Theorem sumcubes 42745

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 7375	. . . . 5 ⊢ (𝑥 = 0 → (1...𝑥) = (1...0))
2	1	sumeq1d 15662	. . . 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 15662	. . . . . 6 ⊢ (𝑥 = 0 → Σ𝑘 ∈ (1...𝑥)𝑘 = Σ𝑘 ∈ (1...0)𝑘)
4	3	oveq2d 7383	. . . . 5 ⊢ (𝑥 = 0 → (1...Σ𝑘 ∈ (1...𝑥)𝑘) = (1...Σ𝑘 ∈ (1...0)𝑘))
5	4	sumeq1d 15662	. . . 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 2752	. . 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 7375	. . . . 5 ⊢ (𝑥 = 𝑦 → (1...𝑥) = (1...𝑦))
8	7	sumeq1d 15662	. . . 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 15662	. . . . . 6 ⊢ (𝑥 = 𝑦 → Σ𝑘 ∈ (1...𝑥)𝑘 = Σ𝑘 ∈ (1...𝑦)𝑘)
10	9	oveq2d 7383	. . . . 5 ⊢ (𝑥 = 𝑦 → (1...Σ𝑘 ∈ (1...𝑥)𝑘) = (1...Σ𝑘 ∈ (1...𝑦)𝑘))
11	10	sumeq1d 15662	. . . 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 2752	. . 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 7375	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (1...𝑥) = (1...(𝑦 + 1)))
14	13	sumeq1d 15662	. . . 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 15662	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → Σ𝑘 ∈ (1...𝑥)𝑘 = Σ𝑘 ∈ (1...(𝑦 + 1))𝑘)
16	15	oveq2d 7383	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (1...Σ𝑘 ∈ (1...𝑥)𝑘) = (1...Σ𝑘 ∈ (1...(𝑦 + 1))𝑘))
17	16	sumeq1d 15662	. . . 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 2752	. . 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 7375	. . . . 5 ⊢ (𝑥 = 𝑁 → (1...𝑥) = (1...𝑁))
20	19	sumeq1d 15662	. . . 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 15662	. . . . . 6 ⊢ (𝑥 = 𝑁 → Σ𝑘 ∈ (1...𝑥)𝑘 = Σ𝑘 ∈ (1...𝑁)𝑘)
22	21	oveq2d 7383	. . . . 5 ⊢ (𝑥 = 𝑁 → (1...Σ𝑘 ∈ (1...𝑥)𝑘) = (1...Σ𝑘 ∈ (1...𝑁)𝑘))
23	22	sumeq1d 15662	. . . 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 2752	. . 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 15683	. . . . 5 ⊢ Σ𝑘 ∈ ∅ Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = 0
26		sum0 15683	. . . . 5 ⊢ Σ𝑚 ∈ ∅ ((2 · 𝑚) − 1) = 0
27	25, 26	eqtr4i 2762	. . . 4 ⊢ Σ𝑘 ∈ ∅ Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ ∅ ((2 · 𝑚) − 1)
28		fz10 13499	. . . . 5 ⊢ (1...0) = ∅
29	28	sumeq1i 15659	. . . 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 15659	. . . . . . . 8 ⊢ Σ𝑘 ∈ (1...0)𝑘 = Σ𝑘 ∈ ∅ 𝑘
31		sum0 15683	. . . . . . . 8 ⊢ Σ𝑘 ∈ ∅ 𝑘 = 0
32	30, 31	eqtri 2759	. . . . . . 7 ⊢ Σ𝑘 ∈ (1...0)𝑘 = 0
33	32	oveq2i 7378	. . . . . 6 ⊢ (1...Σ𝑘 ∈ (1...0)𝑘) = (1...0)
34	33, 28	eqtri 2759	. . . . 5 ⊢ (1...Σ𝑘 ∈ (1...0)𝑘) = ∅
35	34	sumeq1i 15659	. . . 4 ⊢ Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...0)𝑘)((2 · 𝑚) − 1) = Σ𝑚 ∈ ∅ ((2 · 𝑚) − 1)
36	27, 29, 35	3eqtr4i 2769	. . 3 ⊢ Σ𝑘 ∈ (1...0)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...0)𝑘)((2 · 𝑚) − 1)
37		simpr 484	. . . . . 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 13935	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 → (1...𝑦) ∈ Fin)
39		elfznn 13507	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝑦) → 𝑘 ∈ ℕ)
40	39	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑦)) → 𝑘 ∈ ℕ)
41	40	nnnn0d 12498	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑦)) → 𝑘 ∈ ℕ0)
42	38, 41	fsumnn0cl 15698	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℕ0)
43	42	nn0zd 12549	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℤ)
44		nn0p1nn 12476	. . . . . . . . . . 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 12550	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0 → (Σ𝑘 ∈ (1...𝑦)𝑘 + 1) ∈ ℤ)
47		peano2nn0 12477	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ0)
48	47	nn0zd 12549	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℤ)
49	43, 48	zaddcld 12637	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0 → (Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)) ∈ ℤ)
50		2cnd 12259	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 2 ∈ ℂ)
51		elfzelz 13478	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1))) → 𝑚 ∈ ℤ)
52	51	zcnd 12634	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1))) → 𝑚 ∈ ℂ)
53	52	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 𝑚 ∈ ℂ)
54	50, 53	mulcld 11165	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → (2 · 𝑚) ∈ ℂ)
