arisum2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > arisum2		Structured version Visualization version GIF version

Theorem arisum2 15911

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 12501	. 2 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		nnm1nn0 12540	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
3		nn0uz 12895	. . . . . 6 ⊢ ℕ₀ = (ℤ_≥‘0)
4	2, 3	eleqtrdi 2873	. . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ (ℤ_≥‘0))
5		elfznn0 13644	. . . . . . 7 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ₀)
6	5	adantl 486	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℕ₀)
7	6	nn0cnd 12562	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℂ)
8		id 23	. . . . 5 ⊢ (𝑘 = 0 → 𝑘 = 0)
9	4, 7, 8	fsum1p 15800	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = (0 + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))𝑘))
10		1e0p1 12753	. . . . . . . . 9 ⊢ 1 = (0 + 1)
11	10	oveq1i 7420	. . . . . . . 8 ⊢ (1...(𝑁 − 1)) = ((0 + 1)...(𝑁 − 1))
12	11	sumeq1i 15744	. . . . . . 7 ⊢ Σ𝑘 ∈ (1...(𝑁 − 1))𝑘 = Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))𝑘
13	12	oveq2i 7421	. . . . . 6 ⊢ (0 + Σ𝑘 ∈ (1...(𝑁 − 1))𝑘) = (0 + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))𝑘)
14		fzfid 14005	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (1...(𝑁 − 1)) ∈ Fin)
15		elfznn 13577	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...(𝑁 − 1)) → 𝑘 ∈ ℕ)
16	15	adantl 486	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑘 ∈ ℕ)
17	16	nncnd 12244	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑘 ∈ ℂ)
18	14, 17	fsumcl 15780	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (1...(𝑁 − 1))𝑘 ∈ ℂ)
19	18	addlidd 11406	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (0 + Σ𝑘 ∈ (1...(𝑁 − 1))𝑘) = Σ𝑘 ∈ (1...(𝑁 − 1))𝑘)
20	13, 19	eqtr3id 2812	. . . . 5 ⊢ (𝑁 ∈ ℕ → (0 + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))𝑘) = Σ𝑘 ∈ (1...(𝑁 − 1))𝑘)
21		arisum 15910	. . . . . . 7 ⊢ ((𝑁 − 1) ∈ ℕ₀ → Σ𝑘 ∈ (1...(𝑁 − 1))𝑘 = ((((𝑁 − 1)↑2) + (𝑁 − 1)) / 2))
22	2, 21	syl 18	. . . . . 6 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (1...(𝑁 − 1))𝑘 = ((((𝑁 − 1)↑2) + (𝑁 − 1)) / 2))
23		nncn 12236	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
24	23	2timesd 12482	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (2 · 𝑁) = (𝑁 + 𝑁))
25	24	oveq2d 7426	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → ((𝑁↑2) − (2 · 𝑁)) = ((𝑁↑2) − (𝑁 + 𝑁)))
26	23	sqcld 14176	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (𝑁↑2) ∈ ℂ)
27	26, 23, 23	subsub4d 11595	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (((𝑁↑2) − 𝑁) − 𝑁) = ((𝑁↑2) − (𝑁 + 𝑁)))
28	25, 27	eqtr4d 2801	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → ((𝑁↑2) − (2 · 𝑁)) = (((𝑁↑2) − 𝑁) − 𝑁))
29	28	oveq1d 7425	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (((𝑁↑2) − (2 · 𝑁)) + 1) = ((((𝑁↑2) − 𝑁) − 𝑁) + 1))
30		binom2sub1 14253	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1)↑2) = (((𝑁↑2) − (2 · 𝑁)) + 1))
31	23, 30	syl 18	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1)↑2) = (((𝑁↑2) − (2 · 𝑁)) + 1))
32	26, 23	subcld 11564	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → ((𝑁↑2) − 𝑁) ∈ ℂ)
33		1cnd 11197	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ)
34	32, 23, 33	subsubd 11592	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (((𝑁↑2) − 𝑁) − (𝑁 − 1)) = ((((𝑁↑2) − 𝑁) − 𝑁) + 1))
35	29, 31, 34	3eqtr4d 2808	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1)↑2) = (((𝑁↑2) − 𝑁) − (𝑁 − 1)))
36	35	oveq1d 7425	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (((𝑁 − 1)↑2) + (𝑁 − 1)) = ((((𝑁↑2) − 𝑁) − (𝑁 − 1)) + (𝑁 − 1)))
