Theorem arisum2 14593

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 11294	. 2 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		nnm1nn0 11334	. . . . . 6 |- ( N e. NN -> ( N - 1 ) e. NN0 )
3		nn0uz 11722	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
4	2, 3	syl6eleq 2711	. . . . 5 |- ( N e. NN -> ( N - 1 ) e. ( ZZ>= ` 0 ) )
5		elfznn0 12433	. . . . . . 7 |- ( k e. ( 0 ... ( N - 1 ) ) -> k e. NN0 )
6	5	adantl 482	. . . . . 6 |- ( ( N e. NN /\ k e. ( 0 ... ( N - 1 ) ) ) -> k e. NN0 )
7	6	nn0cnd 11353	. . . . 5 |- ( ( N e. NN /\ k e. ( 0 ... ( N - 1 ) ) ) -> k e. CC )
8		id 22	. . . . 5 |- ( k = 0 -> k = 0 )
9	4, 7, 8	fsum1p 14482	. . . 4 |- ( N e. NN -> sum_ k e. ( 0 ... ( N - 1 ) ) k = ( 0 + sum_ k e. ( ( 0 + 1 ) ... ( N - 1 ) ) k ) )
10		1e0p1 11552	. . . . . . . . 9 |- 1 = ( 0 + 1 )
11	10	oveq1i 6660	. . . . . . . 8 |- ( 1 ... ( N - 1 ) ) = ( ( 0 + 1 ) ... ( N - 1 ) )
12	11	sumeq1i 14428	. . . . . . 7 |- sum_ k e. ( 1 ... ( N - 1 ) ) k = sum_ k e. ( ( 0 + 1 ) ... ( N - 1 ) ) k
13	12	oveq2i 6661	. . . . . 6 |- ( 0 + sum_ k e. ( 1 ... ( N - 1 ) ) k ) = ( 0 + sum_ k e. ( ( 0 + 1 ) ... ( N - 1 ) ) k )
14		fzfid 12772	. . . . . . . 8 |- ( N e. NN -> ( 1 ... ( N - 1 ) ) e. Fin )
15		elfznn 12370	. . . . . . . . . 10 |- ( k e. ( 1 ... ( N - 1 ) ) -> k e. NN )
16	15	adantl 482	. . . . . . . . 9 |- ( ( N e. NN /\ k e. ( 1 ... ( N - 1 ) ) ) -> k e. NN )
17	16	nncnd 11036	. . . . . . . 8 |- ( ( N e. NN /\ k e. ( 1 ... ( N - 1 ) ) ) -> k e. CC )
18	14, 17	fsumcl 14464	. . . . . . 7 |- ( N e. NN -> sum_ k e. ( 1 ... ( N - 1 ) ) k e. CC )
19	18	addid2d 10237	. . . . . 6 |- ( N e. NN -> ( 0 + sum_ k e. ( 1 ... ( N - 1 ) ) k ) = sum_ k e. ( 1 ... ( N - 1 ) ) k )
20	13, 19	syl5eqr 2670	. . . . 5 |- ( N e. NN -> ( 0 + sum_ k e. ( ( 0 + 1 ) ... ( N - 1 ) ) k ) = sum_ k e. ( 1 ... ( N - 1 ) ) k )
21		arisum 14592	. . . . . . 7 |- ( ( N - 1 ) e. NN0 -> sum_ k e. ( 1 ... ( N - 1 ) ) k = ( ( ( ( N - 1 ) ^ 2 ) + ( N - 1 ) ) / 2 ) )
22	2, 21	syl 17	. . . . . 6 |- ( N e. NN -> sum_ k e. ( 1 ... ( N - 1 ) ) k = ( ( ( ( N - 1 ) ^ 2 ) + ( N - 1 ) ) / 2 ) )
23		nncn 11028	. . . . . . . . . . . . . 14 |- ( N e. NN -> N e. CC )
24	23	2timesd 11275	. . . . . . . . . . . . 13 |- ( N e. NN -> ( 2 x. N ) = ( N + N ) )
