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Theorem arisum2 11538
Description: Arithmetic series sum of the first  N nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.) (Proof shortened by AV, 2-Aug-2021.)
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 9207 . 2  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 nnm1nn0 9246 . . . . . 6  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
3 nn0uz 9591 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
42, 3eleqtrdi 2282 . . . . 5  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  ( ZZ>= `  0
) )
5 elfznn0 10143 . . . . . . 7  |-  ( k  e.  ( 0 ... ( N  -  1 ) )  ->  k  e.  NN0 )
65adantl 277 . . . . . 6  |-  ( ( N  e.  NN  /\  k  e.  ( 0 ... ( N  - 
1 ) ) )  ->  k  e.  NN0 )
76nn0cnd 9260 . . . . 5  |-  ( ( N  e.  NN  /\  k  e.  ( 0 ... ( N  - 
1 ) ) )  ->  k  e.  CC )
8 id 19 . . . . 5  |-  ( k  =  0  ->  k  =  0 )
94, 7, 8fsum1p 11457 . . . 4  |-  ( N  e.  NN  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  ( 0  +  sum_ k  e.  ( ( 0  +  1 ) ... ( N  -  1 ) ) k ) )
10 1e0p1 9454 . . . . . . . . 9  |-  1  =  ( 0  +  1 )
1110oveq1i 5905 . . . . . . . 8  |-  ( 1 ... ( N  - 
1 ) )  =  ( ( 0  +  1 ) ... ( N  -  1 ) )
1211sumeq1i 11402 . . . . . . 7  |-  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k  =  sum_ k  e.  ( (
0  +  1 ) ... ( N  - 
1 ) ) k
1312oveq2i 5906 . . . . . 6  |-  ( 0  +  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k )  =  ( 0  +  sum_ k  e.  ( ( 0  +  1 ) ... ( N  -  1 ) ) k )
14 1zzd 9309 . . . . . . . . 9  |-  ( N  e.  NN  ->  1  e.  ZZ )
152nn0zd 9402 . . . . . . . . 9  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  ZZ )
1614, 15fzfigd 10461 . . . . . . . 8  |-  ( N  e.  NN  ->  (
1 ... ( N  - 
1 ) )  e. 
Fin )
17 elfznn 10083 . . . . . . . . . 10  |-  ( k  e.  ( 1 ... ( N  -  1 ) )  ->  k  e.  NN )
1817adantl 277 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  k  e.  ( 1 ... ( N  - 
1 ) ) )  ->  k  e.  NN )
1918nncnd 8962 . . . . . . . 8  |-  ( ( N  e.  NN  /\  k  e.  ( 1 ... ( N  - 
1 ) ) )  ->  k  e.  CC )
2016, 19fsumcl 11439 . . . . . . 7  |-  ( N  e.  NN  ->  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k  e.  CC )
2120addlidd 8136 . . . . . 6  |-  ( N  e.  NN  ->  (
0  +  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k )  = 
sum_ k  e.  ( 1 ... ( N  -  1 ) ) k )
2213, 21eqtr3id 2236 . . . . 5  |-  ( N  e.  NN  ->  (
0  +  sum_ k  e.  ( ( 0  +  1 ) ... ( N  -  1 ) ) k )  = 
sum_ k  e.  ( 1 ... ( N  -  1 ) ) k )
23 arisum 11537 . . . . . . 7  |-  ( ( N  -  1 )  e.  NN0  ->  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k  =  ( ( ( ( N  -  1 ) ^
2 )  +  ( N  -  1 ) )  /  2 ) )
242, 23syl 14 . . . . . 6  |-  ( N  e.  NN  ->  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k  =  ( ( ( ( N  -  1 ) ^
2 )  +  ( N  -  1 ) )  /  2 ) )
25 nncn 8956 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  N  e.  CC )
26252timesd 9190 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  (
2  x.  N )  =  ( N  +  N ) )
2726oveq2d 5911 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  (
( N ^ 2 )  -  ( 2  x.  N ) )  =  ( ( N ^ 2 )  -  ( N  +  N
