Theorem arisum2 12210

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 9515	. 2 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		nnm1nn0 9554	. . . . . 6 |- ( N e. NN -> ( N - 1 ) e. NN0 )
3		nn0uz 9907	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
4	2, 3	eleqtrdi 2327	. . . . 5 |- ( N e. NN -> ( N - 1 ) e. ( ZZ>= ` 0 ) )
5		elfznn0 10470	. . . . . . 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 9572	. . . . 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 12129	. . . 4 |- ( N e. NN -> sum_ k e. ( 0 ... ( N - 1 ) ) k = ( 0 + sum_ k e. ( ( 0 + 1 ) ... ( N - 1 ) ) k ) )
10		1e0p1 9768	. . . . . . . . 9 |- 1 = ( 0 + 1 )
11	10	oveq1i 6068	. . . . . . . 8 |- ( 1 ... ( N - 1 ) ) = ( ( 0 + 1 ) ... ( N - 1 ) )
12	11	sumeq1i 12073	. . . . . . 7 |- sum_ k e. ( 1 ... ( N - 1 ) ) k = sum_ k e. ( ( 0 + 1 ) ... ( N - 1 ) ) k
13	12	oveq2i 6069	. . . . . 6 |- ( 0 + sum_ k e. ( 1 ... ( N - 1 ) ) k ) = ( 0 + sum_ k e. ( ( 0 + 1 ) ... ( N - 1 ) ) k )
14		1zzd 9621	. . . . . . . . 9 |- ( N e. NN -> 1 e. ZZ )
15	2	nn0zd 9716	. . . . . . . . 9 |- ( N e. NN -> ( N - 1 ) e. ZZ )
16	14, 15	fzfigd 10817	. . . . . . . 8 |- ( N e. NN -> ( 1 ... ( N - 1 ) ) e. Fin )
17		elfznn 10409	. . . . . . . . . 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 9268	. . . . . . . 8 |- ( ( N e. NN /\ k e. ( 1 ... ( N - 1 ) ) ) -> k e. CC )
20	16, 19	fsumcl 12111	. . . . . . 7 |- ( N e. NN -> sum_ k e. ( 1 ... ( N - 1 ) ) k e. CC )
21	20	addlidd 8439	. . . . . 6 |- ( N e. NN -> ( 0 + sum_ k e. ( 1 ... ( N - 1 ) ) k ) = sum_ k e. ( 1 ... ( N - 1 ) ) k )
22	13, 21	eqtr3id 2281	. . . . 5 |- ( N e. NN -> ( 0 + sum_ k e. ( ( 0 + 1 ) ... ( N - 1 ) ) k ) = sum_ k e. ( 1 ... ( N - 1 ) ) k )
23		arisum 12209	. . . . . . 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 9262	. . . . . . . . . . . . . 14 |- ( N e. NN -> N e. CC )
26	25	2timesd 9498	. . . . . . . . . . . . 13 |- ( N e. NN -> ( 2 x. N ) = ( N + N ) )
27	26	oveq2d 6074	. . . . . . . . . . . 12 |- ( N e. NN -> ( ( N ^ 2 ) - ( 2 x. N ) ) = ( ( N ^ 2 ) - ( N + N ) ) )
28	25	sqcld 11058	. . . . . . . . . . . . 13 |- ( N e. NN -> ( N ^ 2 ) e. CC )
29	28, 25, 25	subsub4d 8631	. . . . . . . . . . . 12 |- ( N e. NN -> ( ( ( N ^ 2 ) - N ) - N ) = ( ( N ^ 2 ) - ( N + N ) ) )
30	27, 29	eqtr4d 2270	. . . . . . . . . . 11 |- ( N e. NN -> ( ( N ^ 2 ) - ( 2 x. N ) ) = ( ( ( N ^ 2 ) - N ) - N ) )
31	30	oveq1d 6073	. . . . . . . . . 10 |- ( N e. NN -> ( ( ( N ^ 2 ) - ( 2 x. N ) ) + 1 ) = ( ( ( ( N ^ 2 ) - N ) - N ) + 1 ) )
32		binom2sub1 11040	. . . . . . . . . . 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 8600	. . . . . . . . . . 11 |- ( N e. NN -> ( ( N ^ 2 ) - N ) e. CC )
35		1cnd 8306	. . . . . . . . . . 11 |- ( N e. NN -> 1 e. CC )
36	34, 25, 35	subsubd 8628	. . . . . . . . . 10 |- ( N e. NN -> ( ( ( N ^ 2 ) - N ) - ( N - 1 ) ) = ( ( ( ( N ^ 2 ) - N ) - N ) + 1 ) )
37	31, 33, 36	3eqtr4d 2277	. . . . . . . . 9 |- ( N e. NN -> ( ( N - 1 ) ^ 2 ) = ( ( ( N ^ 2 ) - N ) - ( N - 1 ) ) )
38	37	oveq1d 6073	. . . . . . . 8 |- ( N e. NN -> ( ( ( N - 1 ) ^ 2 ) + ( N - 1 ) ) = ( ( ( ( N ^ 2 ) - N ) - ( N - 1 ) ) + ( N - 1 ) ) )
