Theorem sumcubes 41904

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 7434	. . . . 5 ⊢ (𝑥 = 0 → (1...𝑥) = (1...0))
2	1	sumeq1d 15687	. . . 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 15687	. . . . . 6 ⊢ (𝑥 = 0 → Σ𝑘 ∈ (1...𝑥)𝑘 = Σ𝑘 ∈ (1...0)𝑘)
4	3	oveq2d 7442	. . . . 5 ⊢ (𝑥 = 0 → (1...Σ𝑘 ∈ (1...𝑥)𝑘) = (1...Σ𝑘 ∈ (1...0)𝑘))
5	4	sumeq1d 15687	. . . 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 2744	. . 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 7434	. . . . 5 ⊢ (𝑥 = 𝑦 → (1...𝑥) = (1...𝑦))
8	7	sumeq1d 15687	. . . 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 15687	. . . . . 6 ⊢ (𝑥 = 𝑦 → Σ𝑘 ∈ (1...𝑥)𝑘 = Σ𝑘 ∈ (1...𝑦)𝑘)
10	9	oveq2d 7442	. . . . 5 ⊢ (𝑥 = 𝑦 → (1...Σ𝑘 ∈ (1...𝑥)𝑘) = (1...Σ𝑘 ∈ (1...𝑦)𝑘))
11	10	sumeq1d 15687	. . . 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 2744	. . 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 7434	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (1...𝑥) = (1...(𝑦 + 1)))
14	13	sumeq1d 15687	. . . 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 15687	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → Σ𝑘 ∈ (1...𝑥)𝑘 = Σ𝑘 ∈ (1...(𝑦 + 1))𝑘)
16	15	oveq2d 7442	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (1...Σ𝑘 ∈ (1...𝑥)𝑘) = (1...Σ𝑘 ∈ (1...(𝑦 + 1))𝑘))
17	16	sumeq1d 15687	. . . 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 2744	. . 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 7434	. . . . 5 ⊢ (𝑥 = 𝑁 → (1...𝑥) = (1...𝑁))
20	19	sumeq1d 15687	. . . 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 15687	. . . . . 6 ⊢ (𝑥 = 𝑁 → Σ𝑘 ∈ (1...𝑥)𝑘 = Σ𝑘 ∈ (1...𝑁)𝑘)
22	21	oveq2d 7442	. . . . 5 ⊢ (𝑥 = 𝑁 → (1...Σ𝑘 ∈ (1...𝑥)𝑘) = (1...Σ𝑘 ∈ (1...𝑁)𝑘))
23	22	sumeq1d 15687	. . . 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 2744	. . 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 15707	. . . . 5 ⊢ Σ𝑘 ∈ ∅ Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = 0
26		sum0 15707	. . . . 5 ⊢ Σ𝑚 ∈ ∅ ((2 · 𝑚) − 1) = 0
27	25, 26	eqtr4i 2759	. . . 4 ⊢ Σ𝑘 ∈ ∅ Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ ∅ ((2 · 𝑚) − 1)
28		fz10 13562	. . . . 5 ⊢ (1...0) = ∅
29	28	sumeq1i 15684	. . . 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 15684	. . . . . . . 8 ⊢ Σ𝑘 ∈ (1...0)𝑘 = Σ𝑘 ∈ ∅ 𝑘
31		sum0 15707	. . . . . . . 8 ⊢ Σ𝑘 ∈ ∅ 𝑘 = 0
32	30, 31	eqtri 2756	. . . . . . 7 ⊢ Σ𝑘 ∈ (1...0)𝑘 = 0
33	32	oveq2i 7437	. . . . . 6 ⊢ (1...Σ𝑘 ∈ (1...0)𝑘) = (1...0)
34	33, 28	eqtri 2756	. . . . 5 ⊢ (1...Σ𝑘 ∈ (1...0)𝑘) = ∅
35	34	sumeq1i 15684	. . . 4 ⊢ Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...0)𝑘)((2 · 𝑚) − 1) = Σ𝑚 ∈ ∅ ((2 · 𝑚) − 1)
36	27, 29, 35	3eqtr4i 2766	. . 3 ⊢ Σ𝑘 ∈ (1...0)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...0)𝑘)((2 · 𝑚) − 1)
37		simpr 483	. . . . . 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 13978	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 → (1...𝑦) ∈ Fin)
39		elfznn 13570	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝑦) → 𝑘 ∈ ℕ)
40	39	adantl 480	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑦)) → 𝑘 ∈ ℕ)
41	40	nnnn0d 12570	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑦)) → 𝑘 ∈ ℕ0)
42	38, 41	fsumnn0cl 15722	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℕ0)
43	42	nn0zd 12622	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℤ)
44		nn0p1nn 12549	. . . . . . . . . . 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 12623	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0 → (Σ𝑘 ∈ (1...𝑦)𝑘 + 1) ∈ ℤ)
47		peano2nn0 12550	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ0)
48	47	nn0zd 12622	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℤ)
49	43, 48	zaddcld 12708	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0 → (Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)) ∈ ℤ)
50		2cnd 12328	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 2 ∈ ℂ)
51		elfzelz 13541	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1))) → 𝑚 ∈ ℤ)
52	51	zcnd 12705	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1))) → 𝑚 ∈ ℂ)
53	52	adantl 480	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 𝑚 ∈ ℂ)