37		ax-1cn 11153	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
38		subcl 11451	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑁 − 1) ∈ ℂ)
39	23, 37, 38	sylancl 597	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℂ)
40	32, 39	npcand 11568	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ((((𝑁↑2) − 𝑁) − (𝑁 − 1)) + (𝑁 − 1)) = ((𝑁↑2) − 𝑁))
41	36, 40	eqtrd 2798	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (((𝑁 − 1)↑2) + (𝑁 − 1)) = ((𝑁↑2) − 𝑁))
42	41	oveq1d 7425	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ((((𝑁 − 1)↑2) + (𝑁 − 1)) / 2) = (((𝑁↑2) − 𝑁) / 2))
43	22, 42	eqtrd 2798	. . . . 5 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (1...(𝑁 − 1))𝑘 = (((𝑁↑2) − 𝑁) / 2))
44	20, 43	eqtrd 2798	. . . 4 ⊢ (𝑁 ∈ ℕ → (0 + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))𝑘) = (((𝑁↑2) − 𝑁) / 2))
45	9, 44	eqtrd 2798	. . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = (((𝑁↑2) − 𝑁) / 2))
46		oveq1 7417	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑁 − 1) = (0 − 1))
47	46	oveq2d 7426	. . . . . . 7 ⊢ (𝑁 = 0 → (0...(𝑁 − 1)) = (0...(0 − 1)))
48		0re 11205	. . . . . . . . 9 ⊢ 0 ∈ ℝ
49		ltm1 12052	. . . . . . . . 9 ⊢ (0 ∈ ℝ → (0 − 1) < 0)
50	48, 49	ax-mp 5	. . . . . . . 8 ⊢ (0 − 1) < 0
51		0z 12597	. . . . . . . . 9 ⊢ 0 ∈ ℤ
52		peano2zm 12632	. . . . . . . . . 10 ⊢ (0 ∈ ℤ → (0 − 1) ∈ ℤ)
53	51, 52	ax-mp 5	. . . . . . . . 9 ⊢ (0 − 1) ∈ ℤ
54		fzn 13563	. . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ (0 − 1) ∈ ℤ) → ((0 − 1) < 0 ↔ (0...(0 − 1)) = ∅))
55	51, 53, 54	mp2an 704	. . . . . . . 8 ⊢ ((0 − 1) < 0 ↔ (0...(0 − 1)) = ∅)
56	50, 55	mpbi 233	. . . . . . 7 ⊢ (0...(0 − 1)) = ∅
57	47, 56	eqtrdi 2814	. . . . . 6 ⊢ (𝑁 = 0 → (0...(𝑁 − 1)) = ∅)
58	57	sumeq1d 15747	. . . . 5 ⊢ (𝑁 = 0 → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = Σ𝑘 ∈ ∅ 𝑘)
59		sum0 15768	. . . . 5 ⊢ Σ𝑘 ∈ ∅ 𝑘 = 0
60	58, 59	eqtrdi 2814	. . . 4 ⊢ (𝑁 = 0 → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = 0)
61		sq0i 14225	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑁↑2) = 0)
62		id 23	. . . . . . . 8 ⊢ (𝑁 = 0 → 𝑁 = 0)
63	61, 62	oveq12d 7428	. . . . . . 7 ⊢ (𝑁 = 0 → ((𝑁↑2) − 𝑁) = (0 − 0))
64		0m0e0 12354	. . . . . . 7 ⊢ (0 − 0) = 0
65	63, 64	eqtrdi 2814	. . . . . 6 ⊢ (𝑁 = 0 → ((𝑁↑2) − 𝑁) = 0)
66	65	oveq1d 7425	. . . . 5 ⊢ (𝑁 = 0 → (((𝑁↑2) − 𝑁) / 2) = (0 / 2))
67		2cn 12311	. . . . . 6 ⊢ 2 ∈ ℂ
68		2ne0 12342	. . . . . 6 ⊢ 2 ≠ 0
69	67, 68	div0i 11944	. . . . 5 ⊢ (0 / 2) = 0
70	66, 69	eqtrdi 2814	. . . 4 ⊢ (𝑁 = 0 → (((𝑁↑2) − 𝑁) / 2) = 0)
71	60, 70	eqtr4d 2801	. . 3 ⊢ (𝑁 = 0 → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = (((𝑁↑2) − 𝑁) / 2))
72	45, 71	jaoi 870	. 2 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = (((𝑁↑2) − 𝑁) / 2))
73	1, 72	sylbi 220	1 ⊢ (𝑁 ∈ ℕ₀ → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = (((𝑁↑2) − 𝑁) / 2))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 = wceq 1570 ∈ wcel 2143 ∅c0 4286 class class class wbr 5109 ‘cfv 6536 (class class class)co 7410 ℂcc 11093 ℝcr 11094 0cc0 11095 1c1 11096 + caddc 11098 · cmul 11100 < clt 11238 − cmin 11436 / cdiv 11866 ℕcn 12228 2c2 12290 ℕ₀cn0 12499 ℤcz 12586 ℤ_≥cuz 12857 ...cfz 13530 ↑cexp 14093 Σcsu 15733

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9398 df-oi 9468 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-n0 12500 df-z 12587 df-uz 12858 df-rp 13012 df-fz 13531 df-fzo 13679 df-seq 14034 df-exp 14094 df-fac 14306 df-bc 14335 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-clim 15535 df-sum 15734

This theorem is referenced by: birthdaylem3 27118

W3C validator