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Theorem arisum2 12595
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 10179 . 2  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 nnm1nn0 10217 . . . . . 6  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
3 nn0uz 10476 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
42, 3syl6eleq 2494 . . . . 5  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  ( ZZ>= `  0
) )
5 elfznn0 11039 . . . . . . 7  |-  ( k  e.  ( 0 ... ( N  -  1 ) )  ->  k  e.  NN0 )
65adantl 453 . . . . . 6  |-  ( ( N  e.  NN  /\  k  e.  ( 0 ... ( N  - 
1 ) ) )  ->  k  e.  NN0 )
76nn0cnd 10232 . . . . 5  |-  ( ( N  e.  NN  /\  k  e.  ( 0 ... ( N  - 
1 ) ) )  ->  k  e.  CC )
8 id 20 . . . . 5  |-  ( k  =  0  ->  k  =  0 )
94, 7, 8fsum1p 12494 . . . 4  |-  ( N  e.  NN  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  ( 0  +  sum_ k  e.  ( ( 0  +  1 ) ... ( N  -  1 ) ) k ) )
10 1e0p1 10366 . . . . . . . . 9  |-  1  =  ( 0  +  1 )
1110oveq1i 6050 . . . . . . . 8  |-  ( 1 ... ( N  - 
1 ) )  =  ( ( 0  +  1 ) ... ( N  -  1 ) )
1211sumeq1i 12447 . . . . . . 7  |-  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k  =  sum_ k  e.  ( (
0  +  1 ) ... ( N  - 
1 ) ) k
1312oveq2i 6051 . . . . . 6  |-  ( 0  +  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k )  =  ( 0  +  sum_ k  e.  ( ( 0  +  1 ) ... ( N  -  1 ) ) k )
14 fzfid 11267 . . . . . . . 8  |-  ( N  e.  NN  ->  (
1 ... ( N  - 
1 ) )  e. 
Fin )
15 elfznn 11036 . . . . . . . . . 10  |-  ( k  e.  ( 1 ... ( N  -  1 ) )  ->  k  e.  NN )
1615adantl 453 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  k  e.  ( 1 ... ( N  - 
1 ) ) )  ->  k  e.  NN )
1716nncnd 9972 . . . . . . . 8  |-  ( ( N  e.  NN  /\  k  e.  ( 1 ... ( N  - 
1 ) ) )  ->  k  e.  CC )
1814, 17fsumcl 12482 . . . . . . 7  |-  ( N  e.  NN  ->  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k  e.  CC )
1918addid2d 9223 . . . . . 6  |-  ( N  e.  NN  ->  (
0  +  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k )  = 
sum_ k  e.  ( 1 ... ( N  -  1 ) ) k )
2013, 19syl5eqr 2450 . . . . 5  |-  ( N  e.  NN  ->  (
0  +  sum_ k  e.  ( ( 0  +  1 ) ... ( N  -  1 ) ) k )  = 
sum_ k  e.  ( 1 ... ( N  -  1 ) ) k )
21 arisum 12594 . . . . . . 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 9964 . . . . . . . . . . . . . . . 16  |-  ( N  e.  NN  ->  N  e.  CC )
2423mulid1d 9061 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN  ->  ( N  x.  1 )  =  N )
2524oveq2d 6056 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  (
2  x.  ( N  x.  1 ) )  =  ( 2  x.  N ) )
26232timesd 10166 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  (
2  x.  N )  =  ( N  +  N ) )
2725, 26eqtrd 2436 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  (
2  x.  ( N  x.  1 ) )  =  ( N  +  N ) )
2827oveq2d 6056 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  (
( N ^ 2 )  -  ( 2  x.  ( N  x.  1 ) ) )  =  ( ( N ^ 2 )  -  ( N  +  N
) ) )
2923sqcld 11476 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  ( N ^ 2 )  e.  CC )
3029, 23, 23subsub4d 9398 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  (
( ( N ^
2 )  -  N
)  -  N )  =  ( ( N ^ 2 )  -  ( N  +  N
) ) )
3128, 30eqtr4d 2439 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
( N ^ 2 )  -  ( 2  x.  ( N  x.  1 ) ) )  =  ( ( ( N ^ 2 )  -  N )  -  N ) )
32 sq1 11431 . . . . . . . . . . . 12  |-  ( 1 ^ 2 )  =  1
3332a1i 11 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
1 ^ 2 )  =  1 )
3431, 33oveq12d 6058 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
( ( N ^
2 )  -  (
2  x.  ( N  x.  1 ) ) )  +  ( 1 ^ 2 ) )  =  ( ( ( ( N ^ 2 )  -  N )  -  N )  +  1 ) )
35 ax-1cn 9004 . . . . . . . . . . 11  |-  1  e.  CC
