Theorem arisum2 11300

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 9003	. 2 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		nnm1nn0 9042	. . . . . 6 |- ( N e. NN -> ( N - 1 ) e. NN0 )
3		nn0uz 9384	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
4	2, 3	eleqtrdi 2233	. . . . 5 |- ( N e. NN -> ( N - 1 ) e. ( ZZ>= ` 0 ) )
5		elfznn0 9925	. . . . . . 7 |- ( k e. ( 0 ... ( N - 1 ) ) -> k e. NN0 )
6	5	adantl 275	. . . . . 6 |- ( ( N e. NN /\ k e. ( 0 ... ( N - 1 ) ) ) -> k e. NN0 )
7	6	nn0cnd 9056	. . . . 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 11219	. . . 4 |- ( N e. NN -> sum_ k e. ( 0 ... ( N - 1 ) ) k = ( 0 + sum_ k e. ( ( 0 + 1 ) ... ( N - 1 ) ) k ) )
10		1e0p1 9247	. . . . . . . . 9 |- 1 = ( 0 + 1 )
11	10	oveq1i 5792	. . . . . . . 8 |- ( 1 ... ( N - 1 ) ) = ( ( 0 + 1 ) ... ( N - 1 ) )
12	11	sumeq1i 11164	. . . . . . 7 |- sum_ k e. ( 1 ... ( N - 1 ) ) k = sum_ k e. ( ( 0 + 1 ) ... ( N - 1 ) ) k
13	12	oveq2i 5793	. . . . . 6 |- ( 0 + sum_ k e. ( 1 ... ( N - 1 ) ) k ) = ( 0 + sum_ k e. ( ( 0 + 1 ) ... ( N - 1 ) ) k )
14		1zzd 9105	. . . . . . . . 9 |- ( N e. NN -> 1 e. ZZ )
15	2	nn0zd 9195	. . . . . . . . 9 |- ( N e. NN -> ( N - 1 ) e. ZZ )
16	14, 15	fzfigd 10235	. . . . . . . 8 |- ( N e. NN -> ( 1 ... ( N - 1 ) ) e. Fin )
17		elfznn 9865	. . . . . . . . . 10 |- ( k e. ( 1 ... ( N - 1 ) ) -> k e. NN )
18	17	adantl 275	. . . . . . . . 9 |- ( ( N e. NN /\ k e. ( 1 ... ( N - 1 ) ) ) -> k e. NN )
19	18	nncnd 8758	. . . . . . . 8 |- ( ( N e. NN /\ k e. ( 1 ... ( N - 1 ) ) ) -> k e. CC )
20	16, 19	fsumcl 11201	. . . . . . 7 |- ( N e. NN -> sum_ k e. ( 1 ... ( N - 1 ) ) k e. CC )
21	20	addid2d 7936	. . . . . 6 |- ( N e. NN -> ( 0 + sum_ k e. ( 1 ... ( N - 1 ) ) k ) = sum_ k e. ( 1 ... ( N - 1 ) ) k )
22	13, 21	syl5eqr 2187	. . . . 5 |- ( N e. NN -> ( 0 + sum_ k e. ( ( 0 + 1 ) ... ( N - 1 ) ) k ) = sum_ k e. ( 1 ... ( N - 1 ) ) k )
23		arisum 11299	. . . . . . 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 8752	. . . . . . . . . . . . . 14 |- ( N e. NN -> N e. CC )
26	25	2timesd 8986	. . . . . . . . . . . . 13 |- ( N e. NN -> ( 2 x. N ) = ( N + N ) )
27	26	oveq2d 5798	. . . . . . . . . . . 12 |- ( N e. NN -> ( ( N ^ 2 ) - ( 2 x. N ) ) = ( ( N ^ 2 ) - ( N + N ) ) )
28	25	sqcld 10453	. . . . . . . . . . . . 13 |- ( N e. NN -> ( N ^ 2 ) e. CC )
