MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fsumm1 Unicode version

Theorem fsumm1 12232
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 10254 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
31, 2syl 15 . . . . . 6  |-  ( ph  ->  N  e.  ZZ )
4 fzsn 10849 . . . . . 6  |-  ( N  e.  ZZ  ->  ( N ... N )  =  { N } )
53, 4syl 15 . . . . 5  |-  ( ph  ->  ( N ... N
)  =  { N } )
65ineq2d 3383 . . . 4  |-  ( ph  ->  ( ( M ... ( N  -  1
) )  i^i  ( N ... N ) )  =  ( ( M ... ( N  - 
1 ) )  i^i 
{ N } ) )
73zred 10133 . . . . . 6  |-  ( ph  ->  N  e.  RR )
87ltm1d 9705 . . . . 5  |-  ( ph  ->  ( N  -  1 )  <  N )
9 fzdisj 10833 . . . . 5  |-  ( ( N  -  1 )  <  N  ->  (
( M ... ( N  -  1 ) )  i^i  ( N ... N ) )  =  (/) )
108, 9syl 15 . . . 4  |-  ( ph  ->  ( ( M ... ( N  -  1
) )  i^i  ( N ... N ) )  =  (/) )
116, 10eqtr3d 2330 . . 3  |-  ( ph  ->  ( ( M ... ( N  -  1
) )  i^i  { N } )  =  (/) )
12 eluzel2 10251 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
131, 12syl 15 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
14 peano2zm 10078 . . . . . . . 8  |-  ( M  e.  ZZ  ->  ( M  -  1 )  e.  ZZ )
1513, 14syl 15 . . . . . . 7  |-  ( ph  ->  ( M  -  1 )  e.  ZZ )
1613zcnd 10134 . . . . . . . . . 10  |-  ( ph  ->  M  e.  CC )
17 ax-1cn 8811 . . . . . . . . . 10  |-  1  e.  CC
18 npcan 9076 . . . . . . . . . 10  |-  ( ( M  e.  CC  /\  1  e.  CC )  ->  ( ( M  - 
1 )  +  1 )  =  M )
1916, 17, 18sylancl 643 . . . . . . . . 9  |-  ( ph  ->  ( ( M  - 
1 )  +  1 )  =  M )
2019fveq2d 5545 . . . . . . . 8  |-  ( ph  ->  ( ZZ>= `  ( ( M  -  1 )  +  1 ) )  =  ( ZZ>= `  M
) )
211, 20eleqtrrd 2373 . . . . . . 7  |-  ( ph  ->  N  e.  ( ZZ>= `  ( ( M  - 
1 )  +  1 ) ) )
22 eluzp1m1 10267 . . . . . . 7  |-  ( ( ( M  -  1 )  e.  ZZ  /\  N  e.  ( ZZ>= `  ( ( M  - 
1 )  +  1 ) ) )  -> 
( N  -  1 )  e.  ( ZZ>= `  ( M  -  1
) ) )
2315, 21, 22syl2anc 642 . . . . . 6  |-  ( ph  ->  ( N  -  1 )  e.  ( ZZ>= `  ( M  -  1
) ) )
24 fzsuc2 10858 . . . . . 6  |-  ( ( M  e.  ZZ  /\  ( N  -  1
)  e.  ( ZZ>= `  ( M  -  1
) ) )  -> 
( M ... (
( N  -  1 )  +  1 ) )  =  ( ( M ... ( N  -  1 ) )  u.  { ( ( N  -  1 )  +  1 ) } ) )
2513, 23, 24syl2anc 642 . . . . 5  |-  ( ph  ->  ( M ... (
( N  -  1 )  +  1 ) )  =  ( ( M ... ( N  -  1 ) )  u.  { ( ( N  -  1 )  +  1 ) } ) )
263zcnd 10134 . . . . . . 7  |-  ( ph  ->  N  e.  CC )
27 npcan 9076 . . . . . . 7  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
2826, 17, 27sylancl 643 . . . . . 6  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
2928oveq2d 5890 . . . . 5  |-  ( ph  ->  ( M ... (
( N  -  1 )  +  1 ) )  =  ( M ... N ) )
3025, 29eqtr3d 2330 . . . 4  |-  ( ph  ->  ( ( M ... ( N  -  1
) )  u.  {
( ( N  - 
1 )  +  1 ) } )  =  ( M ... N
) )
3128sneqd 3666 . . . . 5  |-  ( ph  ->  { ( ( N  -  1 )  +  1 ) }  =  { N } )
3231uneq2d 3342 . . . 4  |-  ( ph  ->  ( ( M ... ( N  -  1
) )  u.  {
( ( N  - 
1 )  +  1 ) } )  =  ( ( M ... ( N  -  1
) )  u.  { N } ) )
3330, 32eqtr3d 2330 . . 3  |-  ( ph  ->  ( M ... N
)  =  ( ( M ... ( N  -  1 ) )  u.  { N }
) )
34 fzfid 11051 . . 3  |-  ( ph  ->  ( M ... N
)  e.  Fin )
35 fsumm1.2 . . 3  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  A  e.  CC )
3611, 33, 34, 35fsumsplit 12228 . 2  |-  ( ph  -> 
sum_ k  e.  ( M ... N ) A  =  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  sum_ k  e.  { N } A ) )
37 eluzfz2 10820 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
381, 37syl 15 . . . . 5  |-  ( ph  ->  N  e.  ( M ... N ) )
3935ralrimiva 2639 . . . . 5  |-  ( ph  ->  A. k  e.  ( M ... N ) A  e.  CC )
40 fsumm1.3 . . . . . . 7  |-  ( k  =  N  ->  A  =  B )
4140eleq1d 2362 . . . . . 6  |-  ( k  =  N  ->  ( A  e.  CC  <->  B  e.  CC ) )
4241rspcv 2893 . . . . 5  |-  ( N  e.  ( M ... N )  ->  ( A. k  e.  ( M ... N ) A  e.  CC  ->  B  e.  CC ) )
4338, 39, 42sylc 56 . . . 4  |-  ( ph  ->  B  e.  CC )
4440sumsn 12229 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  B  e.  CC )  ->  sum_ k  e.  { N } A  =  B )
451, 43, 44syl2anc 642 . . 3  |-  ( ph  -> 
sum_ k  e.  { N } A  =  B )
4645oveq2d 5890 . 2  |-  ( ph  ->  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  sum_ k  e.  { N } A
)  =  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  B ) )
4736, 46eqtrd 2328 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 358    = wceq 1632    e. wcel 1696   A.wral 2556    u. cun 3163    i^i cin 3164   (/)c0 3468   {csn 3653   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   1c1 8754    + caddc 8756    < clt 8883    - cmin 9053   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798   sum_csu 12174
This theorem is referenced by:  fzosump1  12233  fsump1  12235  fsumtscopo  12276  fsumparts  12280  binom1dif  12307  prmreclem4  12982  ovolicc2lem4  18895  dvfsumlem1  19389  abelthlem6  19828  log2ublem2  20259  harmonicbnd4  20320  ftalem1  20326  ftalem5  20330  chpp1  20409  1sgmprm  20454  chtublem  20466  logdivbnd  20721  pntrlog2bndlem1  20742  bpolysum  24860  bpolydiflem  24861  mettrifi  26576  stoweidlem17  27869
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175
  Copyright terms: Public domain W3C validator