sumcubes - Mathbox for Steven Nguyen

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Theorem sumcubes 41211

Description: The sum of the first 𝑁 perfect cubes is the sum of the first 𝑁 nonnegative integers, squared. This is the Proof by Nicomachus from https://proofwiki.org/wiki/Sum_of_Sequence_of_Cubes using induction and index shifting to collect all the odd numbers. (Contributed by SN, 22-Mar-2025.)

**Assertion**
Ref	Expression
sumcubes	⊢ (𝑁 ∈ ℕ₀ → Σ𝑘 ∈ (1...𝑁)(𝑘↑3) = (Σ𝑘 ∈ (1...𝑁)𝑘↑2))

Distinct variable group: 𝑘,𝑁

Proof of Theorem sumcubes Dummy variables 𝑙 𝑚 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7416	. . . . 5 ⊢ (𝑥 = 0 → (1...𝑥) = (1...0))
2	1	sumeq1d 15646	. . . 4 ⊢ (𝑥 = 0 → Σ𝑘 ∈ (1...𝑥)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑘 ∈ (1...0)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)))
3	1	sumeq1d 15646	. . . . . 6 ⊢ (𝑥 = 0 → Σ𝑘 ∈ (1...𝑥)𝑘 = Σ𝑘 ∈ (1...0)𝑘)
4	3	oveq2d 7424	. . . . 5 ⊢ (𝑥 = 0 → (1...Σ𝑘 ∈ (1...𝑥)𝑘) = (1...Σ𝑘 ∈ (1...0)𝑘))
5	4	sumeq1d 15646	. . . 4 ⊢ (𝑥 = 0 → Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑥)𝑘)((2 · 𝑚) − 1) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...0)𝑘)((2 · 𝑚) − 1))
6	2, 5	eqeq12d 2748	. . 3 ⊢ (𝑥 = 0 → (Σ𝑘 ∈ (1...𝑥)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑥)𝑘)((2 · 𝑚) − 1) ↔ Σ𝑘 ∈ (1...0)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...0)𝑘)((2 · 𝑚) − 1)))
7		oveq2 7416	. . . . 5 ⊢ (𝑥 = 𝑦 → (1...𝑥) = (1...𝑦))
8	7	sumeq1d 15646	. . . 4 ⊢ (𝑥 = 𝑦 → Σ𝑘 ∈ (1...𝑥)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑘 ∈ (1...𝑦)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)))
9	7	sumeq1d 15646	. . . . . 6 ⊢ (𝑥 = 𝑦 → Σ𝑘 ∈ (1...𝑥)𝑘 = Σ𝑘 ∈ (1...𝑦)𝑘)
10	9	oveq2d 7424	. . . . 5 ⊢ (𝑥 = 𝑦 → (1...Σ𝑘 ∈ (1...𝑥)𝑘) = (1...Σ𝑘 ∈ (1...𝑦)𝑘))
11	10	sumeq1d 15646	. . . 4 ⊢ (𝑥 = 𝑦 → Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑥)𝑘)((2 · 𝑚) − 1) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑦)𝑘)((2 · 𝑚) − 1))
12	8, 11	eqeq12d 2748	. . 3 ⊢ (𝑥 = 𝑦 → (Σ𝑘 ∈ (1...𝑥)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑥)𝑘)((2 · 𝑚) − 1) ↔ Σ𝑘 ∈ (1...𝑦)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑦)𝑘)((2 · 𝑚) − 1)))
13		oveq2 7416	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (1...𝑥) = (1...(𝑦 + 1)))
14	13	sumeq1d 15646	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → Σ𝑘 ∈ (1...𝑥)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑘 ∈ (1...(𝑦 + 1))Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)))
15	13	sumeq1d 15646	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → Σ𝑘 ∈ (1...𝑥)𝑘 = Σ𝑘 ∈ (1...(𝑦 + 1))𝑘)
16	15	oveq2d 7424	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (1...Σ𝑘 ∈ (1...𝑥)𝑘) = (1...Σ𝑘 ∈ (1...(𝑦 + 1))𝑘))
17	16	sumeq1d 15646	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑥)𝑘)((2 · 𝑚) − 1) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...(𝑦 + 1))𝑘)((2 · 𝑚) − 1))
18	14, 17	eqeq12d 2748	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (Σ𝑘 ∈ (1...𝑥)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑥)𝑘)((2 · 𝑚) − 1) ↔ Σ𝑘 ∈ (1...(𝑦 + 1))Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...(𝑦 + 1))𝑘)((2 · 𝑚) − 1)))
