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Theorem fsumm1 12538
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 10497 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
31, 2syl 16 . . . . . 6  |-  ( ph  ->  N  e.  ZZ )
4 fzsn 11095 . . . . . 6  |-  ( N  e.  ZZ  ->  ( N ... N )  =  { N } )
53, 4syl 16 . . . . 5  |-  ( ph  ->  ( N ... N
)  =  { N } )
65ineq2d 3543 . . . 4  |-  ( ph  ->  ( ( M ... ( N  -  1
) )  i^i  ( N ... N ) )  =  ( ( M ... ( N  - 
1 ) )  i^i 
{ N } ) )
73zred 10376 . . . . . 6  |-  ( ph  ->  N  e.  RR )
87ltm1d 9944 . . . . 5  |-  ( ph  ->  ( N  -  1 )  <  N )
9 fzdisj 11079 . . . . 5  |-  ( ( N  -  1 )  <  N  ->  (
( M ... ( N  -  1 ) )  i^i  ( N ... N ) )  =  (/) )
108, 9syl 16 . . . 4  |-  ( ph  ->  ( ( M ... ( N  -  1
) )  i^i  ( N ... N ) )  =  (/) )
116, 10eqtr3d 2471 . . 3  |-  ( ph  ->  ( ( M ... ( N  -  1
) )  i^i  { N } )  =  (/) )
12 eluzel2 10494 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
131, 12syl 16 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
14 peano2zm 10321 . . . . . . . 8  |-  ( M  e.  ZZ  ->  ( M  -  1 )  e.  ZZ )
1513, 14syl 16 . . . . . . 7  |-  ( ph  ->  ( M  -  1 )  e.  ZZ )
1613zcnd 10377 . . . . . . . . . 10  |-  ( ph  ->  M  e.  CC )
17 ax-1cn 9049 . . . . . . . . . 10  |-  1  e.  CC
18 npcan 9315 . . . . . . . . . 10  |-  ( ( M  e.  CC  /\  1  e.  CC )  ->  ( ( M  - 
1 )  +  1 )  =  M )
1916, 17, 18sylancl 645 . . . . . . . . 9  |-  ( ph  ->  ( ( M  - 
1 )  +  1 )  =  M )
2019fveq2d 5733 . . . . . . . 8  |-  ( ph  ->  ( ZZ>= `  ( ( M  -  1 )  +  1 ) )  =  ( ZZ>= `  M
) )
211, 20eleqtrrd 2514 . . . . . . 7  |-  ( ph  ->  N  e.  ( ZZ>= `  ( ( M  - 
1 )  +  1 ) ) )
22 eluzp1m1 10510 . . . . . . 7  |-  ( ( ( M  -  1 )  e.  ZZ  /\  N  e.  ( ZZ>= `  ( ( M  - 
1 )  +  1 ) ) )  -> 
( N  -  1 )  e.  ( ZZ>= `  ( M  -  1
) ) )
2315, 21, 22syl2anc 644 . . . . . 6  |-  ( ph  ->  ( N  -  1 )  e.  ( ZZ>= `  ( M  -  1
) ) )
24 fzsuc2 11105 . . . . . 6  |-  ( ( M  e.  ZZ  /\  ( N  -  1
)  e.  ( ZZ>= `  ( M  -  1
) ) )  -> 
( M ... (
( N  -  1 )  +  1 ) )  =  ( ( M ... ( N  -  1 ) )  u.  { ( ( N  -  1 )  +  1 ) } ) )
2513, 23, 24syl2anc 644 . . . . 5  |-  ( ph  ->  ( M ... (
( N  -  1 )  +  1 ) )  =  ( ( M ... ( N  -  1 ) )  u.  { ( ( N  -  1 )  +  1 ) } ) )
263zcnd 10377 . . . . . . 7  |-  ( ph  ->  N  e.  CC )
27 npcan 9315 . . . . . . 7  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
2826, 17, 27sylancl 645 . . . . . 6  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
2928oveq2d 6098 . . . . 5  |-  ( ph  ->  ( M ... (
( N  -  1 )  +  1 ) )  =  ( M ... N ) )
3025, 29eqtr3d 2471 . . . 4  |-  ( ph  ->  ( ( M ... ( N  -  1
) )  u.  {
( ( N  - 
1 )  +  1 ) } )  =  ( M ... N
) )
3128sneqd 3828 . . . . 5  |-  ( ph  ->  { ( ( N  -  1 )  +  1 ) }  =  { N } )
