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Theorem fsumm1 11136
Description: Separate out the last term in a finite sum. (Contributed by Mario Carneiro, 26-Apr-2014.)
Hypotheses
Ref Expression
fsumm1.1  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
fsumm1.2  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  A  e.  CC )
fsumm1.3  |-  ( k  =  N  ->  A  =  B )
Assertion
Ref Expression
fsumm1  |-  ( ph  -> 
sum_ k  e.  ( M ... N ) A  =  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  B ) )
Distinct variable groups:    B, k    k, M    k, N    ph, k
Allowed substitution hint:    A( k)

Proof of Theorem fsumm1
StepHypRef Expression
1 fsumm1.1 . . . . . . 7  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
2 eluzelz 9287 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
31, 2syl 14 . . . . . 6  |-  ( ph  ->  N  e.  ZZ )
4 fzsn 9797 . . . . . 6  |-  ( N  e.  ZZ  ->  ( N ... N )  =  { N } )
53, 4syl 14 . . . . 5  |-  ( ph  ->  ( N ... N
)  =  { N } )
65ineq2d 3245 . . . 4  |-  ( ph  ->  ( ( M ... ( N  -  1
) )  i^i  ( N ... N ) )  =  ( ( M ... ( N  - 
1 ) )  i^i 
{ N } ) )
73zred 9127 . . . . . 6  |-  ( ph  ->  N  e.  RR )
87ltm1d 8650 . . . . 5  |-  ( ph  ->  ( N  -  1 )  <  N )
9 fzdisj 9783 . . . . 5  |-  ( ( N  -  1 )  <  N  ->  (
( M ... ( N  -  1 ) )  i^i  ( N ... N ) )  =  (/) )
108, 9syl 14 . . . 4  |-  ( ph  ->  ( ( M ... ( N  -  1
) )  i^i  ( N ... N ) )  =  (/) )
116, 10eqtr3d 2150 . . 3  |-  ( ph  ->  ( ( M ... ( N  -  1
) )  i^i  { N } )  =  (/) )
12 eluzel2 9283 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
131, 12syl 14 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
14 peano2zm 9046 . . . . . . . 8  |-  ( M  e.  ZZ  ->  ( M  -  1 )  e.  ZZ )
1513, 14syl 14 . . . . . . 7  |-  ( ph  ->  ( M  -  1 )  e.  ZZ )
1613zcnd 9128 . . . . . . . . . 10  |-  ( ph  ->  M  e.  CC )
17 ax-1cn 7677 . . . . . . . . . 10  |-  1  e.  CC
18 npcan 7935 . . . . . . . . . 10  |-  ( ( M  e.  CC  /\  1  e.  CC )  ->  ( ( M  - 
1 )  +  1 )  =  M )
1916, 17, 18sylancl 407 . . . . . . . . 9  |-  ( ph  ->  ( ( M  - 
1 )  +  1 )  =  M )
2019fveq2d 5391 . . . . . . . 8  |-  ( ph  ->  ( ZZ>= `  ( ( M  -  1 )  +  1 ) )  =  ( ZZ>= `  M
) )
211, 20eleqtrrd 2195 . . . . . . 7  |-  ( ph  ->  N  e.  ( ZZ>= `  ( ( M  - 
1 )  +  1 ) ) )
22 eluzp1m1 9301 . . . . . . 7  |-  ( ( ( M  -  1 )  e.  ZZ  /\  N  e.  ( ZZ>= `  ( ( M  - 
1 )  +  1 ) ) )  -> 
( N  -  1 )  e.  ( ZZ>= `  ( M  -  1
) ) )
2315, 21, 22syl2anc 406 . . . . . 6  |-  ( ph  ->  ( N  -  1 )  e.  ( ZZ>= `  ( M  -  1
) ) )
24 fzsuc2 9810 . . . . . 6  |-  ( ( M  e.  ZZ  /\  ( N  -  1
)  e.  ( ZZ>= `  ( M  -  1
) ) )  -> 
( M ... (
( N  -  1 )  +  1 ) )  =  ( ( M ... ( N  -  1 ) )  u.  { ( ( N  -  1 )  +  1 ) } ) )
2513, 23, 24syl2anc 406 . . . . 5  |-  ( ph  ->  ( M ... (
( N  -  1 )  +  1 ) )  =  ( ( M ... ( N  -  1 ) )  u.  { ( ( N  -  1 )  +  1 ) } ) )
263zcnd 9128 . . . . . . 7  |-  ( ph  ->  N  e.  CC )
27 npcan 7935 . . . . . . 7  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
2826, 17, 27sylancl 407 . . . . . 6  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
2928oveq2d 5756 . . . . 5  |-  ( ph  ->  ( M ... (
( N  -  1 )  +  1 ) )  =  ( M ... N ) )
3025, 29eqtr3d 2150 . . . 4  |-  ( ph  ->  ( ( M ... ( N  -  1
) )  u.  {
( ( N  - 
1 )  +  1 ) } )  =  ( M ... N
) )
3128sneqd 3508 . . . . 5  |-  ( ph  ->  { ( ( N  -  1 )  +  1 ) }  =  { N } )
3231uneq2d 3198 . . . 4  |-  ( ph  ->  ( ( M ... ( N  -  1
) )  u.  {
( ( N  - 
1 )  +  1 ) } )  =  ( ( M ... ( N  -  1
) )  u.  { N } ) )
3330, 32eqtr3d 2150 . . 3  |-  ( ph  ->  ( M ... N
)  =  ( ( M ... ( N  -  1 ) )  u.  { N }
) )
3413, 3fzfigd 10155 . . 3  |-  ( ph  ->  ( M ... N
)  e.  Fin )
35 fsumm1.2 . . 3  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  A  e.  CC )
3611, 33, 34, 35fsumsplit 11127 . 2  |-  ( ph  -> 
sum_ k  e.  ( M ... N ) A  =  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  sum_ k  e.  { N } A ) )
37 fsumm1.3 . . . . . 6  |-  ( k  =  N  ->  A  =  B )
3837eleq1d 2184 . . . . 5  |-  ( k  =  N  ->  ( A  e.  CC  <->  B  e.  CC ) )
3935ralrimiva 2480 . . . . 5  |-  ( ph  ->  A. k  e.  ( M ... N ) A  e.  CC )
40 eluzfz2 9763 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
411, 40syl 14 . . . . 5  |-  ( ph  ->  N  e.  ( M ... N ) )
4238, 39, 41rspcdva 2766 . . . 4  |-  ( ph  ->  B  e.  CC )
4337sumsn 11131 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  B  e.  CC )  ->  sum_ k  e.  { N } A  =  B )
441, 42, 43syl2anc 406 . . 3  |-  ( ph  -> 
sum_ k  e.  { N } A  =  B )
4544oveq2d 5756 . 2  |-  ( ph  ->  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  sum_ k  e.  { N } A
)  =  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  B ) )
4636, 45eqtrd 2148 1  |-  ( ph  -> 
sum_ k  e.  ( M ... N ) A  =  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463    u. cun 3037    i^i cin 3038   (/)c0 3331   {csn 3495   class class class wbr 3897   ` cfv 5091  (class class class)co 5740   CCcc 7582   1c1 7585    + caddc 7587    < clt 7764    - cmin 7897   ZZcz 9008   ZZ>=cuz 9278   ...cfz 9741   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:  fzosump1  11137  fsump1  11140  telfsumo  11186  fsumparts  11190  binom1dif  11207
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