54	50, 53	mulcld 11272	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → (2 · 𝑚) ∈ ℂ)
55		1cnd 11247	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 1 ∈ ℂ)
56	54, 55	subcld 11609	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → ((2 · 𝑚) − 1) ∈ ℂ)
57		oveq2 7434	. . . . . . . . . 10 ⊢ (𝑚 = (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘) → (2 · 𝑚) = (2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)))
58	57	oveq1d 7441	. . . . . . . . 9 ⊢ (𝑚 = (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘) → ((2 · 𝑚) − 1) = ((2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)) − 1))
59	43, 46, 49, 56, 58	fsumshftm 15767	. . . . . . . 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 13541	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑦) → 𝑘 ∈ ℤ)
61	60	adantl 480	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑦)) → 𝑘 ∈ ℤ)
62	61	zred 12704	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑦)) → 𝑘 ∈ ℝ)
63	38, 62	fsumrecl 15720	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℝ)
64	63	recnd 11280	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℂ)
65		1cnd 11247	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 → 1 ∈ ℂ)
66	64, 65	pncan2d 11611	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0 → ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1) − Σ𝑘 ∈ (1...𝑦)𝑘) = 1)
67	47	nn0cnd 12572	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℂ)
68	64, 67	pncan2d 11611	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0 → ((Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)) − Σ𝑘 ∈ (1...𝑦)𝑘) = (𝑦 + 1))
69	66, 68	oveq12d 7444	. . . . . . . . 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 13541	. . . . . . . . . . 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 12705	. . . . . . . . . 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 12328	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ) → 2 ∈ ℂ)
73		simpr 483	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ) → 𝑙 ∈ ℂ)
74	64	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ) → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℂ)
75	72, 73, 74	adddid 11276	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ) → (2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)) = ((2 · 𝑙) + (2 · Σ𝑘 ∈ (1...𝑦)𝑘)))
76	75	oveq1d 7441	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ) → ((2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)) − 1) = (((2 · 𝑙) + (2 · Σ𝑘 ∈ (1...𝑦)𝑘)) − 1))
77	72, 73	mulcld 11272	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ) → (2 · 𝑙) ∈ ℂ)
78	72, 74	mulcld 11272	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ) → (2 · Σ𝑘 ∈ (1...𝑦)𝑘) ∈ ℂ)
79		1cnd 11247	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ) → 1 ∈ ℂ)
80	77, 78, 79	addsubassd 11629	. . . . . . . . . . 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 11632	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ) → ((2 · 𝑙) + ((2 · Σ𝑘 ∈ (1...𝑦)𝑘) − 1)) = ((2 · Σ𝑘 ∈ (1...𝑦)𝑘) + ((2 · 𝑙) − 1)))
82		arisum 15846	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 = (((𝑦↑2) + 𝑦) / 2))
83	82	oveq2d 7442	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ0 → (2 · Σ𝑘 ∈ (1...𝑦)𝑘) = (2 · (((𝑦↑2) + 𝑦) / 2)))
84		nn0cn 12520	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℂ)
85	84	sqcld 14148	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℕ0 → (𝑦↑2) ∈ ℂ)
86	85, 84	addcld 11271	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ0 → ((𝑦↑2) + 𝑦) ∈ ℂ)
87		2cnd 12328	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ0 → 2 ∈ ℂ)
88		2ne0 12354	. . . . . . . . . . . . . . . . 17 ⊢ 2 ≠ 0
89	88	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ0 → 2 ≠ 0)
90	86, 87, 89	divcan2d 12030	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ0 → (2 · (((𝑦↑2) + 𝑦) / 2)) = ((𝑦↑2) + 𝑦))
91		binom21 14221	. . . . . . . . . . . . . . . . . 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 7441	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ0 → (((𝑦 + 1)↑2) − (𝑦 + 1)) = ((((𝑦↑2) + (2 · 𝑦)) + 1) − (𝑦 + 1)))
94	87, 84	mulcld 11272	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ0 → (2 · 𝑦) ∈ ℂ)