25	24	oveq2d 6666	. . . . . . . . . . . 12 |- ( N e. NN -> ( ( N ^ 2 ) - ( 2 x. N ) ) = ( ( N ^ 2 ) - ( N + N ) ) )
26	23	sqcld 13006	. . . . . . . . . . . . 13 |- ( N e. NN -> ( N ^ 2 ) e. CC )
27	26, 23, 23	subsub4d 10423	. . . . . . . . . . . 12 |- ( N e. NN -> ( ( ( N ^ 2 ) - N ) - N ) = ( ( N ^ 2 ) - ( N + N ) ) )
28	25, 27	eqtr4d 2659	. . . . . . . . . . 11 |- ( N e. NN -> ( ( N ^ 2 ) - ( 2 x. N ) ) = ( ( ( N ^ 2 ) - N ) - N ) )
29	28	oveq1d 6665	. . . . . . . . . 10 |- ( N e. NN -> ( ( ( N ^ 2 ) - ( 2 x. N ) ) + 1 ) = ( ( ( ( N ^ 2 ) - N ) - N ) + 1 ) )
30		binom2sub1 12982	. . . . . . . . . . 11 |- ( N e. CC -> ( ( N - 1 ) ^ 2 ) = ( ( ( N ^ 2 ) - ( 2 x. N ) ) + 1 ) )
31	23, 30	syl 17	. . . . . . . . . 10 |- ( N e. NN -> ( ( N - 1 ) ^ 2 ) = ( ( ( N ^ 2 ) - ( 2 x. N ) ) + 1 ) )
32	26, 23	subcld 10392	. . . . . . . . . . 11 |- ( N e. NN -> ( ( N ^ 2 ) - N ) e. CC )
33		1cnd 10056	. . . . . . . . . . 11 |- ( N e. NN -> 1 e. CC )
34	32, 23, 33	subsubd 10420	. . . . . . . . . 10 |- ( N e. NN -> ( ( ( N ^ 2 ) - N ) - ( N - 1 ) ) = ( ( ( ( N ^ 2 ) - N ) - N ) + 1 ) )
35	29, 31, 34	3eqtr4d 2666	. . . . . . . . 9 |- ( N e. NN -> ( ( N - 1 ) ^ 2 ) = ( ( ( N ^ 2 ) - N ) - ( N - 1 ) ) )
36	35	oveq1d 6665	. . . . . . . 8 |- ( N e. NN -> ( ( ( N - 1 ) ^ 2 ) + ( N - 1 ) ) = ( ( ( ( N ^ 2 ) - N ) - ( N - 1 ) ) + ( N - 1 ) ) )
37		ax-1cn 9994	. . . . . . . . . 10 |- 1 e. CC
38		subcl 10280	. . . . . . . . . 10 |- ( ( N e. CC /\ 1 e. CC ) -> ( N - 1 ) e. CC )
39	23, 37, 38	sylancl 694	. . . . . . . . 9 |- ( N e. NN -> ( N - 1 ) e. CC )
40	32, 39	npcand 10396	. . . . . . . 8 |- ( N e. NN -> ( ( ( ( N ^ 2 ) - N ) - ( N - 1 ) ) + ( N - 1 ) ) = ( ( N ^ 2 ) - N ) )
41	36, 40	eqtrd 2656	. . . . . . 7 |- ( N e. NN -> ( ( ( N - 1 ) ^ 2 ) + ( N - 1 ) ) = ( ( N ^ 2 ) - N ) )
42	41	oveq1d 6665	. . . . . 6 |- ( N e. NN -> ( ( ( ( N - 1 ) ^ 2 ) + ( N - 1 ) ) / 2 ) = ( ( ( N ^ 2 ) - N ) / 2 ) )
43	22, 42	eqtrd 2656	. . . . 5 |- ( N e. NN -> sum_ k e. ( 1 ... ( N - 1 ) ) k = ( ( ( N ^ 2 ) - N ) / 2 ) )
44	20, 43	eqtrd 2656	. . . 4 |- ( N e. NN -> ( 0 + sum_ k e. ( ( 0 + 1 ) ... ( N - 1 ) ) k ) = ( ( ( N ^ 2 ) - N ) / 2 ) )
45	9, 44	eqtrd 2656	. . 3 |- ( N e. NN -> sum_ k e. ( 0 ... ( N - 1 ) ) k = ( ( ( N ^ 2 ) - N ) / 2 ) )