39		ax-1cn 8236	. . . . . . . . . 10 |- 1 e. CC
40		subcl 8488	. . . . . . . . . 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 8604	. . . . . . . 8 |- ( N e. NN -> ( ( ( ( N ^ 2 ) - N ) - ( N - 1 ) ) + ( N - 1 ) ) = ( ( N ^ 2 ) - N ) )
43	38, 42	eqtrd 2267	. . . . . . 7 |- ( N e. NN -> ( ( ( N - 1 ) ^ 2 ) + ( N - 1 ) ) = ( ( N ^ 2 ) - N ) )
44	43	oveq1d 6073	. . . . . 6 |- ( N e. NN -> ( ( ( ( N - 1 ) ^ 2 ) + ( N - 1 ) ) / 2 ) = ( ( ( N ^ 2 ) - N ) / 2 ) )
45	24, 44	eqtrd 2267	. . . . 5 |- ( N e. NN -> sum_ k e. ( 1 ... ( N - 1 ) ) k = ( ( ( N ^ 2 ) - N ) / 2 ) )
46	22, 45	eqtrd 2267	. . . 4 |- ( N e. NN -> ( 0 + sum_ k e. ( ( 0 + 1 ) ... ( N - 1 ) ) k ) = ( ( ( N ^ 2 ) - N ) / 2 ) )
47	9, 46	eqtrd 2267	. . 3 |- ( N e. NN -> sum_ k e. ( 0 ... ( N - 1 ) ) k = ( ( ( N ^ 2 ) - N ) / 2 ) )
48		oveq1 6065	. . . . . . . 8 |- ( N = 0 -> ( N - 1 ) = ( 0 - 1 ) )
49	48	oveq2d 6074	. . . . . . 7 |- ( N = 0 -> ( 0 ... ( N - 1 ) ) = ( 0 ... ( 0 - 1 ) ) )
50		0re 8290	. . . . . . . . 9 |- 0 e. RR
51		ltm1 9137	. . . . . . . . 9 |- ( 0 e. RR -> ( 0 - 1 ) < 0 )
52	50, 51	ax-mp 5	. . . . . . . 8 |- ( 0 - 1 ) < 0
53		0z 9605	. . . . . . . . 9 |- 0 e. ZZ
54		peano2zm 9632	. . . . . . . . . 10 |- ( 0 e. ZZ -> ( 0 - 1 ) e. ZZ )
55	53, 54	ax-mp 5	. . . . . . . . 9 |- ( 0 - 1 ) e. ZZ
56		fzn 10396	. . . . . . . . 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 2283	. . . . . 6 |- ( N = 0 -> ( 0 ... ( N - 1 ) ) = (/) )
60	59	sumeq1d 12076	. . . . 5 |- ( N = 0 -> sum_ k e. ( 0 ... ( N - 1 ) ) k = sum_ k e. (/) k )
61		sum0 12099	. . . . 5 |- sum_ k e. (/) k = 0
62	60, 61	eqtrdi 2283	. . . 4 |- ( N = 0 -> sum_ k e. ( 0 ... ( N - 1 ) ) k = 0 )
63		sq0i 11017	. . . . . . . 8 |- ( N = 0 -> ( N ^ 2 ) = 0 )
64		id 19	. . . . . . . 8 |- ( N = 0 -> N = 0 )
65	63, 64	oveq12d 6076	. . . . . . 7 |- ( N = 0 -> ( ( N ^ 2 ) - N ) = ( 0 - 0 ) )
66		0m0e0 9366	. . . . . . 7 |- ( 0 - 0 ) = 0
67	65, 66	eqtrdi 2283	. . . . . 6 |- ( N = 0 -> ( ( N ^ 2 ) - N ) = 0 )
68	67	oveq1d 6073	. . . . 5 |- ( N = 0 -> ( ( ( N ^ 2 ) - N ) / 2 ) = ( 0 / 2 ) )
69		2cn 9325	. . . . . 6 |- 2 e. CC
70		2ap0 9347	. . . . . 6 |- 2 # 0
71	69, 70	div0api 9037	. . . . 5 |- ( 0 / 2 ) = 0
72	68, 71	eqtrdi 2283	. . . 4 |- ( N = 0 -> ( ( ( N ^ 2 ) - N ) / 2 ) = 0 )
73	62, 72	eqtr4d 2270	. . 3 |- ( N = 0 -> sum_ k e. ( 0 ... ( N - 1 ) ) k = ( ( ( N ^ 2 ) - N ) / 2 ) )
74	47, 73	jaoi 724	. 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 716   = wceq 1398   e. wcel 2205   (/)c0 3512   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   CCcc 8141   RRcr 8142   0cc0 8143   1c1 8144   + caddc 8146   x. cmul 8148   < clt 8324   - cmin 8460   / cdiv 8963   NNcn 9254   2c2 9305   NN0cn0 9513   ZZcz 9594   ZZ>=cuz 9871   ...cfz 10361   ^cexp 10924   sum_csu 12063

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-seqfrec 10834  df-exp 10925  df-fac 11113  df-bc 11135  df-ihash 11164  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-clim 11989  df-sumdc 12064

This theorem is referenced by: (None)