19		oveq2 7416	. . . . 5 ⊢ (𝑥 = 𝑁 → (1...𝑥) = (1...𝑁))
20	19	sumeq1d 15646	. . . 4 ⊢ (𝑥 = 𝑁 → Σ𝑘 ∈ (1...𝑥)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑘 ∈ (1...𝑁)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)))
21	19	sumeq1d 15646	. . . . . 6 ⊢ (𝑥 = 𝑁 → Σ𝑘 ∈ (1...𝑥)𝑘 = Σ𝑘 ∈ (1...𝑁)𝑘)
22	21	oveq2d 7424	. . . . 5 ⊢ (𝑥 = 𝑁 → (1...Σ𝑘 ∈ (1...𝑥)𝑘) = (1...Σ𝑘 ∈ (1...𝑁)𝑘))
23	22	sumeq1d 15646	. . . 4 ⊢ (𝑥 = 𝑁 → Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑥)𝑘)((2 · 𝑚) − 1) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑁)𝑘)((2 · 𝑚) − 1))
24	20, 23	eqeq12d 2748	. . 3 ⊢ (𝑥 = 𝑁 → (Σ𝑘 ∈ (1...𝑥)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑥)𝑘)((2 · 𝑚) − 1) ↔ Σ𝑘 ∈ (1...𝑁)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑁)𝑘)((2 · 𝑚) − 1)))
25		sum0 15666	. . . . 5 ⊢ Σ𝑘 ∈ ∅ Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = 0
26		sum0 15666	. . . . 5 ⊢ Σ𝑚 ∈ ∅ ((2 · 𝑚) − 1) = 0
27	25, 26	eqtr4i 2763	. . . 4 ⊢ Σ𝑘 ∈ ∅ Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ ∅ ((2 · 𝑚) − 1)
28		fz10 13521	. . . . 5 ⊢ (1...0) = ∅
29	28	sumeq1i 15643	. . . 4 ⊢ Σ𝑘 ∈ (1...0)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑘 ∈ ∅ Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1))
30	28	sumeq1i 15643	. . . . . . . 8 ⊢ Σ𝑘 ∈ (1...0)𝑘 = Σ𝑘 ∈ ∅ 𝑘
31		sum0 15666	. . . . . . . 8 ⊢ Σ𝑘 ∈ ∅ 𝑘 = 0
32	30, 31	eqtri 2760	. . . . . . 7 ⊢ Σ𝑘 ∈ (1...0)𝑘 = 0
33	32	oveq2i 7419	. . . . . 6 ⊢ (1...Σ𝑘 ∈ (1...0)𝑘) = (1...0)
34	33, 28	eqtri 2760	. . . . 5 ⊢ (1...Σ𝑘 ∈ (1...0)𝑘) = ∅
35	34	sumeq1i 15643	. . . 4 ⊢ Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...0)𝑘)((2 · 𝑚) − 1) = Σ𝑚 ∈ ∅ ((2 · 𝑚) − 1)
36	27, 29, 35	3eqtr4i 2770	. . 3 ⊢ Σ𝑘 ∈ (1...0)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...0)𝑘)((2 · 𝑚) − 1)
37		simpr 485	. . . . . 6 ⊢ ((𝑦 ∈ ℕ₀ ∧ Σ𝑘 ∈ (1...𝑦)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑦)𝑘)((2 · 𝑚) − 1)) → Σ𝑘 ∈ (1...𝑦)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑦)𝑘)((2 · 𝑚) − 1))
38		fzfid 13937	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → (1...𝑦) ∈ Fin)
39		elfznn 13529	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝑦) → 𝑘 ∈ ℕ)
40	39	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑘 ∈ (1...𝑦)) → 𝑘 ∈ ℕ)
41	40	nnnn0d 12531	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑘 ∈ (1...𝑦)) → 𝑘 ∈ ℕ₀)
42	38, 41	fsumnn0cl 15681	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℕ₀)
43	42	nn0zd 12583	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℤ)
44		nn0p1nn 12510	. . . . . . . . . . 11 ⊢ (Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℕ₀ → (Σ𝑘 ∈ (1...𝑦)𝑘 + 1) ∈ ℕ)
45	42, 44	syl 17	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → (Σ𝑘 ∈ (1...𝑦)𝑘 + 1) ∈ ℕ)
46	45	nnzd 12584	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → (Σ𝑘 ∈ (1...𝑦)𝑘 + 1) ∈ ℤ)
47		peano2nn0 12511	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → (𝑦 + 1) ∈ ℕ₀)
48	47	nn0zd 12583	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → (𝑦 + 1) ∈ ℤ)
