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Theorem arisum2 13651
Description: Arithmetic series sum of the first  N nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)
Assertion
Ref Expression
arisum2  |-  ( N  e.  NN0  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  ( ( ( N ^
2 )  -  N
)  /  2 ) )
Distinct variable group:    k, N

Proof of Theorem arisum2
StepHypRef Expression
1 elnn0 10809 . 2  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 nnm1nn0 10849 . . . . . 6  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
3 nn0uz 11128 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
42, 3syl6eleq 2565 . . . . 5  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  ( ZZ>= `  0
) )
5 elfznn0 11782 . . . . . . 7  |-  ( k  e.  ( 0 ... ( N  -  1 ) )  ->  k  e.  NN0 )
65adantl 466 . . . . . 6  |-  ( ( N  e.  NN  /\  k  e.  ( 0 ... ( N  - 
1 ) ) )  ->  k  e.  NN0 )
76nn0cnd 10866 . . . . 5  |-  ( ( N  e.  NN  /\  k  e.  ( 0 ... ( N  - 
1 ) ) )  ->  k  e.  CC )
8 id 22 . . . . 5  |-  ( k  =  0  ->  k  =  0 )
94, 7, 8fsum1p 13547 . . . 4  |-  ( N  e.  NN  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  ( 0  +  sum_ k  e.  ( ( 0  +  1 ) ... ( N  -  1 ) ) k ) )
10 1e0p1 11016 . . . . . . . . 9  |-  1  =  ( 0  +  1 )
1110oveq1i 6305 . . . . . . . 8  |-  ( 1 ... ( N  - 
1 ) )  =  ( ( 0  +  1 ) ... ( N  -  1 ) )
1211sumeq1i 13499 . . . . . . 7  |-  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k  =  sum_ k  e.  ( (
0  +  1 ) ... ( N  - 
1 ) ) k
1312oveq2i 6306 . . . . . 6  |-  ( 0  +  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k )  =  ( 0  +  sum_ k  e.  ( ( 0  +  1 ) ... ( N  -  1 ) ) k )
14 fzfid 12063 . . . . . . . 8  |-  ( N  e.  NN  ->  (
1 ... ( N  - 
1 ) )  e. 
Fin )
15 elfznn 11726 . . . . . . . . . 10  |-  ( k  e.  ( 1 ... ( N  -  1 ) )  ->  k  e.  NN )
1615adantl 466 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  k  e.  ( 1 ... ( N  - 
1 ) ) )  ->  k  e.  NN )
1716nncnd 10564 . . . . . . . 8  |-  ( ( N  e.  NN  /\  k  e.  ( 1 ... ( N  - 
1 ) ) )  ->  k  e.  CC )
1814, 17fsumcl 13534 . . . . . . 7  |-  ( N  e.  NN  ->  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k  e.  CC )
1918addid2d 9792 . . . . . 6  |-  ( N  e.  NN  ->  (
0  +  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k )  = 
sum_ k  e.  ( 1 ... ( N  -  1 ) ) k )
2013, 19syl5eqr 2522 . . . . 5  |-  ( N  e.  NN  ->  (
0  +  sum_ k  e.  ( ( 0  +  1 ) ... ( N  -  1 ) ) k )  = 
sum_ k  e.  ( 1 ... ( N  -  1 ) ) k )
21 arisum 13650 . . . . . . 7  |-  ( ( N  -  1 )  e.  NN0  ->  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k  =  ( ( ( ( N  -  1 ) ^
2 )  +  ( N  -  1 ) )  /  2 ) )
222, 21syl 16 . . . . . 6  |-  ( N  e.  NN  ->  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k  =  ( ( ( ( N  -  1 ) ^
2 )  +  ( N  -  1 ) )  /  2 ) )
23 nncn 10556 . . . . . . . . . . . . . . . 16  |-  ( N  e.  NN  ->  N  e.  CC )
2423mulid1d 9625 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN  ->  ( N  x.  1 )  =  N )
2524oveq2d 6311 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  (
