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Theorem fsumm1 13331
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 10974 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
31, 2syl 16 . . . . . 6  |-  ( ph  ->  N  e.  ZZ )
4 fzsn 11610 . . . . . 6  |-  ( N  e.  ZZ  ->  ( N ... N )  =  { N } )
53, 4syl 16 . . . . 5  |-  ( ph  ->  ( N ... N
)  =  { N } )
65ineq2d 3653 . . . 4  |-  ( ph  ->  ( ( M ... ( N  -  1
) )  i^i  ( N ... N ) )  =  ( ( M ... ( N  - 
1 ) )  i^i 
{ N } ) )
73zred 10851 . . . . . 6  |-  ( ph  ->  N  e.  RR )
87ltm1d 10369 . . . . 5  |-  ( ph  ->  ( N  -  1 )  <  N )
9 fzdisj 11586 . . . . 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 2494 . . 3  |-  ( ph  ->  ( ( M ... ( N  -  1
) )  i^i  { N } )  =  (/) )
12 eluzel2 10970 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
131, 12syl 16 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
14 peano2zm 10792 . . . . . . . 8  |-  ( M  e.  ZZ  ->  ( M  -  1 )  e.  ZZ )
1513, 14syl 16 . . . . . . 7  |-  ( ph  ->  ( M  -  1 )  e.  ZZ )
1613zcnd 10852 . . . . . . . . . 10  |-  ( ph  ->  M  e.  CC )
17 ax-1cn 9444 . . . . . . . . . 10  |-  1  e.  CC
18 npcan 9723 . . . . . . . . . 10  |-  ( ( M  e.  CC  /\  1  e.  CC )  ->  ( ( M  - 
1 )  +  1 )  =  M )
1916, 17, 18sylancl 662 . . . . . . . . 9  |-  ( ph  ->  ( ( M  - 
1 )  +  1 )  =  M )
2019fveq2d 5796 . . . . . . . 8  |-  ( ph  ->  ( ZZ>= `  ( ( M  -  1 )  +  1 ) )  =  ( ZZ>= `  M
) )
211, 20eleqtrrd 2542 . . . . . . 7  |-  ( ph  ->  N  e.  ( ZZ>= `  ( ( M  - 
1 )  +  1 ) ) )
22 eluzp1m1 10988 . . . . . . 7  |-  ( ( ( M  -  1 )  e.  ZZ  /\  N  e.  ( ZZ>= `  ( ( M  - 
1 )  +  1 ) ) )  -> 
( N  -  1 )  e.  ( ZZ>= `  ( M  -  1
) ) )
2315, 21, 22syl2anc 661 . . . . . 6  |-  ( ph  ->  ( N  -  1 )  e.  ( ZZ>= `  ( M  -  1
) ) )
24 fzsuc2 11624 . . . . . 6  |-  ( ( M  e.  ZZ  /\  ( N  -  1
)  e.  ( ZZ>= `  ( M  -  1
) ) )  -> 
( M ... (
( N  -  1 )  +  1 ) )  =  ( ( M ... ( N  -  1 ) )  u.  { ( ( N  -  1 )  +  1 ) } ) )
2513, 23, 24syl2anc 661 . . . . 5  |-  ( ph  ->  ( M ... (
( N  -  1 )  +  1 ) )  =  ( ( M ... ( N  -  1 ) )  u.  { ( ( N  -  1 )  +  1 ) } ) )
263zcnd 10852 . . . . . . 7  |-  ( ph  ->  N  e.  CC )
27 npcan 9723 . . . . . . 7  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
2826, 17, 27sylancl 662 . . . . . 6  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
2928oveq2d 6209 . . . . 5  |-  ( ph  ->  ( M ... (
( N  -  1 )  +  1 ) )  =  ( M ... N ) )
3025, 29eqtr3d 2494 . . . 4  |-  ( ph  ->  ( ( M ... ( N  -  1
) )  u.  {
( ( N  - 
1 )  +  1 ) } )  =  ( M ... N
) )
3128sneqd 3990 . . . . 5  |-  ( ph  ->  { ( ( N  -  1 )  +  1 ) }  =  { N } )
3231uneq2d 3611 . . . 4  |-  ( ph  ->  ( ( M ... ( N  -  1
) )  u.  {
( ( N  - 
1 )  +  1 ) } )  =  ( ( M ... ( N  -  1
) )  u.  { N } ) )
3330, 32eqtr3d 2494 . . 3  |-  ( ph  ->  ( M ... N
)  =  ( ( M ... ( N  -  1 ) )  u.  { N }
) )
34 fzfid 11905 . . 3  |-  ( ph  ->  ( M ... N
)  e.  Fin )
35 fsumm1.2 . . 3  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  A  e.  CC )
3611, 33, 34, 35fsumsplit 13327 . 2  |-  ( ph  -> 
sum_ k  e.  ( M ... N ) A  =  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  sum_ k  e.  { N } A ) )
37 eluzfz2 11569 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
381, 37syl 16 . . . . 5  |-  ( ph  ->  N  e.  ( M ... N ) )
3935ralrimiva 2825 . . . . 5  |-  ( ph  ->  A. k  e.  ( M ... N ) A  e.  CC )
40 fsumm1.3 . . . . . . 7  |-  ( k  =  N  ->  A  =  B )
4140eleq1d 2520 . . . . . 6  |-  ( k  =  N  ->  ( A  e.  CC  <->  B  e.  CC ) )
4241rspcv 3168 . . . . 5  |-  ( N  e.  ( M ... N )  ->  ( A. k  e.  ( M ... N ) A  e.  CC  ->  B  e.  CC ) )
4338, 39, 42sylc 60 . . . 4  |-  ( ph  ->  B  e.  CC )
4440sumsn 13328 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  B  e.  CC )  ->  sum_ k  e.  { N } A  =  B )
451, 43, 44syl2anc 661 . . 3  |-  ( ph  -> 
sum_ k  e.  { N } A  =  B )
4645oveq2d 6209 . 2  |-  ( ph  ->  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  sum_ k  e.  { N } A
)  =  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  B ) )
4736, 46eqtrd 2492 1  |-  ( ph  -> 
sum_ k  e.  ( M ... N ) A  =  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795    u. cun 3427    i^i cin 3428   (/)c0 3738   {csn 3978   class class class wbr 4393   ` cfv 5519  (class class class)co 6193   CCcc 9384   1c1 9387    + caddc 9389    < clt 9522    - cmin 9699   ZZcz 10750   ZZ>=cuz 10965   ...cfz 11547   sum_csu 13274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-sup 7795  df-oi 7828  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-n0 10684  df-z 10751  df-uz 10966  df-rp 11096  df-fz 11548  df-fzo 11659  df-seq 11917  df-exp 11976  df-hash 12214  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-clim 13077  df-sum 13275
This theorem is referenced by:  fzosump1  13332  fsump1  13334  fsumtscopo  13376  fsumparts  13380  binom1dif  13407  prmreclem4  14091  ovolicc2lem4  21128  dvfsumlem1  21624  abelthlem6  22027  log2ublem2  22468  harmonicbnd4  22530  ftalem1  22536  ftalem5  22540  chpp1  22619  1sgmprm  22664  chtublem  22676  logdivbnd  22931  pntrlog2bndlem1  22952  bpolysum  28333  bpolydiflem  28334  mettrifi  28794  stoweidlem17  29953
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