49	43, 48	zaddcld 12669	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → (Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)) ∈ ℤ)
50		2cnd 12289	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 2 ∈ ℂ)
51		elfzelz 13500	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1))) → 𝑚 ∈ ℤ)
52	51	zcnd 12666	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1))) → 𝑚 ∈ ℂ)
53	52	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 𝑚 ∈ ℂ)
54	50, 53	mulcld 11233	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → (2 · 𝑚) ∈ ℂ)
55		1cnd 11208	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 1 ∈ ℂ)
56	54, 55	subcld 11570	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → ((2 · 𝑚) − 1) ∈ ℂ)
57		oveq2 7416	. . . . . . . . . 10 ⊢ (𝑚 = (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘) → (2 · 𝑚) = (2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)))
58	57	oveq1d 7423	. . . . . . . . 9 ⊢ (𝑚 = (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘) → ((2 · 𝑚) − 1) = ((2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)) − 1))
59	43, 46, 49, 56, 58	fsumshftm 15726	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → Σ𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))((2 · 𝑚) − 1) = Σ𝑙 ∈ (((Σ𝑘 ∈ (1...𝑦)𝑘 + 1) − Σ𝑘 ∈ (1...𝑦)𝑘)...((Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)) − Σ𝑘 ∈ (1...𝑦)𝑘))((2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)) − 1))
60		elfzelz 13500	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑦) → 𝑘 ∈ ℤ)
61	60	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑘 ∈ (1...𝑦)) → 𝑘 ∈ ℤ)
62	61	zred 12665	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑘 ∈ (1...𝑦)) → 𝑘 ∈ ℝ)
63	38, 62	fsumrecl 15679	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ₀ → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℝ)
64	63	recnd 11241	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℂ)
65		1cnd 11208	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → 1 ∈ ℂ)
66	64, 65	pncan2d 11572	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1) − Σ𝑘 ∈ (1...𝑦)𝑘) = 1)
67	47	nn0cnd 12533	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ₀ → (𝑦 + 1) ∈ ℂ)
68	64, 67	pncan2d 11572	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → ((Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)) − Σ𝑘 ∈ (1...𝑦)𝑘) = (𝑦 + 1))
69	66, 68	oveq12d 7426	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → (((Σ𝑘 ∈ (1...𝑦)𝑘 + 1) − Σ𝑘 ∈ (1...𝑦)𝑘)...((Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)) − Σ𝑘 ∈ (1...𝑦)𝑘)) = (1...(𝑦 + 1)))
70		elfzelz 13500	. . . . . . . . . . 11 ⊢ (𝑙 ∈ (((Σ𝑘 ∈ (1...𝑦)𝑘 + 1) − Σ𝑘 ∈ (1...𝑦)𝑘)...((Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)) − Σ𝑘 ∈ (1...𝑦)𝑘)) → 𝑙 ∈ ℤ)
71	70	zcnd 12666	. . . . . . . . . 10 ⊢ (𝑙 ∈ (((Σ𝑘 ∈ (1...𝑦)𝑘 + 1) − Σ𝑘 ∈ (1...𝑦)𝑘)...((Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)) − Σ𝑘 ∈ (1...𝑦)𝑘)) → 𝑙 ∈ ℂ)
72		2cnd 12289	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑙 ∈ ℂ) → 2 ∈ ℂ)
73		simpr 485	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑙 ∈ ℂ) → 𝑙 ∈ ℂ)
74	64	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑙 ∈ ℂ) → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℂ)
