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

Step	Hyp	Ref	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 )
4	2, 3	eleqtrdi 2282	. . . . 5 |- ( N e. NN -> ( N - 1 ) e. ( ZZ>= ` 0 ) )
5		elfznn0 10143	. . . . . . 7 |- ( k e. ( 0 ... ( N - 1 ) ) -> k e. NN0 )
6	5	adantl 277	. . . . . 6 |- ( ( N e. NN /\ k e. ( 0 ... ( N - 1 ) ) ) -> k e. NN0 )
7	6	nn0cnd 9260	. . . . 5 |- ( ( N e. NN /\ k e. ( 0 ... ( N - 1 ) ) ) -> k e. CC )
8		id 19	. . . . 5 |- ( k = 0 -> k = 0 )
9	4, 7, 8	fsum1p 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 )
11	10	oveq1i 5905	. . . . . . . 8 |- ( 1 ... ( N - 1 ) ) = ( ( 0 + 1 ) ... ( N - 1 ) )
12	11	sumeq1i 11402	. . . . . . 7 |- sum_ k e. ( 1 ... ( N - 1 ) ) k = sum_ k e. ( ( 0 + 1 ) ... ( N - 1 ) ) k
13	12	oveq2i 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 )
15	2	nn0zd 9402	. . . . . . . . 9 |- ( N e. NN -> ( N - 1 ) e. ZZ )
16	14, 15	fzfigd 10461	. . . . . . . 8 |- ( N e. NN -> ( 1 ... ( N - 1 ) ) e. Fin )
17		elfznn 10083	. . . . . . . . . 10 |- ( k e. ( 1 ... ( N - 1 ) ) -> k e. NN )
18	17	adantl 277	. . . . . . . . 9 |- ( ( N e. NN /\ k e. ( 1 ... ( N - 1 ) ) ) -> k e. NN )
19	18	nncnd 8962	. . . . . . . 8 |- ( ( N e. NN /\ k e. ( 1 ... ( N - 1 ) ) ) -> k e. CC )
20	16, 19	fsumcl 11439	. . . . . . 7 |- ( N e. NN -> sum_ k e. ( 1 ... ( N - 1 ) ) k e. CC )
21	20	addlidd 8136	. . . . . 6 |- ( N e. NN -> ( 0 + sum_ k e. ( 1 ... ( N - 1 ) ) k ) = sum_ k e. ( 1 ... ( N - 1 ) ) k )
22	13, 21	eqtr3id 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 ) )
24	2, 23	syl 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 )
26	25	2timesd 9190	. . . . . . . . . . . . 13 |- ( N e. NN -> ( 2 x. N ) = ( N + N ) )
27	26	oveq2d 5911	. . . . . . . . . . . 12 |- ( N e. NN -> ( ( N ^ 2 ) - ( 2 x. N ) ) = ( ( N ^ 2 ) - ( N + N ) ) )
28	25	sqcld 10682	. . . . . . . . . . . . 13 |- ( N e. NN -> ( N ^ 2 ) e. CC )
29	28, 25, 25	subsub4d 8328	. . . . . . . . . . . 12 |- ( N e. NN -> ( ( ( N ^ 2 ) - N ) - N ) = ( ( N ^ 2 ) - ( N + N ) ) )
30	27, 29	eqtr4d 2225	. . . . . . . . . . 11 |- ( N e. NN -> ( ( N ^ 2 ) - ( 2 x. N ) ) = ( ( ( N ^ 2 ) - N ) - N ) )
31	30	oveq1d 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 ) )
33	25, 32	syl 14	. . . . . . . . . 10 |- ( N e. NN -> ( ( N - 1 ) ^ 2 ) = ( ( ( N ^ 2 ) - ( 2 x. N ) ) + 1 ) )
34	28, 25	subcld 8297	. . . . . . . . . . 11 |- ( N e. NN -> ( ( N ^ 2 ) - N ) e. CC )
35		1cnd 8002	. . . . . . . . . . 11 |- ( N e. NN -> 1 e. CC )
36	34, 25, 35	subsubd 8325	. . . . . . . . . 10 |- ( N e. NN -> ( ( ( N ^ 2 ) - N ) - ( N - 1 ) ) = ( ( ( ( N ^ 2 ) - N ) - N ) + 1 ) )