95	85, 94	addcld 11271	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℕ0 → ((𝑦↑2) + (2 · 𝑦)) ∈ ℂ)
96	95, 84, 65	pnpcan2d 11647	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ0 → ((((𝑦↑2) + (2 · 𝑦)) + 1) − (𝑦 + 1)) = (((𝑦↑2) + (2 · 𝑦)) − 𝑦))
97	85, 94, 84	addsubassd 11629	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℕ0 → (((𝑦↑2) + (2 · 𝑦)) − 𝑦) = ((𝑦↑2) + ((2 · 𝑦) − 𝑦)))
98	84	2timesd 12493	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℕ0 → (2 · 𝑦) = (𝑦 + 𝑦))
99	84, 84, 98	mvrladdd 11665	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ0 → ((2 · 𝑦) − 𝑦) = 𝑦)
100	99	oveq2d 7442	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℕ0 → ((𝑦↑2) + ((2 · 𝑦) − 𝑦)) = ((𝑦↑2) + 𝑦))
101	97, 100	eqtrd 2768	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ0 → (((𝑦↑2) + (2 · 𝑦)) − 𝑦) = ((𝑦↑2) + 𝑦))
102	93, 96, 101	3eqtrrd 2773	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ0 → ((𝑦↑2) + 𝑦) = (((𝑦 + 1)↑2) − (𝑦 + 1)))
103	83, 90, 102	3eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℕ0 → (2 · Σ𝑘 ∈ (1...𝑦)𝑘) = (((𝑦 + 1)↑2) − (𝑦 + 1)))
104	103	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ) → (2 · Σ𝑘 ∈ (1...𝑦)𝑘) = (((𝑦 + 1)↑2) − (𝑦 + 1)))
105	104	oveq1d 7441	. . . . . . . . . . . 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 2768	. . . . . . . . . . 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 2772	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ) → ((2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)) − 1) = ((((𝑦 + 1)↑2) − (𝑦 + 1)) + ((2 · 𝑙) − 1)))
108	71, 107	sylan2 591	. . . . . . . . 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 15692	. . . . . . . 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 2769	. . . . . . 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 479	. . . . . 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 7444	. . . . 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 13978	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑘 ∈ (1...(𝑦 + 1))) → (1...𝑘) ∈ Fin)
115		elfzelz 13541	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...(𝑦 + 1)) → 𝑘 ∈ ℤ)
116	115	zcnd 12705	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...(𝑦 + 1)) → 𝑘 ∈ ℂ)
117	116	sqcld 14148	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...(𝑦 + 1)) → (𝑘↑2) ∈ ℂ)
118	117, 116	subcld 11609	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...(𝑦 + 1)) → ((𝑘↑2) − 𝑘) ∈ ℂ)
119		2cnd 12328	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ (1...𝑘) → 2 ∈ ℂ)
120		elfzelz 13541	. . . . . . . . . . . . 13 ⊢ (𝑙 ∈ (1...𝑘) → 𝑙 ∈ ℤ)
121	120	zcnd 12705	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ (1...𝑘) → 𝑙 ∈ ℂ)
122	119, 121	mulcld 11272	. . . . . . . . . . 11 ⊢ (𝑙 ∈ (1...𝑘) → (2 · 𝑙) ∈ ℂ)
123		1cnd 11247	. . . . . . . . . . 11 ⊢ (𝑙 ∈ (1...𝑘) → 1 ∈ ℂ)
124	122, 123	subcld 11609	. . . . . . . . . 10 ⊢ (𝑙 ∈ (1...𝑘) → ((2 · 𝑙) − 1) ∈ ℂ)
125		addcl 11228	. . . . . . . . . 10 ⊢ ((((𝑘↑2) − 𝑘) ∈ ℂ ∧ ((2 · 𝑙) − 1) ∈ ℂ) → (((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) ∈ ℂ)
126	118, 124, 125	syl2an 594	. . . . . . . . 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 15719	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑘 ∈ (1...(𝑦 + 1))) → Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) ∈ ℂ)
129		oveq2 7434	. . . . . . . 8 ⊢ (𝑘 = (𝑦 + 1) → (1...𝑘) = (1...(𝑦 + 1)))
130		oveq1 7433	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑦 + 1) → (𝑘↑2) = ((𝑦 + 1)↑2))
131		id 22	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑦 + 1) → 𝑘 = (𝑦 + 1))
132	130, 131	oveq12d 7444	. . . . . . . . . 10 ⊢ (𝑘 = (𝑦 + 1) → ((𝑘↑2) − 𝑘) = (((𝑦 + 1)↑2) − (𝑦 + 1)))
133	132	oveq1d 7441	. . . . . . . . 9 ⊢ (𝑘 = (𝑦 + 1) → (((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = ((((𝑦 + 1)↑2) − (𝑦 + 1)) + ((2 · 𝑙) − 1)))
134	133	adantr 479	. . . . . . . 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 15692	. . . . . . 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 41901	. . . . . 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 479	. . . . 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 480	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑘 ∈ (1...(𝑦 + 1))) → 𝑘 ∈ ℂ)
