arisum2 - Intuitionistic Logic Explorer

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Theorem arisum2 12291

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 9576	. 2 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		nnm1nn0 9615	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
3		nn0uz 9973	. . . . . 6 ⊢ ℕ₀ = (ℤ_≥‘0)
4	2, 3	eleqtrdi 2331	. . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ (ℤ_≥‘0))
5		elfznn0 10538	. . . . . . 7 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ₀)
6	5	adantl 277	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℕ₀)
7	6	nn0cnd 9633	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℂ)
8		id 19	. . . . 5 ⊢ (𝑘 = 0 → 𝑘 = 0)
9	4, 7, 8	fsum1p 12210	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = (0 + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))𝑘))
10		1e0p1 9834	. . . . . . . . 9 ⊢ 1 = (0 + 1)
11	10	oveq1i 6098	. . . . . . . 8 ⊢ (1...(𝑁 − 1)) = ((0 + 1)...(𝑁 − 1))
12	11	sumeq1i 12154	. . . . . . 7 ⊢ Σ𝑘 ∈ (1...(𝑁 − 1))𝑘 = Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))𝑘
13	12	oveq2i 6099	. . . . . 6 ⊢ (0 + Σ𝑘 ∈ (1...(𝑁 − 1))𝑘) = (0 + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))𝑘)
14		1zzd 9682	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℤ)
15	2	nn0zd 9777	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℤ)
16	14, 15	fzfigd 10889	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (1...(𝑁 − 1)) ∈ Fin)
17		elfznn 10477	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...(𝑁 − 1)) → 𝑘 ∈ ℕ)
18	17	adantl 277	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑘 ∈ ℕ)
19	18	nncnd 9327	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑘 ∈ ℂ)
20	16, 19	fsumcl 12192	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (1...(𝑁 − 1))𝑘 ∈ ℂ)
21	20	addlidd 8483	. . . . . 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 12290	. . . . . . 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 9321	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
26	25	2timesd 9559	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (2 · 𝑁) = (𝑁 + 𝑁))
27	26	oveq2d 6104	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → ((𝑁↑2) − (2 · 𝑁)) = ((𝑁↑2) − (𝑁 + 𝑁)))
28	25	sqcld 11130	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (𝑁↑2) ∈ ℂ)
29	28, 25, 25	subsub4d 8675	. . . . . . . . . . . 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 6103	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (((𝑁↑2) − (2 · 𝑁)) + 1) = ((((𝑁↑2) − 𝑁) − 𝑁) + 1))
32		binom2sub1 11112	. . . . . . . . . . 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 8644	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → ((𝑁↑2) − 𝑁) ∈ ℂ)
35		1cnd 8348	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ)
36	34, 25, 35	subsubd 8672	. . . . . . . . . 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 6103	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (((𝑁 − 1)↑2) + (𝑁 − 1)) = ((((𝑁↑2) − 𝑁) − (𝑁 − 1)) + (𝑁 − 1)))
39		ax-1cn 8278	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
40		subcl 8532	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑁 − 1) ∈ ℂ)
41	25, 39, 40	sylancl 417	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℂ)
42	34, 41	npcand 8648	. . . . . . . 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 6103	. . . . . 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 6095	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑁 − 1) = (0 − 1))
49	48	oveq2d 6104	. . . . . . 7 ⊢ (𝑁 = 0 → (0...(𝑁 − 1)) = (0...(0 − 1)))
50		0re 8332	. . . . . . . . 9 ⊢ 0 ∈ ℝ
51		ltm1 9184	. . . . . . . . 9 ⊢ (0 ∈ ℝ → (0 − 1) < 0)
52	50, 51	ax-mp 5	. . . . . . . 8 ⊢ (0 − 1) < 0
53		0z 9666	. . . . . . . . 9 ⊢ 0 ∈ ℤ
54		peano2zm 9693	. . . . . . . . . 10 ⊢ (0 ∈ ℤ → (0 − 1) ∈ ℤ)
55	53, 54	ax-mp 5	. . . . . . . . 9 ⊢ (0 − 1) ∈ ℤ
56		fzn 10463	. . . . . . . . 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 12157	. . . . 5 ⊢ (𝑁 = 0 → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = Σ𝑘 ∈ ∅ 𝑘)
61		sum0 12180	. . . . 5 ⊢ Σ𝑘 ∈ ∅ 𝑘 = 0
62	60, 61	eqtrdi 2287	. . . 4 ⊢ (𝑁 = 0 → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = 0)
63		sq0i 11089	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑁↑2) = 0)
64		id 19	. . . . . . . 8 ⊢ (𝑁 = 0 → 𝑁 = 0)
65	63, 64	oveq12d 6106	. . . . . . 7 ⊢ (𝑁 = 0 → ((𝑁↑2) − 𝑁) = (0 − 0))
66		0m0e0 9425	. . . . . . 7 ⊢ (0 − 0) = 0
67	65, 66	eqtrdi 2287	. . . . . 6 ⊢ (𝑁 = 0 → ((𝑁↑2) − 𝑁) = 0)
68	67	oveq1d 6103	. . . . 5 ⊢ (𝑁 = 0 → (((𝑁↑2) − 𝑁) / 2) = (0 / 2))
69		2cn 9384	. . . . . 6 ⊢ 2 ∈ ℂ
70		2ap0 9406	. . . . . 6 ⊢ 2 # 0
71	69, 70	div0api 9084	. . . . 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 4131 ‘cfv 5380 (class class class)co 6088 ℂcc 8183 ℝcr 8184 0cc0 8185 1c1 8186 + caddc 8188 · cmul 8190 < clt 8366 − cmin 8504 / cdiv 9010 ℕcn 9313 2c2 9364 ℕ₀cn0 9574 ℤcz 9655 ℤ_≥cuz 9937 ...cfz 10428 ↑cexp 10996 Σcsu 12144

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 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-mulrcl 8284 ax-addcom 8285 ax-mulcom 8286 ax-addass 8287 ax-mulass 8288 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-1rid 8292 ax-0id 8293 ax-rnegex 8294 ax-precex 8295 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-ltwlin 8298 ax-pre-lttrn 8299 ax-pre-apti 8300 ax-pre-ltadd 8301 ax-pre-mulgt0 8302 ax-pre-mulext 8303 ax-arch 8304 ax-caucvg 8305

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 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-isom 5389 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-irdg 6645 df-frec 6666 df-1o 6691 df-oadd 6695 df-er 6811 df-en 7027 df-dom 7028 df-fin 7029 df-pnf 8368 df-mnf 8369 df-xr 8370 df-ltxr 8371 df-le 8372 df-sub 8506 df-neg 8507 df-reap 8911 df-ap 8918 df-div 9011 df-inn 9314 df-2 9372 df-3 9373 df-4 9374 df-n0 9575 df-z 9656 df-uz 9938 df-q 10036 df-rp 10072 df-fz 10429 df-fzo 10567 df-seqfrec 10906 df-exp 10997 df-fac 11186 df-bc 11208 df-ihash 11237 df-cj 11629 df-re 11630 df-im 11631 df-rsqrt 11786 df-abs 11787 df-clim 12070 df-sumdc 12145

This theorem is used by: birthdaylem3 16227

W3C validator