arisum2 - Intuitionistic Logic Explorer

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Theorem arisum2 12266

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 9565	. 2 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		nnm1nn0 9604	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
3		nn0uz 9957	. . . . . 6 ⊢ ℕ₀ = (ℤ_≥‘0)
4	2, 3	eleqtrdi 2331	. . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ (ℤ_≥‘0))
5		elfznn0 10521	. . . . . . 7 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ₀)
6	5	adantl 277	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℕ₀)
7	6	nn0cnd 9622	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℂ)
8		id 19	. . . . 5 ⊢ (𝑘 = 0 → 𝑘 = 0)
9	4, 7, 8	fsum1p 12185	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = (0 + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))𝑘))
10		1e0p1 9818	. . . . . . . . 9 ⊢ 1 = (0 + 1)
11	10	oveq1i 6095	. . . . . . . 8 ⊢ (1...(𝑁 − 1)) = ((0 + 1)...(𝑁 − 1))
12	11	sumeq1i 12129	. . . . . . 7 ⊢ Σ𝑘 ∈ (1...(𝑁 − 1))𝑘 = Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))𝑘
13	12	oveq2i 6096	. . . . . 6 ⊢ (0 + Σ𝑘 ∈ (1...(𝑁 − 1))𝑘) = (0 + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))𝑘)
14		1zzd 9671	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℤ)
15	2	nn0zd 9766	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℤ)
16	14, 15	fzfigd 10868	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (1...(𝑁 − 1)) ∈ Fin)
17		elfznn 10460	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...(𝑁 − 1)) → 𝑘 ∈ ℕ)
18	17	adantl 277	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑘 ∈ ℕ)
19	18	nncnd 9318	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑘 ∈ ℂ)
20	16, 19	fsumcl 12167	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (1...(𝑁 − 1))𝑘 ∈ ℂ)
21	20	addlidd 8476	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (0 + Σ𝑘 ∈ (1...(𝑁 − 1))𝑘) = Σ𝑘 ∈ (1...(𝑁 − 1))𝑘)
22	13, 21	eqtr3id 2285	. . . . 5 ⊢ (𝑁 ∈ ℕ → (0 + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))𝑘) = Σ𝑘 ∈ (1...(𝑁 − 1))𝑘)
23		arisum 12265	. . . . . . 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 9312	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
26	25	2timesd 9548	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (2 · 𝑁) = (𝑁 + 𝑁))
27	26	oveq2d 6101	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → ((𝑁↑2) − (2 · 𝑁)) = ((𝑁↑2) − (𝑁 + 𝑁)))
28	25	sqcld 11109	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (𝑁↑2) ∈ ℂ)
29	28, 25, 25	subsub4d 8668	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (((𝑁↑2) − 𝑁) − 𝑁) = ((𝑁↑2) − (𝑁 + 𝑁)))
30	27, 29	eqtr4d 2274	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → ((𝑁↑2) − (2 · 𝑁)) = (((𝑁↑2) − 𝑁) − 𝑁))
31	30	oveq1d 6100	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (((𝑁↑2) − (2 · 𝑁)) + 1) = ((((𝑁↑2) − 𝑁) − 𝑁) + 1))
32		binom2sub1 11091	. . . . . . . . . . 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 8637	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → ((𝑁↑2) − 𝑁) ∈ ℂ)
35		1cnd 8342	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ)
36	34, 25, 35	subsubd 8665	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (((𝑁↑2) − 𝑁) − (𝑁 − 1)) = ((((𝑁↑2) − 𝑁) − 𝑁) + 1))
37	31, 33, 36	3eqtr4d 2281	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1)↑2) = (((𝑁↑2) − 𝑁) − (𝑁 − 1)))
38	37	oveq1d 6100	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (((𝑁 − 1)↑2) + (𝑁 − 1)) = ((((𝑁↑2) − 𝑁) − (𝑁 − 1)) + (𝑁 − 1)))
39		ax-1cn 8272	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
40		subcl 8525	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑁 − 1) ∈ ℂ)
41	25, 39, 40	sylancl 417	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℂ)
42	34, 41	npcand 8641	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ((((𝑁↑2) − 𝑁) − (𝑁 − 1)) + (𝑁 − 1)) = ((𝑁↑2) − 𝑁))
43	38, 42	eqtrd 2271	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (((𝑁 − 1)↑2) + (𝑁 − 1)) = ((𝑁↑2) − 𝑁))
44	43	oveq1d 6100	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ((((𝑁 − 1)↑2) + (𝑁 − 1)) / 2) = (((𝑁↑2) − 𝑁) / 2))
45	24, 44	eqtrd 2271	. . . . 5 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (1...(𝑁 − 1))𝑘 = (((𝑁↑2) − 𝑁) / 2))
46	22, 45	eqtrd 2271	. . . 4 ⊢ (𝑁 ∈ ℕ → (0 + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))𝑘) = (((𝑁↑2) − 𝑁) / 2))
47	9, 46	eqtrd 2271	. . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = (((𝑁↑2) − 𝑁) / 2))
48		oveq1 6092	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑁 − 1) = (0 − 1))
49	48	oveq2d 6101	. . . . . . 7 ⊢ (𝑁 = 0 → (0...(𝑁 − 1)) = (0...(0 − 1)))
50		0re 8326	. . . . . . . . 9 ⊢ 0 ∈ ℝ
51		ltm1 9176	. . . . . . . . 9 ⊢ (0 ∈ ℝ → (0 − 1) < 0)
52	50, 51	ax-mp 5	. . . . . . . 8 ⊢ (0 − 1) < 0
53		0z 9655	. . . . . . . . 9 ⊢ 0 ∈ ℤ
54		peano2zm 9682	. . . . . . . . . 10 ⊢ (0 ∈ ℤ → (0 − 1) ∈ ℤ)
55	53, 54	ax-mp 5	. . . . . . . . 9 ⊢ (0 − 1) ∈ ℤ
56		fzn 10446	. . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ (0 − 1) ∈ ℤ) → ((0 − 1) < 0 ↔ (0...(0 − 1)) = ∅))
57	53, 55, 56	mp2an 430	. . . . . . . 8 ⊢ ((0 − 1) < 0 ↔ (0...(0 − 1)) = ∅)
58	52, 57	mpbi 145	. . . . . . 7 ⊢ (0...(0 − 1)) = ∅
59	49, 58	eqtrdi 2287	. . . . . 6 ⊢ (𝑁 = 0 → (0...(𝑁 − 1)) = ∅)
60	59	sumeq1d 12132	. . . . 5 ⊢ (𝑁 = 0 → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = Σ𝑘 ∈ ∅ 𝑘)
61		sum0 12155	. . . . 5 ⊢ Σ𝑘 ∈ ∅ 𝑘 = 0
62	60, 61	eqtrdi 2287	. . . 4 ⊢ (𝑁 = 0 → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = 0)
63		sq0i 11068	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑁↑2) = 0)
64		id 19	. . . . . . . 8 ⊢ (𝑁 = 0 → 𝑁 = 0)
65	63, 64	oveq12d 6103	. . . . . . 7 ⊢ (𝑁 = 0 → ((𝑁↑2) − 𝑁) = (0 − 0))
66		0m0e0 9416	. . . . . . 7 ⊢ (0 − 0) = 0
67	65, 66	eqtrdi 2287	. . . . . 6 ⊢ (𝑁 = 0 → ((𝑁↑2) − 𝑁) = 0)
68	67	oveq1d 6100	. . . . 5 ⊢ (𝑁 = 0 → (((𝑁↑2) − 𝑁) / 2) = (0 / 2))
69		2cn 9375	. . . . . 6 ⊢ 2 ∈ ℂ
70		2ap0 9397	. . . . . 6 ⊢ 2 # 0
71	69, 70	div0api 9076	. . . . 5 ⊢ (0 / 2) = 0
72	68, 71	eqtrdi 2287	. . . 4 ⊢ (𝑁 = 0 → (((𝑁↑2) − 𝑁) / 2) = 0)
73	62, 72	eqtr4d 2274	. . 3 ⊢ (𝑁 = 0 → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = (((𝑁↑2) − 𝑁) / 2))
74	47, 73	jaoi 728	. 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