75	72, 73, 74	adddid 11237	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑙 ∈ ℂ) → (2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)) = ((2 · 𝑙) + (2 · Σ𝑘 ∈ (1...𝑦)𝑘)))
76	75	oveq1d 7423	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑙 ∈ ℂ) → ((2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)) − 1) = (((2 · 𝑙) + (2 · Σ𝑘 ∈ (1...𝑦)𝑘)) − 1))
77	72, 73	mulcld 11233	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑙 ∈ ℂ) → (2 · 𝑙) ∈ ℂ)
78	72, 74	mulcld 11233	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑙 ∈ ℂ) → (2 · Σ𝑘 ∈ (1...𝑦)𝑘) ∈ ℂ)
79		1cnd 11208	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑙 ∈ ℂ) → 1 ∈ ℂ)
80	77, 78, 79	addsubassd 11590	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑙 ∈ ℂ) → (((2 · 𝑙) + (2 · Σ𝑘 ∈ (1...𝑦)𝑘)) − 1) = ((2 · 𝑙) + ((2 · Σ𝑘 ∈ (1...𝑦)𝑘) − 1)))
81	77, 78, 79	addsub12d 11593	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑙 ∈ ℂ) → ((2 · 𝑙) + ((2 · Σ𝑘 ∈ (1...𝑦)𝑘) − 1)) = ((2 · Σ𝑘 ∈ (1...𝑦)𝑘) + ((2 · 𝑙) − 1)))
82		arisum 15805	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ₀ → Σ𝑘 ∈ (1...𝑦)𝑘 = (((𝑦↑2) + 𝑦) / 2))
83	82	oveq2d 7424	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ₀ → (2 · Σ𝑘 ∈ (1...𝑦)𝑘) = (2 · (((𝑦↑2) + 𝑦) / 2)))
84		nn0cn 12481	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ ℂ)
85	84	sqcld 14108	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℕ₀ → (𝑦↑2) ∈ ℂ)
86	85, 84	addcld 11232	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ₀ → ((𝑦↑2) + 𝑦) ∈ ℂ)
87		2cnd 12289	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ₀ → 2 ∈ ℂ)
88		2ne0 12315	. . . . . . . . . . . . . . . . 17 ⊢ 2 ≠ 0
89	88	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ₀ → 2 ≠ 0)
90	86, 87, 89	divcan2d 11991	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ₀ → (2 · (((𝑦↑2) + 𝑦) / 2)) = ((𝑦↑2) + 𝑦))
91		binom21 14181	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℂ → ((𝑦 + 1)↑2) = (((𝑦↑2) + (2 · 𝑦)) + 1))
92	84, 91	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℕ₀ → ((𝑦 + 1)↑2) = (((𝑦↑2) + (2 · 𝑦)) + 1))
93	92	oveq1d 7423	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ₀ → (((𝑦 + 1)↑2) − (𝑦 + 1)) = ((((𝑦↑2) + (2 · 𝑦)) + 1) − (𝑦 + 1)))
94	87, 84	mulcld 11233	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ₀ → (2 · 𝑦) ∈ ℂ)
95	85, 94	addcld 11232	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℕ₀ → ((𝑦↑2) + (2 · 𝑦)) ∈ ℂ)
96	95, 84, 65	pnpcan2d 11608	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ₀ → ((((𝑦↑2) + (2 · 𝑦)) + 1) − (𝑦 + 1)) = (((𝑦↑2) + (2 · 𝑦)) − 𝑦))
97	85, 94, 84	addsubassd 11590	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℕ₀ → (((𝑦↑2) + (2 · 𝑦)) − 𝑦) = ((𝑦↑2) + ((2 · 𝑦) − 𝑦)))
98	84	2timesd 12454	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℕ₀ → (2 · 𝑦) = (𝑦 + 𝑦))
99	84, 84, 98	mvrladdd 11626	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ₀ → ((2 · 𝑦) − 𝑦) = 𝑦)
100	99	oveq2d 7424	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℕ₀ → ((𝑦↑2) + ((2 · 𝑦) − 𝑦)) = ((𝑦↑2) + 𝑦))