46		oveq1 6657	. . . . . . . 8 |- ( N = 0 -> ( N - 1 ) = ( 0 - 1 ) )
47	46	oveq2d 6666	. . . . . . 7 |- ( N = 0 -> ( 0 ... ( N - 1 ) ) = ( 0 ... ( 0 - 1 ) ) )
48		0re 10040	. . . . . . . . 9 |- 0 e. RR
49		ltm1 10863	. . . . . . . . 9 |- ( 0 e. RR -> ( 0 - 1 ) < 0 )
50	48, 49	ax-mp 5	. . . . . . . 8 |- ( 0 - 1 ) < 0
51		0z 11388	. . . . . . . . 9 |- 0 e. ZZ
52		peano2zm 11420	. . . . . . . . . 10 |- ( 0 e. ZZ -> ( 0 - 1 ) e. ZZ )
53	51, 52	ax-mp 5	. . . . . . . . 9 |- ( 0 - 1 ) e. ZZ
54		fzn 12357	. . . . . . . . 9 |- ( ( 0 e. ZZ /\ ( 0 - 1 ) e. ZZ ) -> ( ( 0 - 1 ) < 0 <-> ( 0 ... ( 0 - 1 ) ) = (/) ) )
55	51, 53, 54	mp2an 708	. . . . . . . 8 |- ( ( 0 - 1 ) < 0 <-> ( 0 ... ( 0 - 1 ) ) = (/) )
56	50, 55	mpbi 220	. . . . . . 7 |- ( 0 ... ( 0 - 1 ) ) = (/)
57	47, 56	syl6eq 2672	. . . . . 6 |- ( N = 0 -> ( 0 ... ( N - 1 ) ) = (/) )
58	57	sumeq1d 14431	. . . . 5 |- ( N = 0 -> sum_ k e. ( 0 ... ( N - 1 ) ) k = sum_ k e. (/) k )
59		sum0 14452	. . . . 5 |- sum_ k e. (/) k = 0
60	58, 59	syl6eq 2672	. . . 4 |- ( N = 0 -> sum_ k e. ( 0 ... ( N - 1 ) ) k = 0 )
61		sq0i 12956	. . . . . . . 8 |- ( N = 0 -> ( N ^ 2 ) = 0 )
62		id 22	. . . . . . . 8 |- ( N = 0 -> N = 0 )
63	61, 62	oveq12d 6668	. . . . . . 7 |- ( N = 0 -> ( ( N ^ 2 ) - N ) = ( 0 - 0 ) )
64		0m0e0 11130	. . . . . . 7 |- ( 0 - 0 ) = 0
65	63, 64	syl6eq 2672	. . . . . 6 |- ( N = 0 -> ( ( N ^ 2 ) - N ) = 0 )
66	65	oveq1d 6665	. . . . 5 |- ( N = 0 -> ( ( ( N ^ 2 ) - N ) / 2 ) = ( 0 / 2 ) )
67		2cn 11091	. . . . . 6 |- 2 e. CC
68		2ne0 11113	. . . . . 6 |- 2 =/= 0
69	67, 68	div0i 10759	. . . . 5 |- ( 0 / 2 ) = 0
70	66, 69	syl6eq 2672	. . . 4 |- ( N = 0 -> ( ( ( N ^ 2 ) - N ) / 2 ) = 0 )
71	60, 70	eqtr4d 2659	. . 3 |- ( N = 0 -> sum_ k e. ( 0 ... ( N - 1 ) ) k = ( ( ( N ^ 2 ) - N ) / 2 ) )
72	45, 71	jaoi 394	. 2 |- ( ( N e. NN \/ N = 0 ) -> sum_ k e. ( 0 ... ( N - 1 ) ) k = ( ( ( N ^ 2 ) - N ) / 2 ) )
73	1, 72	sylbi 207	1 |- ( N e. NN0 -> sum_ k e. ( 0 ... ( N - 1 ) ) k = ( ( ( N ^ 2 ) - N ) / 2 ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   (/)c0 3915   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   ^cexp 12860   sum_csu 14416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417

This theorem is referenced by:  birthdaylem3  24680