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 720 = wceq 1402 ∈ wcel 2209 ∅c0 3520 class class class wbr 4130 ‘cfv 5377 (class class class)co 6085 ℂcc 8177 ℝcr 8178 0cc0 8179 1c1 8180 + caddc 8182 · cmul 8184 < clt 8360 − cmin 8497 / cdiv 9002 ℕcn 9304 2c2 9355 ℕ₀cn0 9563 ℤcz 9644 ℤ_≥cuz 9921 ...cfz 10411 ↑cexp 10975 Σcsu 12119

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297 ax-arch 8298 ax-caucvg 8299

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-isom 5386 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-frec 6662 df-1o 6687 df-oadd 6691 df-er 6807 df-en 7023 df-dom 7024 df-fin 7025 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8903 df-ap 8910 df-div 9003 df-inn 9305 df-2 9363 df-3 9364 df-4 9365 df-n0 9564 df-z 9645 df-uz 9922 df-q 10020 df-rp 10055 df-fz 10412 df-fzo 10550 df-seqfrec 10885 df-exp 10976 df-fac 11164 df-bc 11186 df-ihash 11215 df-cj 11607 df-re 11608 df-im 11609 df-rsqrt 11764 df-abs 11765 df-clim 12045 df-sumdc 12120

This theorem is used by: birthdaylem3 16089

W3C validator