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Theorem fsum3ser 11117
Description: A finite sum expressed in terms of a partial sum of an infinite series. The recursive definition follows as fsum1 11132 and fsump1 11140, which should make our notation clear and from which, along with closure fsumcl 11120, we will derive the basic properties of finite sums. (Contributed by NM, 11-Dec-2005.) (Revised by Jim Kingdon, 1-Oct-2022.)
Hypotheses
Ref Expression
fsum3ser.1  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  A )
fsum3ser.2  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
fsum3ser.3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  A  e.  CC )
Assertion
Ref Expression
fsum3ser  |-  ( ph  -> 
sum_ k  e.  ( M ... N ) A  =  (  seq M (  +  ,  F ) `  N
) )
Distinct variable groups:    k, F    k, M    k, N    ph, k
Allowed substitution hint:    A( k)

Proof of Theorem fsum3ser
Dummy variables  m  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2115 . . . . 5  |-  ( m  e.  ( ZZ>= `  M
)  |->  if ( m  e.  ( M ... N ) ,  ( F `  m ) ,  0 ) )  =  ( m  e.  ( ZZ>= `  M )  |->  if ( m  e.  ( M ... N
) ,  ( F `
 m ) ,  0 ) )
2 eleq1w 2176 . . . . . 6  |-  ( m  =  k  ->  (
m  e.  ( M ... N )  <->  k  e.  ( M ... N ) ) )
3 fveq2 5387 . . . . . 6  |-  ( m  =  k  ->  ( F `  m )  =  ( F `  k ) )
42, 3ifbieq1d 3462 . . . . 5  |-  ( m  =  k  ->  if ( m  e.  ( M ... N ) ,  ( F `  m
) ,  0 )  =  if ( k  e.  ( M ... N ) ,  ( F `  k ) ,  0 ) )
5 simpr 109 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  k  e.  ( ZZ>= `  M )
)
6 fsum3ser.1 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  A )
7 fsum3ser.3 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  A  e.  CC )
86, 7eqeltrd 2192 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
98adantr 272 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  e.  CC )
10 0cnd 7723 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  -.  k  e.  ( M ... N
) )  ->  0  e.  CC )
11 eluzelz 9287 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
12 eluzel2 9283 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
13 fsum3ser.2 . . . . . . . . 9  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
14 eluzelz 9287 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
1513, 14syl 14 . . . . . . . 8  |-  ( ph  ->  N  e.  ZZ )
1615adantr 272 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  N  e.  ZZ )
17 fzdcel 9771 . . . . . . 7  |-  ( ( k  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  k  e.  ( M ... N ) )
1811, 12, 16, 17syl2an23an 1260 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  -> DECID  k  e.  ( M ... N ) )
199, 10, 18ifcldadc 3469 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  if (
k  e.  ( M ... N ) ,  ( F `  k
) ,  0 )  e.  CC )
201, 4, 5, 19fvmptd3 5480 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
m  e.  ( ZZ>= `  M )  |->  if ( m  e.  ( M ... N ) ,  ( F `  m
) ,  0 ) ) `  k )  =  if ( k  e.  ( M ... N ) ,  ( F `  k ) ,  0 ) )
216ifeq1d 3457 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  if (
k  e.  ( M ... N ) ,  ( F `  k
) ,  0 )  =  if ( k  e.  ( M ... N ) ,  A ,  0 ) )
2220, 21eqtrd 2148 . . 3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
m  e.  ( ZZ>= `  M )  |->  if ( m  e.  ( M ... N ) ,  ( F `  m
) ,  0 ) ) `  k )  =  if ( k  e.  ( M ... N ) ,  A ,  0 ) )
23 elfzuz 9753 . . . 4  |-  ( k  e.  ( M ... N )  ->  k  e.  ( ZZ>= `  M )
)
2423, 7sylan2 282 . . 3  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  A  e.  CC )
25 ssidd 3086 . . 3  |-  ( ph  ->  ( M ... N
)  C_  ( M ... N ) )
2622, 13, 24, 18, 25fsumsersdc 11115 . 2  |-  ( ph  -> 
sum_ k  e.  ( M ... N ) A  =  (  seq M (  +  , 
( m  e.  (
ZZ>= `  M )  |->  if ( m  e.  ( M ... N ) ,  ( F `  m ) ,  0 ) ) ) `  N ) )
2723, 20sylan2 282 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( (
m  e.  ( ZZ>= `  M )  |->  if ( m  e.  ( M ... N ) ,  ( F `  m