55		1cnd 11139	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 1 ∈ ℂ)
56	54, 55	subcld 11505	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → ((2 · 𝑚) − 1) ∈ ℂ)
57		oveq2 7375	. . . . . . . . . 10 ⊢ (𝑚 = (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘) → (2 · 𝑚) = (2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)))
58	57	oveq1d 7382	. . . . . . . . 9 ⊢ (𝑚 = (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘) → ((2 · 𝑚) − 1) = ((2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)) − 1))
59	43, 46, 49, 56, 58	fsumshftm 15743	. . . . . . . 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 13478	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑦) → 𝑘 ∈ ℤ)
61	60	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑦)) → 𝑘 ∈ ℤ)
62	61	zred 12633	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑦)) → 𝑘 ∈ ℝ)
63	38, 62	fsumrecl 15696	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℝ)
64	63	recnd 11173	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℂ)
65		1cnd 11139	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 → 1 ∈ ℂ)
66	64, 65	pncan2d 11507	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0 → ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1) − Σ𝑘 ∈ (1...𝑦)𝑘) = 1)
67	47	nn0cnd 12500	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℂ)
68	64, 67	pncan2d 11507	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0 → ((Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)) − Σ𝑘 ∈ (1...𝑦)𝑘) = (𝑦 + 1))
69	66, 68	oveq12d 7385	. . . . . . . . 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 13478	. . . . . . . . . . 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 12634	. . . . . . . . . 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 12259	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ) → 2 ∈ ℂ)
73		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ) → 𝑙 ∈ ℂ)
74	64	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ) → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℂ)
75	72, 73, 74	adddid 11169	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ) → (2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)) = ((2 · 𝑙) + (2 · Σ𝑘 ∈ (1...𝑦)𝑘)))
76	75	oveq1d 7382	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ) → ((2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)) − 1) = (((2 · 𝑙) + (2 · Σ𝑘 ∈ (1...𝑦)𝑘)) − 1))
77	72, 73	mulcld 11165	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ) → (2 · 𝑙) ∈ ℂ)
78	72, 74	mulcld 11165	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ) → (2 · Σ𝑘 ∈ (1...𝑦)𝑘) ∈ ℂ)
79		1cnd 11139	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ) → 1 ∈ ℂ)
80	77, 78, 79	addsubassd 11525	. . . . . . . . . . 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 11528	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ) → ((2 · 𝑙) + ((2 · Σ𝑘 ∈ (1...𝑦)𝑘) − 1)) = ((2 · Σ𝑘 ∈ (1...𝑦)𝑘) + ((2 · 𝑙) − 1)))
82		arisum 15825	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 = (((𝑦↑2) + 𝑦) / 2))
83	82	oveq2d 7383	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ0 → (2 · Σ𝑘 ∈ (1...𝑦)𝑘) = (2 · (((𝑦↑2) + 𝑦) / 2)))
84		nn0cn 12447	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℂ)
85	84	sqcld 14106	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℕ0 → (𝑦↑2) ∈ ℂ)
86	85, 84	addcld 11164	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ0 → ((𝑦↑2) + 𝑦) ∈ ℂ)
87		2cnd 12259	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ0 → 2 ∈ ℂ)
88		2ne0 12285	. . . . . . . . . . . . . . . . 17 ⊢ 2 ≠ 0
89	88	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ0 → 2 ≠ 0)
90	86, 87, 89	divcan2d 11933	. . . . . . . . . . . . . . 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 7382	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ0 → (((𝑦 + 1)↑2) − (𝑦 + 1)) = ((((𝑦↑2) + (2 · 𝑦)) + 1) − (𝑦 + 1)))
94	87, 84	mulcld 11165	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ0 → (2 · 𝑦) ∈ ℂ)
95	85, 94	addcld 11164	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℕ0 → ((𝑦↑2) + (2 · 𝑦)) ∈ ℂ)
96	95, 84, 65	pnpcan2d 11543	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ0 → ((((𝑦↑2) + (2 · 𝑦)) + 1) − (𝑦 + 1)) = (((𝑦↑2) + (2 · 𝑦)) − 𝑦))