37	31, 33, 36	3eqtr4d 2232	. . . . . . . . 9 |- ( N e. NN -> ( ( N - 1 ) ^ 2 ) = ( ( ( N ^ 2 ) - N ) - ( N - 1 ) ) )
38	37	oveq1d 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 )
41	25, 39, 40	sylancl 413	. . . . . . . . 9 |- ( N e. NN -> ( N - 1 ) e. CC )
42	34, 41	npcand 8301	. . . . . . . 8 |- ( N e. NN -> ( ( ( ( N ^ 2 ) - N ) - ( N - 1 ) ) + ( N - 1 ) ) = ( ( N ^ 2 ) - N ) )
43	38, 42	eqtrd 2222	. . . . . . 7 |- ( N e. NN -> ( ( ( N - 1 ) ^ 2 ) + ( N - 1 ) ) = ( ( N ^ 2 ) - N ) )
44	43	oveq1d 5910	. . . . . 6 |- ( N e. NN -> ( ( ( ( N - 1 ) ^ 2 ) + ( N - 1 ) ) / 2 ) = ( ( ( N ^ 2 ) - N ) / 2 ) )
45	24, 44	eqtrd 2222	. . . . 5 |- ( N e. NN -> sum_ k e. ( 1 ... ( N - 1 ) ) k = ( ( ( N ^ 2 ) - N ) / 2 ) )
46	22, 45	eqtrd 2222	. . . 4 |- ( N e. NN -> ( 0 + sum_ k e. ( ( 0 + 1 ) ... ( N - 1 ) ) k ) = ( ( ( N ^ 2 ) - N ) / 2 ) )
47	9, 46	eqtrd 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 ) )
49	48	oveq2d 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 )
52	50, 51	ax-mp 5	. . . . . . . 8 |- ( 0 - 1 ) < 0
53		0z 9293	. . . . . . . . 9 |- 0 e. ZZ
54		peano2zm 9320	. . . . . . . . . 10 |- ( 0 e. ZZ -> ( 0 - 1 ) e. ZZ )
55	53, 54	ax-mp 5	. . . . . . . . 9 |- ( 0 - 1 ) e. ZZ
56		fzn 10071	. . . . . . . . 9 |- ( ( 0 e. ZZ /\ ( 0 - 1 ) e. ZZ ) -> ( ( 0 - 1 ) < 0 <-> ( 0 ... ( 0 - 1 ) ) = (/) ) )
57	53, 55, 56	mp2an 426	. . . . . . . 8 |- ( ( 0 - 1 ) < 0 <-> ( 0 ... ( 0 - 1 ) ) = (/) )
58	52, 57	mpbi 145	. . . . . . 7 |- ( 0 ... ( 0 - 1 ) ) = (/)
59	49, 58	eqtrdi 2238	. . . . . 6 |- ( N = 0 -> ( 0 ... ( N - 1 ) ) = (/) )
60	59	sumeq1d 11405	. . . . 5 |- ( N = 0 -> sum_ k e. ( 0 ... ( N - 1 ) ) k = sum_ k e. (/) k )
61		sum0 11427	. . . . 5 |- sum_ k e. (/) k = 0
62	60, 61	eqtrdi 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 )
65	63, 64	oveq12d 5913	. . . . . . 7 |- ( N = 0 -> ( ( N ^ 2 ) - N ) = ( 0 - 0 ) )
66		0m0e0 9060	. . . . . . 7 |- ( 0 - 0 ) = 0
67	65, 66	eqtrdi 2238	. . . . . 6 |- ( N = 0 -> ( ( N ^ 2 ) - N ) = 0 )
68	67	oveq1d 5910	. . . . 5 |- ( N = 0 -> ( ( ( N ^ 2 ) - N ) / 2 ) = ( 0 / 2 ) )
69		2cn 9019	. . . . . 6 |- 2 e. CC
70		2ap0 9041	. . . . . 6 |- 2 # 0
71	69, 70	div0api 8732	. . . . 5 |- ( 0 / 2 ) = 0
72	68, 71	eqtrdi 2238	. . . 4 |- ( N = 0 -> ( ( ( N ^ 2 ) - N ) / 2 ) = 0 )
73	62, 72	eqtr4d 2225	. . 3 |- ( N = 0 -> sum_ k e. ( 0 ... ( N - 1 ) ) k = ( ( ( N ^ 2 ) - N ) / 2 ) )
74	47, 73	jaoi 717	. 2 |- ( ( N e. NN \/ N = 0 ) -> sum_ k e. ( 0 ... ( N - 1 ) ) k = ( ( ( N ^ 2 ) - N ) / 2 ) )
75	1, 74	sylbi 121	1 |- ( N e. NN0 -> sum_ k e. ( 0 ... ( N - 1 ) ) k = ( ( ( N ^ 2 ) - N ) / 2 ) )

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)