101	97, 100	eqtrd 2772	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ₀ → (((𝑦↑2) + (2 · 𝑦)) − 𝑦) = ((𝑦↑2) + 𝑦))
102	93, 96, 101	3eqtrrd 2777	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ₀ → ((𝑦↑2) + 𝑦) = (((𝑦 + 1)↑2) − (𝑦 + 1)))
103	83, 90, 102	3eqtrd 2776	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℕ₀ → (2 · Σ𝑘 ∈ (1...𝑦)𝑘) = (((𝑦 + 1)↑2) − (𝑦 + 1)))
104	103	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑙 ∈ ℂ) → (2 · Σ𝑘 ∈ (1...𝑦)𝑘) = (((𝑦 + 1)↑2) − (𝑦 + 1)))
105	104	oveq1d 7423	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑙 ∈ ℂ) → ((2 · Σ𝑘 ∈ (1...𝑦)𝑘) + ((2 · 𝑙) − 1)) = ((((𝑦 + 1)↑2) − (𝑦 + 1)) + ((2 · 𝑙) − 1)))
106	81, 105	eqtrd 2772	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑙 ∈ ℂ) → ((2 · 𝑙) + ((2 · Σ𝑘 ∈ (1...𝑦)𝑘) − 1)) = ((((𝑦 + 1)↑2) − (𝑦 + 1)) + ((2 · 𝑙) − 1)))
107	76, 80, 106	3eqtrd 2776	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑙 ∈ ℂ) → ((2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)) − 1) = ((((𝑦 + 1)↑2) − (𝑦 + 1)) + ((2 · 𝑙) − 1)))
108	71, 107	sylan2 593	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑙 ∈ (((Σ𝑘 ∈ (1...𝑦)𝑘 + 1) − Σ𝑘 ∈ (1...𝑦)𝑘)...((Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)) − Σ𝑘 ∈ (1...𝑦)𝑘))) → ((2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)) − 1) = ((((𝑦 + 1)↑2) − (𝑦 + 1)) + ((2 · 𝑙) − 1)))
109	69, 108	sumeq12dv 15651	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → Σ𝑙 ∈ (((Σ𝑘 ∈ (1...𝑦)𝑘 + 1) − Σ𝑘 ∈ (1...𝑦)𝑘)...((Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)) − Σ𝑘 ∈ (1...𝑦)𝑘))((2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)) − 1) = Σ𝑙 ∈ (1...(𝑦 + 1))((((𝑦 + 1)↑2) − (𝑦 + 1)) + ((2 · 𝑙) − 1)))
110	59, 109	eqtr2d 2773	. . . . . . 7 ⊢ (𝑦 ∈ ℕ₀ → Σ𝑙 ∈ (1...(𝑦 + 1))((((𝑦 + 1)↑2) − (𝑦 + 1)) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))((2 · 𝑚) − 1))
111	110	adantr 481	. . . . . 6 ⊢ ((𝑦 ∈ ℕ₀ ∧ Σ𝑘 ∈ (1...𝑦)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑦)𝑘)((2 · 𝑚) − 1)) → Σ𝑙 ∈ (1...(𝑦 + 1))((((𝑦 + 1)↑2) − (𝑦 + 1)) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))((2 · 𝑚) − 1))
112	37, 111	oveq12d 7426	. . . . 5 ⊢ ((𝑦 ∈ ℕ₀ ∧ Σ𝑘 ∈ (1...𝑦)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑦)𝑘)((2 · 𝑚) − 1)) → (Σ𝑘 ∈ (1...𝑦)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) + Σ𝑙 ∈ (1...(𝑦 + 1))((((𝑦 + 1)↑2) − (𝑦 + 1)) + ((2 · 𝑙) − 1))) = (Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑦)𝑘)((2 · 𝑚) − 1) + Σ𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))((2 · 𝑚) − 1)))
113		id 22	. . . . . . 7 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ ℕ₀)
114		fzfid 13937	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑘 ∈ (1...(𝑦 + 1))) → (1...𝑘) ∈ Fin)
115		elfzelz 13500	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...(𝑦 + 1)) → 𝑘 ∈ ℤ)
116	115	zcnd 12666	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...(𝑦 + 1)) → 𝑘 ∈ ℂ)
117	116	sqcld 14108	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...(𝑦 + 1)) → (𝑘↑2) ∈ ℂ)
118	117, 116	subcld 11570	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...(𝑦 + 1)) → ((𝑘↑2) − 𝑘) ∈ ℂ)
119		2cnd 12289	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ (1...𝑘) → 2 ∈ ℂ)