2  x.  ( N  x.  1 ) )  =  ( 2  x.  N ) )
26232timesd 10793 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  (
2  x.  N )  =  ( N  +  N ) )
2725, 26eqtrd 2508 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  (
2  x.  ( N  x.  1 ) )  =  ( N  +  N ) )
2827oveq2d 6311 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  (
( N ^ 2 )  -  ( 2  x.  ( N  x.  1 ) ) )  =  ( ( N ^ 2 )  -  ( N  +  N
) ) )
2923sqcld 12288 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  ( N ^ 2 )  e.  CC )
3029, 23, 23subsub4d 9973 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  (
( ( N ^
2 )  -  N
)  -  N )  =  ( ( N ^ 2 )  -  ( N  +  N
) ) )
3128, 30eqtr4d 2511 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
( N ^ 2 )  -  ( 2  x.  ( N  x.  1 ) ) )  =  ( ( ( N ^ 2 )  -  N )  -  N ) )
32 sq1 12242 . . . . . . . . . . . 12  |-  ( 1 ^ 2 )  =  1
3332a1i 11 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
1 ^ 2 )  =  1 )
3431, 33oveq12d 6313 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
( ( N ^
2 )  -  (
2  x.  ( N  x.  1 ) ) )  +  ( 1 ^ 2 ) )  =  ( ( ( ( N ^ 2 )  -  N )  -  N )  +  1 ) )
35 ax-1cn 9562 . . . . . . . . . . 11  |-  1  e.  CC
36 binom2sub 12265 . . . . . . . . . . 11  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 ) ^ 2 )  =  ( ( ( N ^ 2 )  -  ( 2  x.  ( N  x.  1 ) ) )  +  ( 1 ^ 2 ) ) )
3723, 35, 36sylancl 662 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
( N  -  1 ) ^ 2 )  =  ( ( ( N ^ 2 )  -  ( 2  x.  ( N  x.  1 ) ) )  +  ( 1 ^ 2 ) ) )
3829, 23subcld 9942 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
( N ^ 2 )  -  N )  e.  CC )
3935a1i 11 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  1  e.  CC )
4038, 23, 39subsubd 9970 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
( ( N ^
2 )  -  N
)  -  ( N  -  1 ) )  =  ( ( ( ( N ^ 2 )  -  N )  -  N )  +  1 ) )
4134, 37, 403eqtr4d 2518 . . . . . . . . 9  |-  ( N  e.  NN  ->  (
( N  -  1 ) ^ 2 )  =  ( ( ( N ^ 2 )  -  N )  -  ( N  -  1
) ) )
4241oveq1d 6310 . . . . . . . 8  |-  ( N  e.  NN  ->  (
( ( N  - 
1 ) ^ 2 )  +  ( N  -  1 ) )  =  ( ( ( ( N ^ 2 )  -  N )  -  ( N  - 
1 ) )  +  ( N  -  1 ) ) )
43 subcl 9831 . . . . . . . . . 10  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( N  -  1 )  e.  CC )
4423, 35, 43sylancl 662 . . . . . . . . 9  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  CC )
4538, 44npcand 9946 . . . . . . . 8  |-  ( N  e.  NN  ->  (
( ( ( N ^ 2 )  -  N )  -  ( N  -  1 ) )  +  ( N  -  1 ) )  =  ( ( N ^ 2 )  -  N ) )
4642, 45eqtrd 2508 . . . . . . 7  |-  ( N  e.  NN  ->  (
( ( N  - 
1 ) ^ 2 )  +  ( N  -  1 ) )  =  ( ( N ^ 2 )  -  N ) )
4746oveq1d 6310 . . . . . 6  |-  ( N  e.  NN  ->  (
( ( ( N  -  1 ) ^
2 )  +  ( N  -  1 ) )  /  2 )  =  ( ( ( N ^ 2 )  -  N )  / 
2 ) )
4822, 47eqtrd 2508 . . . . 5  |-  ( N  e.  NN  ->  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k  =  ( ( ( N ^
2 )  -  N
)  /  2 ) )
4920, 48eqtrd 2508 . . . 4  |-  ( N  e.  NN  ->  (
0  +  sum_ k  e.  ( ( 0  +  1 ) ... ( N  -  1 ) ) k )  =  ( ( ( N ^ 2 )  -  N )  /  2
) )