97	85, 94, 84	addsubassd 11525	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℕ0 → (((𝑦↑2) + (2 · 𝑦)) − 𝑦) = ((𝑦↑2) + ((2 · 𝑦) − 𝑦)))
98	84	2timesd 12420	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℕ0 → (2 · 𝑦) = (𝑦 + 𝑦))
99	84, 84, 98	mvrladdd 11563	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ0 → ((2 · 𝑦) − 𝑦) = 𝑦)
100	99	oveq2d 7383	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℕ0 → ((𝑦↑2) + ((2 · 𝑦) − 𝑦)) = ((𝑦↑2) + 𝑦))
101	97, 100	eqtrd 2771	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ0 → (((𝑦↑2) + (2 · 𝑦)) − 𝑦) = ((𝑦↑2) + 𝑦))
102	93, 96, 101	3eqtrrd 2776	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ0 → ((𝑦↑2) + 𝑦) = (((𝑦 + 1)↑2) − (𝑦 + 1)))
103	83, 90, 102	3eqtrd 2775	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℕ0 → (2 · Σ𝑘 ∈ (1...𝑦)𝑘) = (((𝑦 + 1)↑2) − (𝑦 + 1)))
104	103	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ) → (2 · Σ𝑘 ∈ (1...𝑦)𝑘) = (((𝑦 + 1)↑2) − (𝑦 + 1)))
105	104	oveq1d 7382	. . . . . . . . . . . 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 2771	. . . . . . . . . . 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 2775	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ) → ((2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)) − 1) = ((((𝑦 + 1)↑2) − (𝑦 + 1)) + ((2 · 𝑙) − 1)))
108	71, 107	sylan2 594	. . . . . . . . 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 15668	. . . . . . . 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 2772	. . . . . . 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 480	. . . . . 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 7385	. . . . 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 13935	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑘 ∈ (1...(𝑦 + 1))) → (1...𝑘) ∈ Fin)
115		elfzelz 13478	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...(𝑦 + 1)) → 𝑘 ∈ ℤ)
116	115	zcnd 12634	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...(𝑦 + 1)) → 𝑘 ∈ ℂ)
117	116	sqcld 14106	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...(𝑦 + 1)) → (𝑘↑2) ∈ ℂ)
118	117, 116	subcld 11505	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...(𝑦 + 1)) → ((𝑘↑2) − 𝑘) ∈ ℂ)
119		2cnd 12259	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ (1...𝑘) → 2 ∈ ℂ)
120		elfzelz 13478	. . . . . . . . . . . . 13 ⊢ (𝑙 ∈ (1...𝑘) → 𝑙 ∈ ℤ)
121	120	zcnd 12634	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ (1...𝑘) → 𝑙 ∈ ℂ)
122	119, 121	mulcld 11165	. . . . . . . . . . 11 ⊢ (𝑙 ∈ (1...𝑘) → (2 · 𝑙) ∈ ℂ)
123		1cnd 11139	. . . . . . . . . . 11 ⊢ (𝑙 ∈ (1...𝑘) → 1 ∈ ℂ)
124	122, 123	subcld 11505	. . . . . . . . . 10 ⊢ (𝑙 ∈ (1...𝑘) → ((2 · 𝑙) − 1) ∈ ℂ)
125		addcl 11120	. . . . . . . . . 10 ⊢ ((((𝑘↑2) − 𝑘) ∈ ℂ ∧ ((2 · 𝑙) − 1) ∈ ℂ) → (((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) ∈ ℂ)
126	118, 124, 125	syl2an 597	. . . . . . . . 9 ⊢ ((𝑘 ∈ (1...(𝑦 + 1)) ∧ 𝑙 ∈ (1...𝑘)) → (((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) ∈ ℂ)
127	126	adantll 715	. . . . . . . 8 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝑘 ∈ (1...(𝑦 + 1))) ∧ 𝑙 ∈ (1...𝑘)) → (((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) ∈ ℂ)
128	114, 127	fsumcl 15695	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑘 ∈ (1...(𝑦 + 1))) → Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) ∈ ℂ)
129		oveq2 7375	. . . . . . . 8 ⊢ (𝑘 = (𝑦 + 1) → (1...𝑘) = (1...(𝑦 + 1)))
130		oveq1 7374	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑦 + 1) → (𝑘↑2) = ((𝑦 + 1)↑2))
131		id 22	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑦 + 1) → 𝑘 = (𝑦 + 1))
132	130, 131	oveq12d 7385	. . . . . . . . . 10 ⊢ (𝑘 = (𝑦 + 1) → ((𝑘↑2) − 𝑘) = (((𝑦 + 1)↑2) − (𝑦 + 1)))
133	132	oveq1d 7382	. . . . . . . . 9 ⊢ (𝑘 = (𝑦 + 1) → (((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = ((((𝑦 + 1)↑2) − (𝑦 + 1)) + ((2 · 𝑙) − 1)))
134	133	adantr 480	. . . . . . . 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 15668	. . . . . . 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 42742	. . . . . 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 480	. . . . 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 481	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑘 ∈ (1...(𝑦 + 1))) → 𝑘 ∈ ℂ)