120		elfzelz 13500	. . . . . . . . . . . . 13 ⊢ (𝑙 ∈ (1...𝑘) → 𝑙 ∈ ℤ)
121	120	zcnd 12666	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ (1...𝑘) → 𝑙 ∈ ℂ)
122	119, 121	mulcld 11233	. . . . . . . . . . 11 ⊢ (𝑙 ∈ (1...𝑘) → (2 · 𝑙) ∈ ℂ)
123		1cnd 11208	. . . . . . . . . . 11 ⊢ (𝑙 ∈ (1...𝑘) → 1 ∈ ℂ)
124	122, 123	subcld 11570	. . . . . . . . . 10 ⊢ (𝑙 ∈ (1...𝑘) → ((2 · 𝑙) − 1) ∈ ℂ)
125		addcl 11191	. . . . . . . . . 10 ⊢ ((((𝑘↑2) − 𝑘) ∈ ℂ ∧ ((2 · 𝑙) − 1) ∈ ℂ) → (((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) ∈ ℂ)
126	118, 124, 125	syl2an 596	. . . . . . . . 9 ⊢ ((𝑘 ∈ (1...(𝑦 + 1)) ∧ 𝑙 ∈ (1...𝑘)) → (((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) ∈ ℂ)
127	126	adantll 712	. . . . . . . 8 ⊢ (((𝑦 ∈ ℕ₀ ∧ 𝑘 ∈ (1...(𝑦 + 1))) ∧ 𝑙 ∈ (1...𝑘)) → (((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) ∈ ℂ)
128	114, 127	fsumcl 15678	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑘 ∈ (1...(𝑦 + 1))) → Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) ∈ ℂ)
129		oveq2 7416	. . . . . . . 8 ⊢ (𝑘 = (𝑦 + 1) → (1...𝑘) = (1...(𝑦 + 1)))
130		oveq1 7415	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑦 + 1) → (𝑘↑2) = ((𝑦 + 1)↑2))
131		id 22	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑦 + 1) → 𝑘 = (𝑦 + 1))
132	130, 131	oveq12d 7426	. . . . . . . . . 10 ⊢ (𝑘 = (𝑦 + 1) → ((𝑘↑2) − 𝑘) = (((𝑦 + 1)↑2) − (𝑦 + 1)))
133	132	oveq1d 7423	. . . . . . . . 9 ⊢ (𝑘 = (𝑦 + 1) → (((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = ((((𝑦 + 1)↑2) − (𝑦 + 1)) + ((2 · 𝑙) − 1)))
134	133	adantr 481	. . . . . . . 8 ⊢ ((𝑘 = (𝑦 + 1) ∧ 𝑙 ∈ (1...𝑘)) → (((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = ((((𝑦 + 1)↑2) − (𝑦 + 1)) + ((2 · 𝑙) − 1)))
135	129, 134	sumeq12dv 15651	. . . . . . 7 ⊢ (𝑘 = (𝑦 + 1) → Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑙 ∈ (1...(𝑦 + 1))((((𝑦 + 1)↑2) − (𝑦 + 1)) + ((2 · 𝑙) − 1)))
136	113, 128, 135	fz1sump1 41208	. . . . . 6 ⊢ (𝑦 ∈ ℕ₀ → Σ𝑘 ∈ (1...(𝑦 + 1))Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = (Σ𝑘 ∈ (1...𝑦)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) + Σ𝑙 ∈ (1...(𝑦 + 1))((((𝑦 + 1)↑2) − (𝑦 + 1)) + ((2 · 𝑙) − 1))))
137	136	adantr 481	. . . . 5 ⊢ ((𝑦 ∈ ℕ₀ ∧ Σ𝑘 ∈ (1...𝑦)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑦)𝑘)((2 · 𝑚) − 1)) → Σ𝑘 ∈ (1...(𝑦 + 1))Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = (Σ𝑘 ∈ (1...𝑦)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) + Σ𝑙 ∈ (1...(𝑦 + 1))((((𝑦 + 1)↑2) − (𝑦 + 1)) + ((2 · 𝑙) − 1))))
138	116	adantl 482	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑘 ∈ (1...(𝑦 + 1))) → 𝑘 ∈ ℂ)
139	113, 138, 131	fz1sump1 41208	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → Σ𝑘 ∈ (1...(𝑦 + 1))𝑘 = (Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))
140	139	adantr 481	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ₀ ∧ Σ𝑘 ∈ (1...𝑦)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑦)𝑘)((2 · 𝑚) − 1)) → Σ𝑘 ∈ (1...(𝑦 + 1))𝑘 = (Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))