) ) )
2825sqcld 10682 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  ( N ^ 2 )  e.  CC )
2928, 25, 25subsub4d 8328 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  (
( ( N ^
2 )  -  N
)  -  N )  =  ( ( N ^ 2 )  -  ( N  +  N
) ) )
3027, 29eqtr4d 2225 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
( N ^ 2 )  -  ( 2  x.  N ) )  =  ( ( ( N ^ 2 )  -  N )  -  N ) )
3130oveq1d 5910 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
( ( N ^
2 )  -  (
2  x.  N ) )  +  1 )  =  ( ( ( ( N ^ 2 )  -  N )  -  N )  +  1 ) )
32 binom2sub1 10665 . . . . . . . . . . 11  |-  ( N  e.  CC  ->  (
( N  -  1 ) ^ 2 )  =  ( ( ( N ^ 2 )  -  ( 2  x.  N ) )  +  1 ) )
3325, 32syl 14 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
( N  -  1 ) ^ 2 )  =  ( ( ( N ^ 2 )  -  ( 2  x.  N ) )  +  1 ) )
3428, 25subcld 8297 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
( N ^ 2 )  -  N )  e.  CC )
35 1cnd 8002 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  1  e.  CC )
3634, 25, 35subsubd 8325 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
( ( N ^
2 )  -  N
)  -  ( N  -  1 ) )  =  ( ( ( ( N ^ 2 )  -  N )  -  N )  +  1 ) )
3731, 33, 363eqtr4d 2232 . . . . . . . . 9  |-  ( N  e.  NN  ->  (
( N  -  1 ) ^ 2 )  =  ( ( ( N ^ 2 )  -  N )  -  ( N  -  1
) ) )
3837oveq1d 5910 . . . . . . . 8  |-  ( N  e.  NN  ->  (
( ( N  - 
1 ) ^ 2 )  +  ( N  -  1 ) )  =  ( ( ( ( N ^ 2 )  -  N )  -  ( N  - 
1 ) )  +  ( N  -  1 ) ) )
39 ax-1cn 7933 . . . . . . . . . 10  |-  1  e.  CC
40 subcl 8185 . . . . . . . . . 10  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( N  -  1 )  e.  CC )
4125, 39, 40sylancl 413 . . . . . . . . 9  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  CC )
4234, 41npcand 8301 . . . . . . . 8  |-  ( N  e.  NN  ->  (
( ( ( N ^ 2 )  -  N )  -  ( N  -  1 ) )  +  ( N  -  1 ) )  =  ( ( N ^ 2 )  -  N ) )
4338, 42eqtrd 2222 . . . . . . 7  |-  ( N  e.  NN  ->  (
( ( N  - 
1 ) ^ 2 )  +  ( N  -  1 ) )  =  ( ( N ^ 2 )  -  N ) )
4443oveq1d 5910 . . . . . 6  |-  ( N  e.  NN  ->  (
( ( ( N  -  1 ) ^
2 )  +  ( N  -  1 ) )  /  2 )  =  ( ( ( N ^ 2 )  -  N )  / 
2 ) )
4524, 44eqtrd 2222 . . . . 5  |-  ( N  e.  NN  ->  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k  =  ( ( ( N ^
2 )  -  N
)  /  2 ) )
4622, 45eqtrd 2222 . . . 4  |-  ( N  e.  NN  ->  (
0  +  sum_ k  e.  ( ( 0  +  1 ) ... ( N  -  1 ) ) k )  =  ( ( ( N ^ 2 )  -  N )  /  2
) )
479, 46eqtrd 2222 . . 3  |-  ( N  e.  NN  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  ( ( ( N ^
2 )  -  N
)  /  2 ) )
48 oveq1 5902 . . . . . . . 8  |-  ( N  =  0  ->  ( N  -  1 )  =  ( 0  -  1 ) )
4948oveq2d 5911 . . . . . . 7  |-  ( N  =  0  ->  (
0 ... ( N  - 
1 ) )  =  ( 0 ... (
0  -  1 ) ) )
50 0re 7986 . . . . . . . . 9  |-  0  e.  RR
51 ltm1 8832 . . . . . . . . 9  |-  ( 0  e.  RR  ->  (
0  -  1 )  <  0 )
5250, 51ax-mp 5 . . . . . . . 8  |-  ( 0  -  1 )  <  0
53 0z 9293 . . . . . . . . 9  |-  0  e.  ZZ
54 peano2zm 9320 . . . . . . . . . 10  |-  ( 0  e.  ZZ  ->  (
0  -  1 )  e.  ZZ )
5553, 54ax-mp 5 . . . . . . . . 9  |-  ( 0  -  1 )  e.  ZZ
56 fzn 10071 . . . . . . . . 9  |-  ( ( 0  e.  ZZ  /\  ( 0  -  1 )  e.  ZZ )  ->  ( ( 0  -  1 )  <  0  <->  ( 0 ... ( 0  -  1 ) )  =  (/) ) )