139	113, 138, 131	fz1sump1 41901	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...(𝑦 + 1))𝑘 = (Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))
140	139	adantr 479	. . . . . . . 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 7442	. . . . . . 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 15687	. . . . . 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 12182	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 < (Σ𝑘 ∈ (1...𝑦)𝑘 + 1))
144		fzdisj 13568	. . . . . . . . 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 12903	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
147	45, 146	eleqtrdi 2839	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0 → (Σ𝑘 ∈ (1...𝑦)𝑘 + 1) ∈ (ℤ≥‘1))
148	43	uzidd 12876	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ (ℤ≥‘Σ𝑘 ∈ (1...𝑦)𝑘))
149		uzaddcl 12926	. . . . . . . . . 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 582	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0 → (Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)) ∈ (ℤ≥‘Σ𝑘 ∈ (1...𝑦)𝑘))
151		fzsplit2 13566	. . . . . . . . 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 582	. . . . . . . 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 13978	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ0 → (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1))) ∈ Fin)
154		2cnd 12328	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 2 ∈ ℂ)
155		elfzelz 13541	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1))) → 𝑚 ∈ ℤ)
156	155	zcnd 12705	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1))) → 𝑚 ∈ ℂ)
157	156	adantl 480	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 𝑚 ∈ ℂ)
158	154, 157	mulcld 11272	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → (2 · 𝑚) ∈ ℂ)
159		1cnd 11247	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 1 ∈ ℂ)
160	158, 159	subcld 11609	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → ((2 · 𝑚) − 1) ∈ ℂ)
161	145, 152, 153, 160	fsumsplit 15727	. . . . . . 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 479	. . . . . 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 2768	. . . . 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 2778	. . . 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 411	. . 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 12695	. 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 13572	. . . . . . 7 ⊢ (1...𝑁) ⊆ ℕ
168		nnssnn0 12513	. . . . . . 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 41903	. . . 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 15689	. 2 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑘 ∈ (1...𝑁)(𝑘↑3))
175		fzfid 13978	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ∈ Fin)
176	175, 171	fsumnn0cl 15722	. . 3 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)𝑘 ∈ ℕ0)
177		oddnumth 41902	. . 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 2776	1 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)(𝑘↑3) = (Σ𝑘 ∈ (1...𝑁)𝑘↑2))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ≠ wne 2937   ∪ cun 3947   ∩ cin 3948   ⊆ wss 3949  ∅c0 4326   class class class wbr 5152  ‘cfv 6553  (class class class)co 7426  ℂcc 11144  0cc0 11146  1c1 11147   + caddc 11149   · cmul 11151   < clt 11286   − cmin 11482   / cdiv 11909  ℕcn 12250  2c2 12305  3c3 12306  ℕ₀cn0 12510  ℤcz 12596  ℤ_≥cuz 12860  ...cfz 13524  ↑cexp 14066  Σcsu 15672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9672  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-pre-sup 11224

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-isom 6562  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-sup 9473  df-oi 9541  df-card 9970  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-div 11910  df-nn 12251  df-2 12313  df-3 12314  df-n0 12511  df-z 12597  df-uz 12861  df-rp 13015  df-fz 13525  df-fzo 13668  df-seq 14007  df-exp 14067  df-fac 14273  df-bc 14302  df-hash 14330  df-cj 15086  df-re 15087  df-im 15088  df-sqrt 15222  df-abs 15223  df-clim 15472  df-sum 15673

This theorem is referenced by:  sum9cubes  42127