3231uneq2d 3502 . . . 4  |-  ( ph  ->  ( ( M ... ( N  -  1
) )  u.  {
( ( N  - 
1 )  +  1 ) } )  =  ( ( M ... ( N  -  1
) )  u.  { N } ) )
3330, 32eqtr3d 2471 . . 3  |-  ( ph  ->  ( M ... N
)  =  ( ( M ... ( N  -  1 ) )  u.  { N }
) )
34 fzfid 11313 . . 3  |-  ( ph  ->  ( M ... N
)  e.  Fin )
35 fsumm1.2 . . 3  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  A  e.  CC )
3611, 33, 34, 35fsumsplit 12534 . 2  |-  ( ph  -> 
sum_ k  e.  ( M ... N ) A  =  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  sum_ k  e.  { N } A ) )
37 eluzfz2 11066 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
381, 37syl 16 . . . . 5  |-  ( ph  ->  N  e.  ( M ... N ) )
3935ralrimiva 2790 . . . . 5  |-  ( ph  ->  A. k  e.  ( M ... N ) A  e.  CC )
40 fsumm1.3 . . . . . . 7  |-  ( k  =  N  ->  A  =  B )
4140eleq1d 2503 . . . . . 6  |-  ( k  =  N  ->  ( A  e.  CC  <->  B  e.  CC ) )
4241rspcv 3049 . . . . 5  |-  ( N  e.  ( M ... N )  ->  ( A. k  e.  ( M ... N ) A  e.  CC  ->  B  e.  CC ) )
4338, 39, 42sylc 59 . . . 4  |-  ( ph  ->  B  e.  CC )
4440sumsn 12535 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  B  e.  CC )  ->  sum_ k  e.  { N } A  =  B )
451, 43, 44syl2anc 644 . . 3  |-  ( ph  -> 
sum_ k  e.  { N } A  =  B )
4645oveq2d 6098 . 2  |-  ( ph  ->  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  sum_ k  e.  { N } A
)  =  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  B ) )
4736, 46eqtrd 2469 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 360    = wceq 1653    e. wcel 1726   A.wral 2706    u. cun 3319    i^i cin 3320   (/)c0 3629   {csn 3815   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   CCcc 8989   1c1 8992    + caddc 8994    < clt 9121    - cmin 9292   ZZcz 10283   ZZ>=cuz 10489   ...cfz 11044   sum_csu 12480
This theorem is referenced by:  fzosump1  12539  fsump1  12541  fsumtscopo  12582  fsumparts  12586  binom1dif  12613  prmreclem4  13288  ovolicc2lem4  19417  dvfsumlem1  19911  abelthlem6  20353  log2ublem2  20788  harmonicbnd4  20850  ftalem1  20856  ftalem5  20860  chpp1  20939  1sgmprm  20984  chtublem  20996  logdivbnd  21251  pntrlog2bndlem1  21272  bpolysum  26100  bpolydiflem  26101  mettrifi  26464  stoweidlem17  27743
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-inf2 7597  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-oadd 6729  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-sup 7447  df-oi 7480  df-card 7827  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-n0 10223  df-z 10284  df-uz 10490  df-rp 10614  df-fz 11045  df-fzo 11137  df-seq 11325  df-exp 11384  df-hash 11620  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042  df-clim 12283  df-sum 12481
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