509, 49eqtrd 2508 . . 3  |-  ( N  e.  NN  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  ( ( ( N ^
2 )  -  N
)  /  2 ) )
51 oveq1 6302 . . . . . . . 8  |-  ( N  =  0  ->  ( N  -  1 )  =  ( 0  -  1 ) )
5251oveq2d 6311 . . . . . . 7  |-  ( N  =  0  ->  (
0 ... ( N  - 
1 ) )  =  ( 0 ... (
0  -  1 ) ) )
53 0re 9608 . . . . . . . . 9  |-  0  e.  RR
54 ltm1 10394 . . . . . . . . 9  |-  ( 0  e.  RR  ->  (
0  -  1 )  <  0 )
5553, 54ax-mp 5 . . . . . . . 8  |-  ( 0  -  1 )  <  0
56 0z 10887 . . . . . . . . 9  |-  0  e.  ZZ
57 peano2zm 10918 . . . . . . . . . 10  |-  ( 0  e.  ZZ  ->  (
0  -  1 )  e.  ZZ )
5856, 57ax-mp 5 . . . . . . . . 9  |-  ( 0  -  1 )  e.  ZZ
59 fzn 11714 . . . . . . . . 9  |-  ( ( 0  e.  ZZ  /\  ( 0  -  1 )  e.  ZZ )  ->  ( ( 0  -  1 )  <  0  <->  ( 0 ... ( 0  -  1 ) )  =  (/) ) )
6056, 58, 59mp2an 672 . . . . . . . 8  |-  ( ( 0  -  1 )  <  0  <->  ( 0 ... ( 0  -  1 ) )  =  (/) )
6155, 60mpbi 208 . . . . . . 7  |-  ( 0 ... ( 0  -  1 ) )  =  (/)
6252, 61syl6eq 2524 . . . . . 6  |-  ( N  =  0  ->  (
0 ... ( N  - 
1 ) )  =  (/) )
6362sumeq1d 13502 . . . . 5  |-  ( N  =  0  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  sum_ k  e.  (/)  k )
64 sum0 13522 . . . . 5  |-  sum_ k  e.  (/)  k  =  0
6563, 64syl6eq 2524 . . . 4  |-  ( N  =  0  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  0 )
66 sq0i 12240 . . . . . . . 8  |-  ( N  =  0  ->  ( N ^ 2 )  =  0 )
67 id 22 . . . . . . . 8  |-  ( N  =  0  ->  N  =  0 )
6866, 67oveq12d 6313 . . . . . . 7  |-  ( N  =  0  ->  (
( N ^ 2 )  -  N )  =  ( 0  -  0 ) )
69 0m0e0 10657 . . . . . . 7  |-  ( 0  -  0 )  =  0
7068, 69syl6eq 2524 . . . . . 6  |-  ( N  =  0  ->  (
( N ^ 2 )  -  N )  =  0 )
7170oveq1d 6310 . . . . 5  |-  ( N  =  0  ->  (
( ( N ^
2 )  -  N
)  /  2 )  =  ( 0  / 
2 ) )
72 2cn 10618 . . . . . 6  |-  2  e.  CC
73 2ne0 10640 . . . . . 6  |-  2  =/=  0
7472, 73div0i 10290 . . . . 5  |-  ( 0  /  2 )  =  0
7571, 74syl6eq 2524 . . . 4  |-  ( N  =  0  ->  (
( ( N ^
2 )  -  N
)  /  2 )  =  0 )
7665, 75eqtr4d 2511 . . 3  |-  ( N  =  0  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  ( ( ( N ^
2 )  -  N
)  /  2 ) )
7750, 76jaoi 379 . 2  |-  ( ( N  e.  NN  \/  N  =  0 )  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  ( ( ( N ^ 2 )  -  N )  /  2 ) )
781, 77sylbi 195 1  |-  ( N  e.  NN0  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  ( ( ( N ^
2 )  -  N
)  /  2 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   (/)c0 3790   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   CCcc 9502   RRcr 9503   0cc0 9504   1c1 9505    + caddc 9507    x. cmul 9509    < clt 9640    - cmin 9817    / cdiv 10218   NNcn 10548   2c2 10597   NN0cn0 10807   ZZcz 10876   ZZ>=cuz 11094   ...cfz 11684   ^cexp 12146   sum_csu 13487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fz 11685  df-fzo 11805  df-seq 12088  df-exp 12147  df-fac 12334  df-bc 12361  df-hash 12386  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-clim 13290  df-sum 13488
This theorem is referenced by:  birthdaylem3  23147
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