arisum2 - Intuitionistic Logic Explorer

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Theorem arisum2 11491

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 9167	. 2 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		nnm1nn0 9206	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
3		nn0uz 9551	. . . . . 6 ⊢ ℕ₀ = (ℤ_≥‘0)
4	2, 3	eleqtrdi 2270	. . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ (ℤ_≥‘0))
5		elfznn0 10100	. . . . . . 7 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ₀)
6	5	adantl 277	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℕ₀)
7	6	nn0cnd 9220	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℂ)
8		id 19	. . . . 5 ⊢ (𝑘 = 0 → 𝑘 = 0)
9	4, 7, 8	fsum1p 11410	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = (0 + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))𝑘))
10		1e0p1 9414	. . . . . . . . 9 ⊢ 1 = (0 + 1)
11	10	oveq1i 5879	. . . . . . . 8 ⊢ (1...(𝑁 − 1)) = ((0 + 1)...(𝑁 − 1))
12	11	sumeq1i 11355	. . . . . . 7 ⊢ Σ𝑘 ∈ (1...(𝑁 − 1))𝑘 = Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))𝑘
13	12	oveq2i 5880	. . . . . 6 ⊢ (0 + Σ𝑘 ∈ (1...(𝑁 − 1))𝑘) = (0 + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))𝑘)
14		1zzd 9269	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℤ)
15	2	nn0zd 9362	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℤ)
16	14, 15	fzfigd 10417	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (1...(𝑁 − 1)) ∈ Fin)
17		elfznn 10040	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...(𝑁 − 1)) → 𝑘 ∈ ℕ)
18	17	adantl 277	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑘 ∈ ℕ)
19	18	nncnd 8922	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑘 ∈ ℂ)
20	16, 19	fsumcl 11392	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (1...(𝑁 − 1))𝑘 ∈ ℂ)
21	20	addid2d 8097	. . . . . 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 11490	. . . . . . 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 8916	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
26	25	2timesd 9150	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (2 · 𝑁) = (𝑁 + 𝑁))
27	26	oveq2d 5885	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → ((𝑁↑2) − (2 · 𝑁)) = ((𝑁↑2) − (𝑁 + 𝑁)))
28	25	sqcld 10637	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (𝑁↑2) ∈ ℂ)
29	28, 25, 25	subsub4d 8289	. . . . . . . . . . . 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 5884	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (((𝑁↑2) − (2 · 𝑁)) + 1) = ((((𝑁↑2) − 𝑁) − 𝑁) + 1))
32		binom2sub1 10620	. . . . . . . . . . 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 8258	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → ((𝑁↑2) − 𝑁) ∈ ℂ)
35		1cnd 7964	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ)
36	34, 25, 35	subsubd 8286	. . . . . . . . . 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 5884	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (((𝑁 − 1)↑2) + (𝑁 − 1)) = ((((𝑁↑2) − 𝑁) − (𝑁 − 1)) + (𝑁 − 1)))
39		ax-1cn 7895	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
40		subcl 8146	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑁 − 1) ∈ ℂ)
41	25, 39, 40	sylancl 413	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℂ)
42	34, 41	npcand 8262	. . . . . . . 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 5884	. . . . . 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 5876	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑁 − 1) = (0 − 1))
49	48	oveq2d 5885	. . . . . . 7 ⊢ (𝑁 = 0 → (0...(𝑁 − 1)) = (0...(0 − 1)))
50		0re 7948	. . . . . . . . 9 ⊢ 0 ∈ ℝ
51		ltm1 8792	. . . . . . . . 9 ⊢ (0 ∈ ℝ → (0 − 1) < 0)
52	50, 51	ax-mp 5	. . . . . . . 8 ⊢ (0 − 1) < 0
53		0z 9253	. . . . . . . . 9 ⊢ 0 ∈ ℤ
54		peano2zm 9280	. . . . . . . . . 10 ⊢ (0 ∈ ℤ → (0 − 1) ∈ ℤ)
55	53, 54	ax-mp 5	. . . . . . . . 9 ⊢ (0 − 1) ∈ ℤ
56		fzn 10028	. . . . . . . . 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 11358	. . . . 5 ⊢ (𝑁 = 0 → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = Σ𝑘 ∈ ∅ 𝑘)
61		sum0 11380	. . . . 5 ⊢ Σ𝑘 ∈ ∅ 𝑘 = 0
62	60, 61	eqtrdi 2226	. . . 4 ⊢ (𝑁 = 0 → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = 0)
63		sq0i 10597	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑁↑2) = 0)
64		id 19	. . . . . . . 8 ⊢ (𝑁 = 0 → 𝑁 = 0)
65	63, 64	oveq12d 5887	. . . . . . 7 ⊢ (𝑁 = 0 → ((𝑁↑2) − 𝑁) = (0 − 0))
66		0m0e0 9020	. . . . . . 7 ⊢ (0 − 0) = 0
67	65, 66	eqtrdi 2226	. . . . . 6 ⊢ (𝑁 = 0 → ((𝑁↑2) − 𝑁) = 0)
68	67	oveq1d 5884	. . . . 5 ⊢ (𝑁 = 0 → (((𝑁↑2) − 𝑁) / 2) = (0 / 2))
69		2cn 8979	. . . . . 6 ⊢ 2 ∈ ℂ
70		2ap0 9001	. . . . . 6 ⊢ 2 # 0
71	69, 70	div0api 8692	. . . . 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 4000 ‘cfv 5212 (class class class)co 5869 ℂcc 7800 ℝcr 7801 0cc0 7802 1c1 7803 + caddc 7805 · cmul 7807 < clt 7982 − cmin 8118 / cdiv 8618 ℕcn 8908 2c2 8959 ℕ₀cn0 9165 ℤcz 9242 ℤ_≥cuz 9517 ...cfz 9995 ↑cexp 10505 Σcsu 11345

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 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 ax-arch 7921 ax-caucvg 7922

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 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-isom 5221 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-irdg 6365 df-frec 6386 df-1o 6411 df-oadd 6415 df-er 6529 df-en 6735 df-dom 6736 df-fin 6737 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-inn 8909 df-2 8967 df-3 8968 df-4 8969 df-n0 9166 df-z 9243 df-uz 9518 df-q 9609 df-rp 9641 df-fz 9996 df-fzo 10129 df-seqfrec 10432 df-exp 10506 df-fac 10690 df-bc 10712 df-ihash 10740 df-cj 10835 df-re 10836 df-im 10837 df-rsqrt 10991 df-abs 10992 df-clim 11271 df-sumdc 11346

This theorem is referenced by: (None)

W3C validator