36 binom2sub 11453 . . . . . . . . . . 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 9367 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
( N ^ 2 )  -  N )  e.  CC )
3935a1i 11 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  1  e.  CC )
4038, 23, 39subsubd 9395 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
( ( N ^
2 )  -  N
)  -  ( N  -  1 ) )  =  ( ( ( ( N ^ 2 )  -  N )  -  N )  +  1 ) )
4134, 37, 403eqtr4d 2446 . . . . . . . . 9  |-  ( N  e.  NN  ->  (
( N  -  1 ) ^ 2 )  =  ( ( ( N ^ 2 )  -  N )  -  ( N  -  1
) ) )
4241oveq1d 6055 . . . . . . . 8  |-  ( N  e.  NN  ->  (
( ( N  - 
1 ) ^ 2 )  +  ( N  -  1 ) )  =  ( ( ( ( N ^ 2 )  -  N )  -  ( N  - 
1 ) )  +  ( N  -  1 ) ) )
43 subcl 9261 . . . . . . . . . 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 9371 . . . . . . . 8  |-  ( N  e.  NN  ->  (
( ( ( N ^ 2 )  -  N )  -  ( N  -  1 ) )  +  ( N  -  1 ) )  =  ( ( N ^ 2 )  -  N ) )
4642, 45eqtrd 2436 . . . . . . 7  |-  ( N  e.  NN  ->  (
( ( N  - 
1 ) ^ 2 )  +  ( N  -  1 ) )  =  ( ( N ^ 2 )  -  N ) )
4746oveq1d 6055 . . . . . 6  |-  ( N  e.  NN  ->  (
( ( ( N  -  1 ) ^
2 )  +  ( N  -  1 ) )  /  2 )  =  ( ( ( N ^ 2 )  -  N )  / 
2 ) )
4822, 47eqtrd 2436 . . . . 5  |-  ( N  e.  NN  ->  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k  =  ( ( ( N ^
2 )  -  N
)  /  2 ) )
4920, 48eqtrd 2436 . . . 4  |-  ( N  e.  NN  ->  (
0  +  sum_ k  e.  ( ( 0  +  1 ) ... ( N  -  1 ) ) k )  =  ( ( ( N ^ 2 )  -  N )  /  2
) )
509, 49eqtrd 2436 . . 3  |-  ( N  e.  NN  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  ( ( ( N ^
2 )  -  N
)  /  2 ) )
51 oveq1 6047 . . . . . . . 8  |-  ( N  =  0  ->  ( N  -  1 )  =  ( 0  -  1 ) )
5251oveq2d 6056 . . . . . . 7  |-  ( N  =  0  ->  (
0 ... ( N  - 
1 ) )  =  ( 0 ... (
0  -  1 ) ) )
53 0re 9047 . . . . . . . . 9  |-  0  e.  RR
54 ltm1 9806 . . . . . . . . 9  |-  ( 0  e.  RR  ->  (
0  -  1 )  <  0 )
5553, 54ax-mp 8 . . . . . . . 8  |-  ( 0  -  1 )  <  0
56 0z 10249 . . . . . . . . 9  |-  0  e.  ZZ
57 peano2zm 10276 . . . . . . . . . 10  |-  ( 0  e.  ZZ  ->  (
0  -  1 )  e.  ZZ )
5856, 57ax-mp 8 . . . . . . . . 9  |-  ( 0  -  1 )  e.  ZZ
59 fzn 11027 . . . . . . . . 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 2452 . . . . . 6  |-  ( N  =  0  ->  (
0 ... ( N  - 
1 ) )  =  (/) )
6362sumeq1d 12450 . . . . 5  |-  ( N  =  0  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  sum_ k  e.  (/)  k )
64 sum0 12470 . . . . 5  |-  sum_ k  e.  (/)  k  =  0
6563, 64syl6eq 2452 . . . 4  |-  ( N  =  0  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  0 )
66 sq0i 11429 . . . . . . . 8  |-  ( N  =  0  ->  ( N ^ 2 )  =  0 )
67 id 20 . . . . . . . 8  |-  ( N  =  0  ->  N  =  0 )
6866, 67oveq12d 6058 . . . . . . 7  |-  ( N  =  0  ->  (
( N ^ 2 )  -  N )  =  ( 0  -  0 ) )
69 0cn 9040 . . . . . . . 8  |-  0  e.  CC
7069subidi 9327 . . . . . . 7  |-  ( 0  -  0 )  =  0
7168, 70syl6eq 2452 . . . . . 6  |-  ( N  =  0  ->  (
( N ^ 2 )  -  N )  =  0 )
7271oveq1d 6055 . . . . 5  |-  ( N  =  0  ->  (
( ( N ^
2 )  -  N
)  /  2 )  =  ( 0  / 
2 ) )
73 2cn 10026 . . . . . 6  |-  2  e.  CC
74 2ne0 10039 . . . . . 6  |-  2  =/=  0
7573, 74div0i 9704 . . . . 5  |-  ( 0  /  2 )  =  0
7672, 75syl6eq 2452 . . . 4  |-  ( N  =  0  ->  (
( ( N ^
2 )  -  N
)  /  2 )  =  0 )
7765, 76eqtr4d 2439 . . 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 1649    e. wcel 1721   (/)c0 3588   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    < clt 9076    - cmin 9247    / cdiv 9633   NNcn 9956   2c2 10005   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999   ^cexp 11337   sum_csu 12434
This theorem is referenced by:  birthdaylem3  20745
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435
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