Theorem arisum2 11664

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 9251	. 2 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		nnm1nn0 9290	. . . . . 6 |- ( N e. NN -> ( N - 1 ) e. NN0 )
3		nn0uz 9636	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
4	2, 3	eleqtrdi 2289	. . . . 5 |- ( N e. NN -> ( N - 1 ) e. ( ZZ>= ` 0 ) )
5		elfznn0 10189	. . . . . . 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 9304	. . . . 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 11583	. . . 4 |- ( N e. NN -> sum_ k e. ( 0 ... ( N - 1 ) ) k = ( 0 + sum_ k e. ( ( 0 + 1 ) ... ( N - 1 ) ) k ) )
10		1e0p1 9498	. . . . . . . . 9 |- 1 = ( 0 + 1 )
11	10	oveq1i 5932	. . . . . . . 8 |- ( 1 ... ( N - 1 ) ) = ( ( 0 + 1 ) ... ( N - 1 ) )
12	11	sumeq1i 11528	. . . . . . 7 |- sum_ k e. ( 1 ... ( N - 1 ) ) k = sum_ k e. ( ( 0 + 1 ) ... ( N - 1 ) ) k
13	12	oveq2i 5933	. . . . . 6 |- ( 0 + sum_ k e. ( 1 ... ( N - 1 ) ) k ) = ( 0 + sum_ k e. ( ( 0 + 1 ) ... ( N - 1 ) ) k )
14		1zzd 9353	. . . . . . . . 9 |- ( N e. NN -> 1 e. ZZ )
15	2	nn0zd 9446	. . . . . . . . 9 |- ( N e. NN -> ( N - 1 ) e. ZZ )
16	14, 15	fzfigd 10523	. . . . . . . 8 |- ( N e. NN -> ( 1 ... ( N - 1 ) ) e. Fin )
17		elfznn 10129	. . . . . . . . . 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 9004	. . . . . . . 8 |- ( ( N e. NN /\ k e. ( 1 ... ( N - 1 ) ) ) -> k e. CC )
20	16, 19	fsumcl 11565	. . . . . . 7 |- ( N e. NN -> sum_ k e. ( 1 ... ( N - 1 ) ) k e. CC )
21	20	addlidd 8176	. . . . . 6 |- ( N e. NN -> ( 0 + sum_ k e. ( 1 ... ( N - 1 ) ) k ) = sum_ k e. ( 1 ... ( N - 1 ) ) k )
22	13, 21	eqtr3id 2243	. . . . 5 |- ( N e. NN -> ( 0 + sum_ k e. ( ( 0 + 1 ) ... ( N - 1 ) ) k ) = sum_ k e. ( 1 ... ( N - 1 ) ) k )
23		arisum 11663	. . . . . . 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 8998	. . . . . . . . . . . . . 14 |- ( N e. NN -> N e. CC )
26	25	2timesd 9234	. . . . . . . . . . . . 13 |- ( N e. NN -> ( 2 x. N ) = ( N + N ) )
27	26	oveq2d 5938	. . . . . . . . . . . 12 |- ( N e. NN -> ( ( N ^ 2 ) - ( 2 x. N ) ) = ( ( N ^ 2 ) - ( N + N ) ) )
28	25	sqcld 10763	. . . . . . . . . . . . 13 |- ( N e. NN -> ( N ^ 2 ) e. CC )
29	28, 25, 25	subsub4d 8368	. . . . . . . . . . . 12 |- ( N e. NN -> ( ( ( N ^ 2 ) - N ) - N ) = ( ( N ^ 2 ) - ( N + N ) ) )
30	27, 29	eqtr4d 2232	. . . . . . . . . . 11 |- ( N e. NN -> ( ( N ^ 2 ) - ( 2 x. N ) ) = ( ( ( N ^ 2 ) - N ) - N ) )
31	30	oveq1d 5937	. . . . . . . . . 10 |- ( N e. NN -> ( ( ( N ^ 2 ) - ( 2 x. N ) ) + 1 ) = ( ( ( ( N ^ 2 ) - N ) - N ) + 1 ) )
32		binom2sub1 10746	. . . . . . . . . . 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 8337	. . . . . . . . . . 11 |- ( N e. NN -> ( ( N ^ 2 ) - N ) e. CC )
35		1cnd 8042	. . . . . . . . . . 11 |- ( N e. NN -> 1 e. CC )
36	34, 25, 35	subsubd 8365	. . . . . . . . . 10 |- ( N e. NN -> ( ( ( N ^ 2 ) - N ) - ( N - 1 ) ) = ( ( ( ( N ^ 2 ) - N ) - N ) + 1 ) )
37	31, 33, 36	3eqtr4d 2239	. . . . . . . . 9 |- ( N e. NN -> ( ( N - 1 ) ^ 2 ) = ( ( ( N ^ 2 ) - N ) - ( N - 1 ) ) )
38	37	oveq1d 5937	. . . . . . . 8 |- ( N e. NN -> ( ( ( N - 1 ) ^ 2 ) + ( N - 1 ) ) = ( ( ( ( N ^ 2 ) - N ) - ( N - 1 ) ) + ( N - 1 ) ) )