139	113, 138, 131	fz1sump1 42742	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...(𝑦 + 1))𝑘 = (Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))
140	139	adantr 480	. . . . . . . 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 7383	. . . . . . 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 15662	. . . . . 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 12086	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 < (Σ𝑘 ∈ (1...𝑦)𝑘 + 1))
144		fzdisj 13505	. . . . . . . . 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 12827	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
147	45, 146	eleqtrdi 2846	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0 → (Σ𝑘 ∈ (1...𝑦)𝑘 + 1) ∈ (ℤ≥‘1))
148	43	uzidd 12804	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ (ℤ≥‘Σ𝑘 ∈ (1...𝑦)𝑘))
149		uzaddcl 12854	. . . . . . . . . 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 585	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0 → (Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)) ∈ (ℤ≥‘Σ𝑘 ∈ (1...𝑦)𝑘))
151		fzsplit2 13503	. . . . . . . . 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 585	. . . . . . . 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 13935	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ0 → (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1))) ∈ Fin)
154		2cnd 12259	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 2 ∈ ℂ)
155		elfzelz 13478	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1))) → 𝑚 ∈ ℤ)
156	155	zcnd 12634	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1))) → 𝑚 ∈ ℂ)
157	156	adantl 481	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 𝑚 ∈ ℂ)
158	154, 157	mulcld 11165	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → (2 · 𝑚) ∈ ℂ)
159		1cnd 11139	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 1 ∈ ℂ)
160	158, 159	subcld 11505	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → ((2 · 𝑚) − 1) ∈ ℂ)
161	145, 152, 153, 160	fsumsplit 15703	. . . . . . 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 480	. . . . . 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 2771	. . . . 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 2781	. . . 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 412	. . 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 12624	. 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 13509	. . . . . . 7 ⊢ (1...𝑁) ⊆ ℕ
168		nnssnn0 12440	. . . . . . 7 ⊢ ℕ ⊆ ℕ0
169	167, 168	sstri 3931	. . . . . 6 ⊢ (1...𝑁) ⊆ ℕ0
170	169	a1i 11	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ⊆ ℕ0)
171	170	sselda 3921	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℕ0)
172		nicomachus 42744	. . . 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 15664	. 2 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑘 ∈ (1...𝑁)(𝑘↑3))
175		fzfid 13935	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ∈ Fin)
176	175, 171	fsumnn0cl 15698	. . 3 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)𝑘 ∈ ℕ0)
177		oddnumth 42743	. . 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 2779	1 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)(𝑘↑3) = (Σ𝑘 ∈ (1...𝑁)𝑘↑2))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2932   ∪ cun 3887   ∩ cin 3888   ⊆ wss 3889  ∅c0 4273   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  ℂcc 11036  0cc0 11038  1c1 11039   + caddc 11041   · cmul 11043   < clt 11179   − cmin 11377   / cdiv 11807  ℕcn 12174  2c2 12236  3c3 12237  ℕ₀cn0 12437  ℤcz 12524  ℤ_≥cuz 12788  ...cfz 13461  ↑cexp 14023  Σcsu 15648

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-inf2 9562  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-sup 9355  df-oi 9425  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-3 12245  df-n0 12438  df-z 12525  df-uz 12789  df-rp 12943  df-fz 13462  df-fzo 13609  df-seq 13964  df-exp 14024  df-fac 14236  df-bc 14265  df-hash 14293  df-cj 15061  df-re 15062  df-im 15063  df-sqrt 15197  df-abs 15198  df-clim 15450  df-sum 15649

This theorem is referenced by:  sum9cubes  43105