5753, 55, 56mp2an 426 . . . . . . . 8  |-  ( ( 0  -  1 )  <  0  <->  ( 0 ... ( 0  -  1 ) )  =  (/) )
5852, 57mpbi 145 . . . . . . 7  |-  ( 0 ... ( 0  -  1 ) )  =  (/)
5949, 58eqtrdi 2238 . . . . . 6  |-  ( N  =  0  ->  (
0 ... ( N  - 
1 ) )  =  (/) )
6059sumeq1d 11405 . . . . 5  |-  ( N  =  0  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  sum_ k  e.  (/)  k )
61 sum0 11427 . . . . 5  |-  sum_ k  e.  (/)  k  =  0
6260, 61eqtrdi 2238 . . . 4  |-  ( N  =  0  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  0 )
63 sq0i 10642 . . . . . . . 8  |-  ( N  =  0  ->  ( N ^ 2 )  =  0 )
64 id 19 . . . . . . . 8  |-  ( N  =  0  ->  N  =  0 )
6563, 64oveq12d 5913 . . . . . . 7  |-  ( N  =  0  ->  (
( N ^ 2 )  -  N )  =  ( 0  -  0 ) )
66 0m0e0 9060 . . . . . . 7  |-  ( 0  -  0 )  =  0
6765, 66eqtrdi 2238 . . . . . 6  |-  ( N  =  0  ->  (
( N ^ 2 )  -  N )  =  0 )
6867oveq1d 5910 . . . . 5  |-  ( N  =  0  ->  (
( ( N ^
2 )  -  N
)  /  2 )  =  ( 0  / 
2 ) )
69 2cn 9019 . . . . . 6  |-  2  e.  CC
70 2ap0 9041 . . . . . 6  |-  2 #  0
7169, 70div0api 8732 . . . . 5  |-  ( 0  /  2 )  =  0
7268, 71eqtrdi 2238 . . . 4  |-  ( N  =  0  ->  (
( ( N ^
2 )  -  N
)  /  2 )  =  0 )
7362, 72eqtr4d 2225 . . 3  |-  ( N  =  0  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  ( ( ( N ^
2 )  -  N
)  /  2 ) )
7447, 73jaoi 717 . 2  |-  ( ( N  e.  NN  \/  N  =  0 )  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  ( ( ( N ^ 2 )  -  N )  /  2 ) )
751, 74sylbi 121 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    /\ wa 104    <-> wb 105    \/ wo 709    = wceq 1364    e. wcel 2160   (/)c0 3437   class class class wbr 4018   ` cfv 5235  (class class class)co 5895   CCcc 7838   RRcr 7839   0cc0 7840   1c1 7841    + caddc 7843    x. cmul 7845    < clt 8021    - cmin 8157    / cdiv 8658   NNcn 8948   2c2 8999   NN0cn0 9205   ZZcz 9282   ZZ>=cuz 9557   ...cfz 10037   ^cexp 10549   sum_csu 11392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-mulrcl 7939  ax-addcom 7940  ax-mulcom 7941  ax-addass 7942  ax-mulass 7943  ax-distr 7944  ax-i2m1 7945  ax-0lt1 7946  ax-1rid 7947  ax-0id 7948  ax-rnegex 7949  ax-precex 7950  ax-cnre 7951  ax-pre-ltirr 7952  ax-pre-ltwlin 7953  ax-pre-lttrn 7954  ax-pre-apti 7955  ax-pre-ltadd 7956  ax-pre-mulgt0 7957  ax-pre-mulext 7958  ax-arch 7959  ax-caucvg 7960
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-isom 5244  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-recs 6329  df-irdg 6394  df-frec 6415  df-1o 6440  df-oadd 6444  df-er 6558  df-en 6766  df-dom 6767  df-fin 6768  df-pnf 8023  df-mnf 8024  df-xr 8025  df-ltxr 8026  df-le 8027  df-sub 8159  df-neg 8160  df-reap 8561  df-ap 8568  df-div 8659  df-inn 8949  df-2 9007  df-3 9008  df-4 9009  df-n0 9206  df-z 9283  df-uz 9558  df-q 9649  df-rp 9683  df-fz 10038  df-fzo 10172  df-seqfrec 10476  df-exp 10550  df-fac 10737  df-bc 10759  df-ihash 10787  df-cj 10882  df-re 10883  df-im 10884  df-rsqrt 11038  df-abs 11039  df-clim 11318  df-sumdc 11393
This theorem is referenced by: (None)
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