39		ax-1cn 7972	. . . . . . . . . 10 |- 1 e. CC
40		subcl 8225	. . . . . . . . . 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 8341	. . . . . . . 8 |- ( N e. NN -> ( ( ( ( N ^ 2 ) - N ) - ( N - 1 ) ) + ( N - 1 ) ) = ( ( N ^ 2 ) - N ) )
43	38, 42	eqtrd 2229	. . . . . . 7 |- ( N e. NN -> ( ( ( N - 1 ) ^ 2 ) + ( N - 1 ) ) = ( ( N ^ 2 ) - N ) )
44	43	oveq1d 5937	. . . . . 6 |- ( N e. NN -> ( ( ( ( N - 1 ) ^ 2 ) + ( N - 1 ) ) / 2 ) = ( ( ( N ^ 2 ) - N ) / 2 ) )
45	24, 44	eqtrd 2229	. . . . 5 |- ( N e. NN -> sum_ k e. ( 1 ... ( N - 1 ) ) k = ( ( ( N ^ 2 ) - N ) / 2 ) )
46	22, 45	eqtrd 2229	. . . 4 |- ( N e. NN -> ( 0 + sum_ k e. ( ( 0 + 1 ) ... ( N - 1 ) ) k ) = ( ( ( N ^ 2 ) - N ) / 2 ) )
47	9, 46	eqtrd 2229	. . 3 |- ( N e. NN -> sum_ k e. ( 0 ... ( N - 1 ) ) k = ( ( ( N ^ 2 ) - N ) / 2 ) )
48		oveq1 5929	. . . . . . . 8 |- ( N = 0 -> ( N - 1 ) = ( 0 - 1 ) )
49	48	oveq2d 5938	. . . . . . 7 |- ( N = 0 -> ( 0 ... ( N - 1 ) ) = ( 0 ... ( 0 - 1 ) ) )
50		0re 8026	. . . . . . . . 9 |- 0 e. RR
51		ltm1 8873	. . . . . . . . 9 |- ( 0 e. RR -> ( 0 - 1 ) < 0 )
52	50, 51	ax-mp 5	. . . . . . . 8 |- ( 0 - 1 ) < 0
53		0z 9337	. . . . . . . . 9 |- 0 e. ZZ
54		peano2zm 9364	. . . . . . . . . 10 |- ( 0 e. ZZ -> ( 0 - 1 ) e. ZZ )
55	53, 54	ax-mp 5	. . . . . . . . 9 |- ( 0 - 1 ) e. ZZ
56		fzn 10117	. . . . . . . . 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 2245	. . . . . 6 |- ( N = 0 -> ( 0 ... ( N - 1 ) ) = (/) )
60	59	sumeq1d 11531	. . . . 5 |- ( N = 0 -> sum_ k e. ( 0 ... ( N - 1 ) ) k = sum_ k e. (/) k )
61		sum0 11553	. . . . 5 |- sum_ k e. (/) k = 0
62	60, 61	eqtrdi 2245	. . . 4 |- ( N = 0 -> sum_ k e. ( 0 ... ( N - 1 ) ) k = 0 )
63		sq0i 10723	. . . . . . . 8 |- ( N = 0 -> ( N ^ 2 ) = 0 )
64		id 19	. . . . . . . 8 |- ( N = 0 -> N = 0 )
65	63, 64	oveq12d 5940	. . . . . . 7 |- ( N = 0 -> ( ( N ^ 2 ) - N ) = ( 0 - 0 ) )
66		0m0e0 9102	. . . . . . 7 |- ( 0 - 0 ) = 0
67	65, 66	eqtrdi 2245	. . . . . 6 |- ( N = 0 -> ( ( N ^ 2 ) - N ) = 0 )
68	67	oveq1d 5937	. . . . 5 |- ( N = 0 -> ( ( ( N ^ 2 ) - N ) / 2 ) = ( 0 / 2 ) )
69		2cn 9061	. . . . . 6 |- 2 e. CC
70		2ap0 9083	. . . . . 6 |- 2 # 0
71	69, 70	div0api 8773	. . . . 5 |- ( 0 / 2 ) = 0
72	68, 71	eqtrdi 2245	. . . 4 |- ( N = 0 -> ( ( ( N ^ 2 ) - N ) / 2 ) = 0 )
73	62, 72	eqtr4d 2232	. . 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 2167   (/)c0 3450   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   CCcc 7877   RRcr 7878   0cc0 7879   1c1 7880   + caddc 7882   x. cmul 7884   < clt 8061   - cmin 8197   / cdiv 8699   NNcn 8990   2c2 9041   NN0cn0 9249   ZZcz 9326   ZZ>=cuz 9601   ...cfz 10083   ^cexp 10630   sum_csu 11518

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-frec 6449  df-1o 6474  df-oadd 6478  df-er 6592  df-en 6800  df-dom 6801  df-fin 6802  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-fz 10084  df-fzo 10218  df-seqfrec 10540  df-exp 10631  df-fac 10818  df-bc 10840  df-ihash 10868  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-clim 11444  df-sumdc 11519

This theorem is referenced by: (None)