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Theorem arisum2 12632
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 10215 . 2  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 nnm1nn0 10253 . . . . . 6  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
3 nn0uz 10512 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
42, 3syl6eleq 2525 . . . . 5  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  ( ZZ>= `  0
) )
5 elfznn0 11075 . . . . . . 7  |-  ( k  e.  ( 0 ... ( N  -  1 ) )  ->  k  e.  NN0 )
65adantl 453 . . . . . 6  |-  ( ( N  e.  NN  /\  k  e.  ( 0 ... ( N  - 
1 ) ) )  ->  k  e.  NN0 )
76nn0cnd 10268 . . . . 5  |-  ( ( N  e.  NN  /\  k  e.  ( 0 ... ( N  - 
1 ) ) )  ->  k  e.  CC )
8 id 20 . . . . 5  |-  ( k  =  0  ->  k  =  0 )
94, 7, 8fsum1p 12531 . . . 4  |-  ( N  e.  NN  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  ( 0  +  sum_ k  e.  ( ( 0  +  1 ) ... ( N  -  1 ) ) k ) )
10 1e0p1 10402 . . . . . . . . 9  |-  1  =  ( 0  +  1 )
1110oveq1i 6083 . . . . . . . 8  |-  ( 1 ... ( N  - 
1 ) )  =  ( ( 0  +  1 ) ... ( N  -  1 ) )
1211sumeq1i 12484 . . . . . . 7  |-  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k  =  sum_ k  e.  ( (
0  +  1 ) ... ( N  - 
1 ) ) k
1312oveq2i 6084 . . . . . 6  |-  ( 0  +  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k )  =  ( 0  +  sum_ k  e.  ( ( 0  +  1 ) ... ( N  -  1 ) ) k )
14 fzfid 11304 . . . . . . . 8  |-  ( N  e.  NN  ->  (
1 ... ( N  - 
1 ) )  e. 
Fin )
15 elfznn 11072 . . . . . . . . . 10  |-  ( k  e.  ( 1 ... ( N  -  1 ) )  ->  k  e.  NN )
1615adantl 453 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  k  e.  ( 1 ... ( N  - 
1 ) ) )  ->  k  e.  NN )
1716nncnd 10008 . . . . . . . 8  |-  ( ( N  e.  NN  /\  k  e.  ( 1 ... ( N  - 
1 ) ) )  ->  k  e.  CC )
1814, 17fsumcl 12519 . . . . . . 7  |-  ( N  e.  NN  ->  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k  e.  CC )
1918addid2d 9259 . . . . . 6  |-  ( N  e.  NN  ->  (
0  +  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k )  = 
sum_ k  e.  ( 1 ... ( N  -  1 ) ) k )
2013, 19syl5eqr 2481 . . . . 5  |-  ( N  e.  NN  ->  (
0  +  sum_ k  e.  ( ( 0  +  1 ) ... ( N  -  1 ) ) k )  = 
sum_ k  e.  ( 1 ... ( N  -  1 ) ) k )
21 arisum 12631 . . . . . . 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 10000 . . . . . . . . . . . . . . . 16  |-  ( N  e.  NN  ->  N  e.  CC )
2423mulid1d 9097 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN  ->  ( N  x.  1 )  =  N )
2524oveq2d 6089 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  (
2  x.  ( N  x.  1 ) )  =  ( 2  x.  N ) )
26232timesd 10202 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  (
2  x.  N )  =  ( N  +  N ) )
2725, 26eqtrd 2467 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  (
2  x.  ( N  x.  1 ) )  =  ( N  +  N ) )
2827oveq2d 6089 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  (