29	28, 25, 25	subsub4d 8128	. . . . . . . . . . . 12 |- ( N e. NN -> ( ( ( N ^ 2 ) - N ) - N ) = ( ( N ^ 2 ) - ( N + N ) ) )
30	27, 29	eqtr4d 2176	. . . . . . . . . . 11 |- ( N e. NN -> ( ( N ^ 2 ) - ( 2 x. N ) ) = ( ( ( N ^ 2 ) - N ) - N ) )
31	30	oveq1d 5797	. . . . . . . . . 10 |- ( N e. NN -> ( ( ( N ^ 2 ) - ( 2 x. N ) ) + 1 ) = ( ( ( ( N ^ 2 ) - N ) - N ) + 1 ) )
32		binom2sub1 10437	. . . . . . . . . . 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 8097	. . . . . . . . . . 11 |- ( N e. NN -> ( ( N ^ 2 ) - N ) e. CC )
35		1cnd 7806	. . . . . . . . . . 11 |- ( N e. NN -> 1 e. CC )
36	34, 25, 35	subsubd 8125	. . . . . . . . . 10 |- ( N e. NN -> ( ( ( N ^ 2 ) - N ) - ( N - 1 ) ) = ( ( ( ( N ^ 2 ) - N ) - N ) + 1 ) )
37	31, 33, 36	3eqtr4d 2183	. . . . . . . . 9 |- ( N e. NN -> ( ( N - 1 ) ^ 2 ) = ( ( ( N ^ 2 ) - N ) - ( N - 1 ) ) )
38	37	oveq1d 5797	. . . . . . . 8 |- ( N e. NN -> ( ( ( N - 1 ) ^ 2 ) + ( N - 1 ) ) = ( ( ( ( N ^ 2 ) - N ) - ( N - 1 ) ) + ( N - 1 ) ) )
39		ax-1cn 7737	. . . . . . . . . 10 |- 1 e. CC
40		subcl 7985	. . . . . . . . . 10 |- ( ( N e. CC /\ 1 e. CC ) -> ( N - 1 ) e. CC )
41	25, 39, 40	sylancl 410	. . . . . . . . 9 |- ( N e. NN -> ( N - 1 ) e. CC )
42	34, 41	npcand 8101	. . . . . . . 8 |- ( N e. NN -> ( ( ( ( N ^ 2 ) - N ) - ( N - 1 ) ) + ( N - 1 ) ) = ( ( N ^ 2 ) - N ) )
43	38, 42	eqtrd 2173	. . . . . . 7 |- ( N e. NN -> ( ( ( N - 1 ) ^ 2 ) + ( N - 1 ) ) = ( ( N ^ 2 ) - N ) )
44	43	oveq1d 5797	. . . . . 6 |- ( N e. NN -> ( ( ( ( N - 1 ) ^ 2 ) + ( N - 1 ) ) / 2 ) = ( ( ( N ^ 2 ) - N ) / 2 ) )
45	24, 44	eqtrd 2173	. . . . 5 |- ( N e. NN -> sum_ k e. ( 1 ... ( N - 1 ) ) k = ( ( ( N ^ 2 ) - N ) / 2 ) )
46	22, 45	eqtrd 2173	. . . 4 |- ( N e. NN -> ( 0 + sum_ k e. ( ( 0 + 1 ) ... ( N - 1 ) ) k ) = ( ( ( N ^ 2 ) - N ) / 2 ) )
47	9, 46	eqtrd 2173	. . 3 |- ( N e. NN -> sum_ k e. ( 0 ... ( N - 1 ) ) k = ( ( ( N ^ 2 ) - N ) / 2 ) )
48		oveq1 5789	. . . . . . . 8 |- ( N = 0 -> ( N - 1 ) = ( 0 - 1 ) )
49	48	oveq2d 5798	. . . . . . 7 |- ( N = 0 -> ( 0 ... ( N - 1 ) ) = ( 0 ... ( 0 - 1 ) ) )
50		0re 7790	. . . . . . . . 9 |- 0 e. RR
51		ltm1 8628	. . . . . . . . 9 |- ( 0 e. RR -> ( 0 - 1 ) < 0 )
52	50, 51	ax-mp 5	. . . . . . . 8 |- ( 0 - 1 ) < 0
53		0z 9089	. . . . . . . . 9 |- 0 e. ZZ
54		peano2zm 9116	. . . . . . . . . 10 |- ( 0 e. ZZ -> ( 0 - 1 ) e. ZZ )
55	53, 54	ax-mp 5	. . . . . . . . 9 |- ( 0 - 1 ) e. ZZ