) ,  0 ) ) `  k )  =  if ( k  e.  ( M ... N ) ,  ( F `  k ) ,  0 ) )
28 iftrue 3447 . . . . 5  |-  ( k  e.  ( M ... N )  ->  if ( k  e.  ( M ... N ) ,  ( F `  k ) ,  0 )  =  ( F `
 k ) )
2928adantl 273 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  if (
k  e.  ( M ... N ) ,  ( F `  k
) ,  0 )  =  ( F `  k ) )
3027, 29eqtrd 2148 . . 3  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( (
m  e.  ( ZZ>= `  M )  |->  if ( m  e.  ( M ... N ) ,  ( F `  m
) ,  0 ) ) `  k )  =  ( F `  k ) )
31 eleq1w 2176 . . . . . 6  |-  ( m  =  x  ->  (
m  e.  ( M ... N )  <->  x  e.  ( M ... N ) ) )
32 fveq2 5387 . . . . . 6  |-  ( m  =  x  ->  ( F `  m )  =  ( F `  x ) )
3331, 32ifbieq1d 3462 . . . . 5  |-  ( m  =  x  ->  if ( m  e.  ( M ... N ) ,  ( F `  m
) ,  0 )  =  if ( x  e.  ( M ... N ) ,  ( F `  x ) ,  0 ) )
34 simpr 109 . . . . 5  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  x  e.  ( ZZ>= `  M )
)
35 fveq2 5387 . . . . . . . 8  |-  ( k  =  x  ->  ( F `  k )  =  ( F `  x ) )
3635eleq1d 2184 . . . . . . 7  |-  ( k  =  x  ->  (
( F `  k
)  e.  CC  <->  ( F `  x )  e.  CC ) )
378ralrimiva 2480 . . . . . . . 8  |-  ( ph  ->  A. k  e.  (
ZZ>= `  M ) ( F `  k )  e.  CC )
3837adantr 272 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  A. k  e.  ( ZZ>= `  M )
( F `  k
)  e.  CC )
3936, 38, 34rspcdva 2766 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  CC )
40 0cnd 7723 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  0  e.  CC )
41 eluzelz 9287 . . . . . . 7  |-  ( x  e.  ( ZZ>= `  M
)  ->  x  e.  ZZ )
42 eluzel2 9283 . . . . . . 7  |-  ( x  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
4315adantr 272 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  N  e.  ZZ )
44 fzdcel 9771 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  x  e.  ( M ... N ) )
4541, 42, 43, 44syl2an23an 1260 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  -> DECID  x  e.  ( M ... N ) )
4639, 40, 45ifcldcd 3475 . . . . 5  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  if (
x  e.  ( M ... N ) ,  ( F `  x
) ,  0 )  e.  CC )
471, 33, 34, 46fvmptd3 5480 . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( (
m  e.  ( ZZ>= `  M )  |->  if ( m  e.  ( M ... N ) ,  ( F `  m
) ,  0 ) ) `  x )  =  if ( x  e.  ( M ... N ) ,  ( F `  x ) ,  0 ) )
4847, 46eqeltrd 2192 . . 3  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( (
m  e.  ( ZZ>= `  M )  |->  if ( m  e.  ( M ... N ) ,  ( F `  m
) ,  0 ) ) `  x )  e.  CC )
4936cbvralv 2629 . . . . 5  |-  ( A. k  e.  ( ZZ>= `  M ) ( F `
 k )  e.  CC  <->  A. x  e.  (
ZZ>= `  M ) ( F `  x )  e.  CC )
5037, 49sylib 121 . . . 4  |-  ( ph  ->  A. x  e.  (
ZZ>= `  M ) ( F `  x )  e.  CC )
5150r19.21bi 2495 . . 3  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  CC )
52 addcl 7709 . . . 4  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  e.  CC )
5352adantl 273 . . 3  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  +  y )  e.  CC )
5413, 30, 48, 51, 53seq3fveq 10195 . 2  |-  ( ph  ->  (  seq M (  +  ,  ( m  e.  ( ZZ>= `  M
)  |->  if ( m  e.  ( M ... N ) ,  ( F `  m ) ,  0 ) ) ) `  N )  =  (  seq M
(  +  ,  F
) `  N )
)
5526, 54eqtrd 2148 1  |-  ( ph  -> 
sum_ k  e.  ( M ... N ) A  =  (  seq M (  +  ,  F ) `  N
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103  DECID wdc 802    = wceq 1314    e. wcel 1463   A.wral 2391   ifcif 3442    |-> cmpt 3957   ` cfv 5091  (class class class)co 5740   CCcc 7582   0cc0 7584    + caddc 7587   ZZcz 9008   ZZ>=cuz 9278   ...cfz 9741    seqcseq 10169   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:  isumclim3  11143  iserabs  11195  isumsplit  11211  trireciplem  11220  geolim  11231  geo2lim  11236  cvgratnnlemseq  11246  mertenslem2  11256  mertensabs  11257  efcvgfsum  11283  effsumlt  11308  cvgcmp2nlemabs  13061
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