arisum2 - Intuitionistic Logic Explorer

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Theorem arisum2 11506

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 ∈ ℕ₀ → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = (((𝑁↑2) − 𝑁) / 2))

Distinct variable group: 𝑘,𝑁

**Proof of Theorem arisum2**
Step	Hyp	Ref	Expression
1		elnn0 9177	. 2 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		nnm1nn0 9216	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
3		nn0uz 9561	. . . . . 6 ⊢ ℕ₀ = (ℤ_≥‘0)
4	2, 3	eleqtrdi 2270	. . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ (ℤ_≥‘0))
5		elfznn0 10113	. . . . . . 7 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ₀)
6	5	adantl 277	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℕ₀)
7	6	nn0cnd 9230	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℂ)
8		id 19	. . . . 5 ⊢ (𝑘 = 0 → 𝑘 = 0)
9	4, 7, 8	fsum1p 11425	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = (0 + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))𝑘))
10		1e0p1 9424	. . . . . . . . 9 ⊢ 1 = (0 + 1)
11	10	oveq1i 5884	. . . . . . . 8 ⊢ (1...(𝑁 − 1)) = ((0 + 1)...(𝑁 − 1))
12	11	sumeq1i 11370	. . . . . . 7 ⊢ Σ𝑘 ∈ (1...(𝑁 − 1))𝑘 = Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))𝑘
13	12	oveq2i 5885	. . . . . 6 ⊢ (0 + Σ𝑘 ∈ (1...(𝑁 − 1))𝑘) = (0 + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))𝑘)
14		1zzd 9279	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℤ)
15	2	nn0zd 9372	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℤ)
16	14, 15	fzfigd 10430	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (1...(𝑁 − 1)) ∈ Fin)
17		elfznn 10053	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...(𝑁 − 1)) → 𝑘 ∈ ℕ)
18	17	adantl 277	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑘 ∈ ℕ)
19	18	nncnd 8932	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑘 ∈ ℂ)
20	16, 19	fsumcl 11407	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (1...(𝑁 − 1))𝑘 ∈ ℂ)
21	20	addid2d 8106	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (0 + Σ𝑘 ∈ (1...(𝑁 − 1))𝑘) = Σ𝑘 ∈ (1...(𝑁 − 1))𝑘)
22	13, 21	eqtr3id 2224	. . . . 5 ⊢ (𝑁 ∈ ℕ → (0 + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))𝑘) = Σ𝑘 ∈ (1...(𝑁 − 1))𝑘)
23		arisum 11505	. . . . . . 7 ⊢ ((𝑁 − 1) ∈ ℕ₀ → Σ𝑘 ∈ (1...(𝑁 − 1))𝑘 = ((((𝑁 − 1)↑2) + (𝑁 − 1)) / 2))
24	2, 23	syl 14	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (1...(𝑁 − 1))𝑘 = ((((𝑁 − 1)↑2) + (𝑁 − 1)) / 2))
25		nncn 8926	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
26	25	2timesd 9160	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (2 · 𝑁) = (𝑁 + 𝑁))
27	26	oveq2d 5890	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → ((𝑁↑2) − (2 · 𝑁)) = ((𝑁↑2) − (𝑁 + 𝑁)))
28	25	sqcld 10651	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (𝑁↑2) ∈ ℂ)
29	28, 25, 25	subsub4d 8298	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (((𝑁↑2) − 𝑁) − 𝑁) = ((𝑁↑2) − (𝑁 + 𝑁)))
30	27, 29	eqtr4d 2213	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → ((𝑁↑2) − (2 · 𝑁)) = (((𝑁↑2) − 𝑁) − 𝑁))
31	30	oveq1d 5889	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (((𝑁↑2) − (2 · 𝑁)) + 1) = ((((𝑁↑2) − 𝑁) − 𝑁) + 1))
32		binom2sub1 10634	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1)↑2) = (((𝑁↑2) − (2 · 𝑁)) + 1))
33	25, 32	syl 14	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1)↑2) = (((𝑁↑2) − (2 · 𝑁)) + 1))
34	28, 25	subcld 8267	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → ((𝑁↑2) − 𝑁) ∈ ℂ)
35		1cnd 7972	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ)
36	34, 25, 35	subsubd 8295	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (((𝑁↑2) − 𝑁) − (𝑁 − 1)) = ((((𝑁↑2) − 𝑁) − 𝑁) + 1))
37	31, 33, 36	3eqtr4d 2220	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1)↑2) = (((𝑁↑2) − 𝑁) − (𝑁 − 1)))
38	37	oveq1d 5889	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (((𝑁 − 1)↑2) + (𝑁 − 1)) = ((((𝑁↑2) − 𝑁) − (𝑁 − 1)) + (𝑁 − 1)))