( N ^ 2 )  -  ( 2  x.  ( N  x.  1 ) ) )  =  ( ( N ^ 2 )  -  ( N  +  N
) ) )
2923sqcld 11513 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  ( N ^ 2 )  e.  CC )
3029, 23, 23subsub4d 9434 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  (
( ( N ^
2 )  -  N
)  -  N )  =  ( ( N ^ 2 )  -  ( N  +  N
) ) )
3128, 30eqtr4d 2470 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
( N ^ 2 )  -  ( 2  x.  ( N  x.  1 ) ) )  =  ( ( ( N ^ 2 )  -  N )  -  N ) )
32 sq1 11468 . . . . . . . . . . . 12  |-  ( 1 ^ 2 )  =  1
3332a1i 11 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
1 ^ 2 )  =  1 )
3431, 33oveq12d 6091 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
( ( N ^
2 )  -  (
2  x.  ( N  x.  1 ) ) )  +  ( 1 ^ 2 ) )  =  ( ( ( ( N ^ 2 )  -  N )  -  N )  +  1 ) )
35 ax-1cn 9040 . . . . . . . . . . 11  |-  1  e.  CC
36 binom2sub 11490 . . . . . . . . . . 11  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 ) ^ 2 )  =  ( ( ( N ^ 2 )  -  ( 2  x.  ( N  x.  1 ) ) )  +  ( 1 ^ 2 ) ) )
3723, 35, 36sylancl 644 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
( N  -  1 ) ^ 2 )  =  ( ( ( N ^ 2 )  -  ( 2  x.  ( N  x.  1 ) ) )  +  ( 1 ^ 2 ) ) )
3829, 23subcld 9403 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
( N ^ 2 )  -  N )  e.  CC )
3935a1i 11 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  1  e.  CC )
4038, 23, 39subsubd 9431 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
( ( N ^
2 )  -  N
)  -  ( N  -  1 ) )  =  ( ( ( ( N ^ 2 )  -  N )  -  N )  +  1 ) )
4134, 37, 403eqtr4d 2477 . . . . . . . . 9  |-  ( N  e.  NN  ->  (
( N  -  1 ) ^ 2 )  =  ( ( ( N ^ 2 )  -  N )  -  ( N  -  1
) ) )
4241oveq1d 6088 . . . . . . . 8  |-  ( N  e.  NN  ->  (
( ( N  - 
1 ) ^ 2 )  +  ( N  -  1 ) )  =  ( ( ( ( N ^ 2 )  -  N )  -  ( N  - 
1 ) )  +  ( N  -  1 ) ) )
43 subcl 9297 . . . . . . . . . 10  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( N  -  1 )  e.  CC )
4423, 35, 43sylancl 644 . . . . . . . . 9  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  CC )
4538, 44npcand 9407 . . . . . . . 8  |-  ( N  e.  NN  ->  (
( ( ( N ^ 2 )  -  N )  -  ( N  -  1 ) )  +  ( N  -  1 ) )  =  ( ( N ^ 2 )  -  N ) )
4642, 45eqtrd 2467 . . . . . . 7  |-  ( N  e.  NN  ->  (
( ( N  - 
1 ) ^ 2 )  +  ( N  -  1 ) )  =  ( ( N ^ 2 )  -  N ) )
4746oveq1d 6088 . . . . . 6  |-  ( N  e.  NN  ->  (
( ( ( N  -  1 ) ^
2 )  +  ( N  -  1 ) )  /  2 )  =  ( ( ( N ^ 2 )  -  N )  / 
2 ) )
4822, 47eqtrd 2467 . . . . 5  |-  ( N  e.  NN  ->  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k  =  ( ( ( N ^
2 )  -  N
)  /  2 ) )
4920, 48eqtrd 2467 . . . 4  |-  ( N  e.  NN  ->  (
0  +  sum_ k  e.  ( ( 0  +  1 ) ... ( N  -  1 ) ) k )  =  ( ( ( N ^ 2 )  -  N )  /  2
) )
509, 49eqtrd 2467 . . 3  |-  ( N  e.  NN  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  ( ( ( N ^
2 )  -  N
)  /  2 ) )