56		fzn 9853	. . . . . . . . 9 |- ( ( 0 e. ZZ /\ ( 0 - 1 ) e. ZZ ) -> ( ( 0 - 1 ) < 0 <-> ( 0 ... ( 0 - 1 ) ) = (/) ) )
57	53, 55, 56	mp2an 423	. . . . . . . 8 |- ( ( 0 - 1 ) < 0 <-> ( 0 ... ( 0 - 1 ) ) = (/) )
58	52, 57	mpbi 144	. . . . . . 7 |- ( 0 ... ( 0 - 1 ) ) = (/)
59	49, 58	eqtrdi 2189	. . . . . 6 |- ( N = 0 -> ( 0 ... ( N - 1 ) ) = (/) )
60	59	sumeq1d 11167	. . . . 5 |- ( N = 0 -> sum_ k e. ( 0 ... ( N - 1 ) ) k = sum_ k e. (/) k )
61		sum0 11189	. . . . 5 |- sum_ k e. (/) k = 0
62	60, 61	eqtrdi 2189	. . . 4 |- ( N = 0 -> sum_ k e. ( 0 ... ( N - 1 ) ) k = 0 )
63		sq0i 10415	. . . . . . . 8 |- ( N = 0 -> ( N ^ 2 ) = 0 )
64		id 19	. . . . . . . 8 |- ( N = 0 -> N = 0 )
65	63, 64	oveq12d 5800	. . . . . . 7 |- ( N = 0 -> ( ( N ^ 2 ) - N ) = ( 0 - 0 ) )
66		0m0e0 8856	. . . . . . 7 |- ( 0 - 0 ) = 0
67	65, 66	eqtrdi 2189	. . . . . 6 |- ( N = 0 -> ( ( N ^ 2 ) - N ) = 0 )
68	67	oveq1d 5797	. . . . 5 |- ( N = 0 -> ( ( ( N ^ 2 ) - N ) / 2 ) = ( 0 / 2 ) )
69		2cn 8815	. . . . . 6 |- 2 e. CC
70		2ap0 8837	. . . . . 6 |- 2 # 0
71	69, 70	div0api 8530	. . . . 5 |- ( 0 / 2 ) = 0
72	68, 71	eqtrdi 2189	. . . 4 |- ( N = 0 -> ( ( ( N ^ 2 ) - N ) / 2 ) = 0 )
73	62, 72	eqtr4d 2176	. . 3 |- ( N = 0 -> sum_ k e. ( 0 ... ( N - 1 ) ) k = ( ( ( N ^ 2 ) - N ) / 2 ) )
74	47, 73	jaoi 706	. 2 |- ( ( N e. NN \/ N = 0 ) -> sum_ k e. ( 0 ... ( N - 1 ) ) k = ( ( ( N ^ 2 ) - N ) / 2 ) )
75	1, 74	sylbi 120	1 |- ( N e. NN0 -> sum_ k e. ( 0 ... ( N - 1 ) ) k = ( ( ( N ^ 2 ) - N ) / 2 ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   = wceq 1332   e. wcel 1481   (/)c0 3368   class class class wbr 3937   ` cfv 5131  (class class class)co 5782   CCcc 7642   RRcr 7643   0cc0 7644   1c1 7645   + caddc 7647   x. cmul 7649   < clt 7824   - cmin 7957   / cdiv 8456   NNcn 8744   2c2 8795   NN0cn0 9001   ZZcz 9078   ZZ>=cuz 9350   ...cfz 9821   ^cexp 10323   sum_csu 11154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-isom 5140  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-frec 6296  df-1o 6321  df-oadd 6325  df-er 6437  df-en 6643  df-dom 6644  df-fin 6645  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-n0 9002  df-z 9079  df-uz 9351  df-q 9439  df-rp 9471  df-fz 9822  df-fzo 9951  df-seqfrec 10250  df-exp 10324  df-fac 10504  df-bc 10526  df-ihash 10554  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803  df-clim 11080  df-sumdc 11155

This theorem is referenced by: (None)