141	140	oveq2d 7424	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ₀ ∧ Σ𝑘 ∈ (1...𝑦)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑦)𝑘)((2 · 𝑚) − 1)) → (1...Σ𝑘 ∈ (1...(𝑦 + 1))𝑘) = (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1))))
142	141	sumeq1d 15646	. . . . . 6 ⊢ ((𝑦 ∈ ℕ₀ ∧ Σ𝑘 ∈ (1...𝑦)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑦)𝑘)((2 · 𝑚) − 1)) → Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...(𝑦 + 1))𝑘)((2 · 𝑚) − 1) = Σ𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))((2 · 𝑚) − 1))
143	63	ltp1d 12143	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → Σ𝑘 ∈ (1...𝑦)𝑘 < (Σ𝑘 ∈ (1...𝑦)𝑘 + 1))
144		fzdisj 13527	. . . . . . . . 9 ⊢ (Σ𝑘 ∈ (1...𝑦)𝑘 < (Σ𝑘 ∈ (1...𝑦)𝑘 + 1) → ((1...Σ𝑘 ∈ (1...𝑦)𝑘) ∩ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) = ∅)
145	143, 144	syl 17	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → ((1...Σ𝑘 ∈ (1...𝑦)𝑘) ∩ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) = ∅)
146		nnuz 12864	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
147	45, 146	eleqtrdi 2843	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → (Σ𝑘 ∈ (1...𝑦)𝑘 + 1) ∈ (ℤ_≥‘1))
148	43	uzidd 12837	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ (ℤ_≥‘Σ𝑘 ∈ (1...𝑦)𝑘))
149		uzaddcl 12887	. . . . . . . . . 10 ⊢ ((Σ𝑘 ∈ (1...𝑦)𝑘 ∈ (ℤ_≥‘Σ𝑘 ∈ (1...𝑦)𝑘) ∧ (𝑦 + 1) ∈ ℕ₀) → (Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)) ∈ (ℤ_≥‘Σ𝑘 ∈ (1...𝑦)𝑘))
150	148, 47, 149	syl2anc 584	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → (Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)) ∈ (ℤ_≥‘Σ𝑘 ∈ (1...𝑦)𝑘))
151		fzsplit2 13525	. . . . . . . . 9 ⊢ (((Σ𝑘 ∈ (1...𝑦)𝑘 + 1) ∈ (ℤ_≥‘1) ∧ (Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)) ∈ (ℤ_≥‘Σ𝑘 ∈ (1...𝑦)𝑘)) → (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1))) = ((1...Σ𝑘 ∈ (1...𝑦)𝑘) ∪ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))))
152	147, 150, 151	syl2anc 584	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1))) = ((1...Σ𝑘 ∈ (1...𝑦)𝑘) ∪ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))))
153		fzfid 13937	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1))) ∈ Fin)
154		2cnd 12289	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 2 ∈ ℂ)
155		elfzelz 13500	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1))) → 𝑚 ∈ ℤ)
156	155	zcnd 12666	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1))) → 𝑚 ∈ ℂ)
157	156	adantl 482	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 𝑚 ∈ ℂ)
158	154, 157	mulcld 11233	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → (2 · 𝑚) ∈ ℂ)
159		1cnd 11208	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 1 ∈ ℂ)
160	158, 159	subcld 11570	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → ((2 · 𝑚) − 1) ∈ ℂ)
161	145, 152, 153, 160	fsumsplit 15686	. . . . . . 7 ⊢ (𝑦 ∈ ℕ₀ → Σ𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))((2 · 𝑚) − 1) = (Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑦)𝑘)((2 · 𝑚) − 1) + Σ𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))((2 · 𝑚) − 1)))