51 oveq1 6080 . . . . . . . 8  |-  ( N  =  0  ->  ( N  -  1 )  =  ( 0  -  1 ) )
5251oveq2d 6089 . . . . . . 7  |-  ( N  =  0  ->  (
0 ... ( N  - 
1 ) )  =  ( 0 ... (
0  -  1 ) ) )
53 0re 9083 . . . . . . . . 9  |-  0  e.  RR
54 ltm1 9842 . . . . . . . . 9  |-  ( 0  e.  RR  ->  (
0  -  1 )  <  0 )
5553, 54ax-mp 8 . . . . . . . 8  |-  ( 0  -  1 )  <  0
56 0z 10285 . . . . . . . . 9  |-  0  e.  ZZ
57 peano2zm 10312 . . . . . . . . . 10  |-  ( 0  e.  ZZ  ->  (
0  -  1 )  e.  ZZ )
5856, 57ax-mp 8 . . . . . . . . 9  |-  ( 0  -  1 )  e.  ZZ
59 fzn 11063 . . . . . . . . 9  |-  ( ( 0  e.  ZZ  /\  ( 0  -  1 )  e.  ZZ )  ->  ( ( 0  -  1 )  <  0  <->  ( 0 ... ( 0  -  1 ) )  =  (/) ) )
6056, 58, 59mp2an 654 . . . . . . . 8  |-  ( ( 0  -  1 )  <  0  <->  ( 0 ... ( 0  -  1 ) )  =  (/) )
6155, 60mpbi 200 . . . . . . 7  |-  ( 0 ... ( 0  -  1 ) )  =  (/)
6252, 61syl6eq 2483 . . . . . 6  |-  ( N  =  0  ->  (
0 ... ( N  - 
1 ) )  =  (/) )
6362sumeq1d 12487 . . . . 5  |-  ( N  =  0  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  sum_ k  e.  (/)  k )
64 sum0 12507 . . . . 5  |-  sum_ k  e.  (/)  k  =  0
6563, 64syl6eq 2483 . . . 4  |-  ( N  =  0  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  0 )
66 sq0i 11466 . . . . . . . 8  |-  ( N  =  0  ->  ( N ^ 2 )  =  0 )
67 id 20 . . . . . . . 8  |-  ( N  =  0  ->  N  =  0 )
6866, 67oveq12d 6091 . . . . . . 7  |-  ( N  =  0  ->  (
( N ^ 2 )  -  N )  =  ( 0  -  0 ) )
69 0cn 9076 . . . . . . . 8  |-  0  e.  CC
7069subidi 9363 . . . . . . 7  |-  ( 0  -  0 )  =  0
7168, 70syl6eq 2483 . . . . . 6  |-  ( N  =  0  ->  (
( N ^ 2 )  -  N )  =  0 )
7271oveq1d 6088 . . . . 5  |-  ( N  =  0  ->  (
( ( N ^
2 )  -  N
)  /  2 )  =  ( 0  / 
2 ) )
73 2cn 10062 . . . . . 6  |-  2  e.  CC
74 2ne0 10075 . . . . . 6  |-  2  =/=  0
7573, 74div0i 9740 . . . . 5  |-  ( 0  /  2 )  =  0
7672, 75syl6eq 2483 . . . 4  |-  ( N  =  0  ->  (
( ( N ^
2 )  -  N
)  /  2 )  =  0 )
7765, 76eqtr4d 2470 . . 3  |-  ( N  =  0  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  ( ( ( N ^
2 )  -  N
)  /  2 ) )
7850, 77jaoi 369 . 2  |-  ( ( N  e.  NN  \/  N  =  0 )  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  ( ( ( N ^ 2 )  -  N )  /  2 ) )
791, 78sylbi 188 1  |-  ( N  e.  NN0  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  ( ( ( N ^
2 )  -  N
)  /  2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   (/)c0 3620   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    < clt 9112    - cmin 9283    / cdiv 9669   NNcn 9992   2c2 10041   NN0cn0 10213   ZZcz 10274   ZZ>=cuz 10480   ...cfz 11035   ^cexp 11374   sum_csu 12471
This theorem is referenced by:  birthdaylem3  20784
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-fzo 11128  df-seq 11316  df-exp 11375  df-fac 11559  df-bc 11586  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-sum 12472
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