39		ax-1cn 7903	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
40		subcl 8155	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑁 − 1) ∈ ℂ)
41	25, 39, 40	sylancl 413	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℂ)
42	34, 41	npcand 8271	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ((((𝑁↑2) − 𝑁) − (𝑁 − 1)) + (𝑁 − 1)) = ((𝑁↑2) − 𝑁))
43	38, 42	eqtrd 2210	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (((𝑁 − 1)↑2) + (𝑁 − 1)) = ((𝑁↑2) − 𝑁))
44	43	oveq1d 5889	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ((((𝑁 − 1)↑2) + (𝑁 − 1)) / 2) = (((𝑁↑2) − 𝑁) / 2))
45	24, 44	eqtrd 2210	. . . . 5 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (1...(𝑁 − 1))𝑘 = (((𝑁↑2) − 𝑁) / 2))
46	22, 45	eqtrd 2210	. . . 4 ⊢ (𝑁 ∈ ℕ → (0 + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))𝑘) = (((𝑁↑2) − 𝑁) / 2))
47	9, 46	eqtrd 2210	. . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = (((𝑁↑2) − 𝑁) / 2))
48		oveq1 5881	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑁 − 1) = (0 − 1))
49	48	oveq2d 5890	. . . . . . 7 ⊢ (𝑁 = 0 → (0...(𝑁 − 1)) = (0...(0 − 1)))
50		0re 7956	. . . . . . . . 9 ⊢ 0 ∈ ℝ
51		ltm1 8802	. . . . . . . . 9 ⊢ (0 ∈ ℝ → (0 − 1) < 0)
52	50, 51	ax-mp 5	. . . . . . . 8 ⊢ (0 − 1) < 0
53		0z 9263	. . . . . . . . 9 ⊢ 0 ∈ ℤ
54		peano2zm 9290	. . . . . . . . . 10 ⊢ (0 ∈ ℤ → (0 − 1) ∈ ℤ)
55	53, 54	ax-mp 5	. . . . . . . . 9 ⊢ (0 − 1) ∈ ℤ
56		fzn 10041	. . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ (0 − 1) ∈ ℤ) → ((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 2226	. . . . . 6 ⊢ (𝑁 = 0 → (0...(𝑁 − 1)) = ∅)
60	59	sumeq1d 11373	. . . . 5 ⊢ (𝑁 = 0 → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = Σ𝑘 ∈ ∅ 𝑘)
61		sum0 11395	. . . . 5 ⊢ Σ𝑘 ∈ ∅ 𝑘 = 0
62	60, 61	eqtrdi 2226	. . . 4 ⊢ (𝑁 = 0 → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = 0)
63		sq0i 10611	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑁↑2) = 0)
64		id 19	. . . . . . . 8 ⊢ (𝑁 = 0 → 𝑁 = 0)
65	63, 64	oveq12d 5892	. . . . . . 7 ⊢ (𝑁 = 0 → ((𝑁↑2) − 𝑁) = (0 − 0))
66		0m0e0 9030	. . . . . . 7 ⊢ (0 − 0) = 0
67	65, 66	eqtrdi 2226	. . . . . 6 ⊢ (𝑁 = 0 → ((𝑁↑2) − 𝑁) = 0)
68	67	oveq1d 5889	. . . . 5 ⊢ (𝑁 = 0 → (((𝑁↑2) − 𝑁) / 2) = (0 / 2))
69		2cn 8989	. . . . . 6 ⊢ 2 ∈ ℂ
70		2ap0 9011	. . . . . 6 ⊢ 2 # 0
71	69, 70	div0api 8702	. . . . 5 ⊢ (0 / 2) = 0
72	68, 71	eqtrdi 2226	. . . 4 ⊢ (𝑁 = 0 → (((𝑁↑2) − 𝑁) / 2) = 0)
73	62, 72	eqtr4d 2213	. . 3 ⊢ (𝑁 = 0 → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = (((𝑁↑2) − 𝑁) / 2))
74	47, 73	jaoi 716	. 2 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = (((𝑁↑2) − 𝑁) / 2))
75	1, 74	sylbi 121	1 ⊢ (𝑁 ∈ ℕ₀ → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = (((𝑁↑2) − 𝑁) / 2))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 708 = wceq 1353 ∈ wcel 2148 ∅c0 3422 class class class wbr 4003 ‘cfv 5216 (class class class)co 5874 ℂcc 7808 ℝcr 7809 0cc0 7810 1c1 7811 + caddc 7813 · cmul 7815 < clt 7991 − cmin 8127 / cdiv 8628 ℕcn 8918 2c2 8969 ℕ₀cn0 9175 ℤcz 9252 ℤ_≥cuz 9527 ...cfz 10007 ↑cexp 10518 Σcsu 11360

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 ax-arch 7929 ax-caucvg 7930

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-isom 5225 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-irdg 6370 df-frec 6391 df-1o 6416 df-oadd 6420 df-er 6534 df-en 6740 df-dom 6741 df-fin 6742 df-pnf 7993 df-mnf 7994 df-xr 7995 df-ltxr 7996 df-le 7997 df-sub 8129 df-neg 8130 df-reap 8531 df-ap 8538 df-div 8629 df-inn 8919 df-2 8977 df-3 8978 df-4 8979 df-n0 9176 df-z 9253 df-uz 9528 df-q 9619 df-rp 9653 df-fz 10008 df-fzo 10142 df-seqfrec 10445 df-exp 10519 df-fac 10705 df-bc 10727 df-ihash 10755 df-cj 10850 df-re 10851 df-im 10852 df-rsqrt 11006 df-abs 11007 df-clim 11286 df-sumdc 11361

This theorem is referenced by: (None)

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