162	161	adantr 481	. . . . . 6 ⊢ ((𝑦 ∈ ℕ₀ ∧ Σ𝑘 ∈ (1...𝑦)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑦)𝑘)((2 · 𝑚) − 1)) → Σ𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))((2 · 𝑚) − 1) = (Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑦)𝑘)((2 · 𝑚) − 1) + Σ𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))((2 · 𝑚) − 1)))
163	142, 162	eqtrd 2772	. . . . 5 ⊢ ((𝑦 ∈ ℕ₀ ∧ Σ𝑘 ∈ (1...𝑦)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑦)𝑘)((2 · 𝑚) − 1)) → Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...(𝑦 + 1))𝑘)((2 · 𝑚) − 1) = (Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑦)𝑘)((2 · 𝑚) − 1) + Σ𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))((2 · 𝑚) − 1)))
164	112, 137, 163	3eqtr4d 2782	. . . 4 ⊢ ((𝑦 ∈ ℕ₀ ∧ Σ𝑘 ∈ (1...𝑦)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑦)𝑘)((2 · 𝑚) − 1)) → Σ𝑘 ∈ (1...(𝑦 + 1))Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...(𝑦 + 1))𝑘)((2 · 𝑚) − 1))
165	164	ex 413	. . 3 ⊢ (𝑦 ∈ ℕ₀ → (Σ𝑘 ∈ (1...𝑦)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑦)𝑘)((2 · 𝑚) − 1) → Σ𝑘 ∈ (1...(𝑦 + 1))Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...(𝑦 + 1))𝑘)((2 · 𝑚) − 1)))
166	6, 12, 18, 24, 36, 165	nn0ind 12656	. 2 ⊢ (𝑁 ∈ ℕ₀ → Σ𝑘 ∈ (1...𝑁)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑁)𝑘)((2 · 𝑚) − 1))
167		fz1ssnn 13531	. . . . . . 7 ⊢ (1...𝑁) ⊆ ℕ
168		nnssnn0 12474	. . . . . . 7 ⊢ ℕ ⊆ ℕ₀
169	167, 168	sstri 3991	. . . . . 6 ⊢ (1...𝑁) ⊆ ℕ₀
170	169	a1i 11	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (1...𝑁) ⊆ ℕ₀)
171	170	sselda 3982	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℕ₀)
172		nicomachus 41210	. . . 4 ⊢ (𝑘 ∈ ℕ₀ → Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = (𝑘↑3))
173	171, 172	syl 17	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ (1...𝑁)) → Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = (𝑘↑3))
174	173	sumeq2dv 15648	. 2 ⊢ (𝑁 ∈ ℕ₀ → Σ𝑘 ∈ (1...𝑁)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑘 ∈ (1...𝑁)(𝑘↑3))
175		fzfid 13937	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → (1...𝑁) ∈ Fin)
176	175, 171	fsumnn0cl 15681	. . 3 ⊢ (𝑁 ∈ ℕ₀ → Σ𝑘 ∈ (1...𝑁)𝑘 ∈ ℕ₀)
177		oddnumth 41209	. . 3 ⊢ (Σ𝑘 ∈ (1...𝑁)𝑘 ∈ ℕ₀ → Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑁)𝑘)((2 · 𝑚) − 1) = (Σ𝑘 ∈ (1...𝑁)𝑘↑2))
178	176, 177	syl 17	. 2 ⊢ (𝑁 ∈ ℕ₀ → Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑁)𝑘)((2 · 𝑚) − 1) = (Σ𝑘 ∈ (1...𝑁)𝑘↑2))
179	166, 174, 178	3eqtr3d 2780	1 ⊢ (𝑁 ∈ ℕ₀ → Σ𝑘 ∈ (1...𝑁)(𝑘↑3) = (Σ𝑘 ∈ (1...𝑁)𝑘↑2))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∪ cun 3946 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 class class class wbr 5148 ‘cfv 6543 (class class class)co 7408 ℂcc 11107 0cc0 11109 1c1 11110 + caddc 11112 · cmul 11114 < clt 11247 − cmin 11443 / cdiv 11870 ℕcn 12211 2c2 12266 3c3 12267 ℕ₀cn0 12471 ℤcz 12557 ℤ_≥cuz 12821 ...cfz 13483 ↑cexp 14026 Σcsu 15631

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-oi 9504 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-z 12558 df-uz 12822 df-rp 12974 df-fz 13484 df-fzo 13627 df-seq 13966 df-exp 14027 df-fac 14233 df-bc 14262 df-hash 14290 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-clim 15431 df-sum 15632

This theorem is referenced by: (None)

W3C validator