  e.  ( M ... j ) ( ( k  e.  ( M ... j ) 
|->  A ) `  m
)
26 ssralv 3129 . . . . . . . . . . 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 11094 . . . . . . . . . 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 2160 . . . . . . . 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 2116 . . . . . . . 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, 4syl6eleq 2208 . . . . . . . 8  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  ( ZZ>= `  M )
)
354eleq2i 2182 . . . . . . . . . 10  |-  ( m  e.  Z  <->  m  e.  ( ZZ>= `  M )
)
3635biimpri 132 . . . . . . . . 9  |-  ( m  e.  ( ZZ>= `  M
)  ->  m  e.  Z )
3717, 36, 9syl2an 285 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  Z )  /\  m  e.  ( ZZ>= `  M )
)  ->  ( (
k  e.  Z  |->  A ) `  m )  e.  CC )
3832, 34, 37fsum3ser 11117 . . . . . . 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 2155 . . . . . 6  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  , 
( k  e.  Z  |->  A ) ) `  j )  =  ( F `  j ) )
404, 15, 1, 5, 39climeq 11019 . . . . 5  |-  ( ph  ->  (  seq M (  +  ,  ( k  e.  Z  |->  A ) )  ~~>  x  <->  F  ~~>  x ) )
4140iotabidv 5077 . . . 4  |-  ( ph  ->  ( iota x  seq M (  +  , 
( k  e.  Z  |->  A ) )  ~~>  x )  =  ( iota x F 
~~>  x ) )
42 df-fv 5099 . . . 4  |-  (  ~~>  `  seq M (  +  , 
( k  e.  Z  |->  A ) ) )  =  ( iota x  seq M (  +  , 
( k  e.  Z  |->  A ) )  ~~>  x )
43 df-fv 5099 . . . 4  |-  (  ~~>  `  F
)  =  ( iota
x F  ~~>  x )
4441, 42, 433eqtr4g 2173 . . 3  |-  ( ph  ->  (  ~~>  `  seq M (  +  ,  ( k  e.  Z  |->  A ) ) )  =  (  ~~>  `
 F ) )
4510, 13, 443eqtr3d 2156 . 2  |-  ( ph  -> 
sum_ k  e.  Z  A  =  (  ~~>  `  F
) )
463, 45breqtrrd 3924 1  |-  ( ph  ->  F  ~~>  sum_ k  e.  Z  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463   A.wral 2391   _Vcvv 2658    C_ wss 3039   class class class wbr 3897    |-> cmpt 3957   dom cdm 4507    |` cres 4509   iotacio 5054   ` cfv 5091  (class class class)co 5740   CCcc 7582    + caddc 7587   ZZcz 9008   ZZ>=cuz 9278   ...cfz 9741    seqcseq 10169    ~~> cli 10998   sum_csu 11073
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703  ax-caucvg 7704
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-isom 5100  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-frec 6254  df-1o 6279  df-oadd 6283  df-er 6395  df-en 6601  df-dom 6602  df-fin 6603  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8396  df-inn 8681  df-2 8739  df-3 8740  df-4 8741  df-n0 8932  df-z 9009  df-uz 9279  df-q 9364  df-rp 9394  df-fz 9742  df-fzo 9871  df-seqfrec 10170  df-exp 10244  df-ihash 10473  df-cj 10565  df-re 10566  df-im 10567  df-rsqrt 10721  df-abs 10722  df-clim 10999  df-sumdc 11074
This theorem is referenced by: (None)
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