Theorem sumcubes 42308

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 7398	. . . . 5 ⊢ (𝑥 = 0 → (1...𝑥) = (1...0))
2	1	sumeq1d 15673	. . . 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 15673	. . . . . 6 ⊢ (𝑥 = 0 → Σ𝑘 ∈ (1...𝑥)𝑘 = Σ𝑘 ∈ (1...0)𝑘)
4	3	oveq2d 7406	. . . . 5 ⊢ (𝑥 = 0 → (1...Σ𝑘 ∈ (1...𝑥)𝑘) = (1...Σ𝑘 ∈ (1...0)𝑘))
5	4	sumeq1d 15673	. . . 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 2746	. . 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 7398	. . . . 5 ⊢ (𝑥 = 𝑦 → (1...𝑥) = (1...𝑦))
8	7	sumeq1d 15673	. . . 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 15673	. . . . . 6 ⊢ (𝑥 = 𝑦 → Σ𝑘 ∈ (1...𝑥)𝑘 = Σ𝑘 ∈ (1...𝑦)𝑘)
10	9	oveq2d 7406	. . . . 5 ⊢ (𝑥 = 𝑦 → (1...Σ𝑘 ∈ (1...𝑥)𝑘) = (1...Σ𝑘 ∈ (1...𝑦)𝑘))
11	10	sumeq1d 15673	. . . 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 2746	. . 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 7398	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (1...𝑥) = (1...(𝑦 + 1)))
14	13	sumeq1d 15673	. . . 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 15673	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → Σ𝑘 ∈ (1...𝑥)𝑘 = Σ𝑘 ∈ (1...(𝑦 + 1))𝑘)
16	15	oveq2d 7406	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (1...Σ𝑘 ∈ (1...𝑥)𝑘) = (1...Σ𝑘 ∈ (1...(𝑦 + 1))𝑘))
17	16	sumeq1d 15673	. . . 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 2746	. . 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 7398	. . . . 5 ⊢ (𝑥 = 𝑁 → (1...𝑥) = (1...𝑁))
20	19	sumeq1d 15673	. . . 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 15673	. . . . . 6 ⊢ (𝑥 = 𝑁 → Σ𝑘 ∈ (1...𝑥)𝑘 = Σ𝑘 ∈ (1...𝑁)𝑘)
22	21	oveq2d 7406	. . . . 5 ⊢ (𝑥 = 𝑁 → (1...Σ𝑘 ∈ (1...𝑥)𝑘) = (1...Σ𝑘 ∈ (1...𝑁)𝑘))
23	22	sumeq1d 15673	. . . 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 2746	. . 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 15694	. . . . 5 ⊢ Σ𝑘 ∈ ∅ Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = 0
26		sum0 15694	. . . . 5 ⊢ Σ𝑚 ∈ ∅ ((2 · 𝑚) − 1) = 0
27	25, 26	eqtr4i 2756	. . . 4 ⊢ Σ𝑘 ∈ ∅ Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ ∅ ((2 · 𝑚) − 1)
28		fz10 13513	. . . . 5 ⊢ (1...0) = ∅
29	28	sumeq1i 15670	. . . 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 15670	. . . . . . . 8 ⊢ Σ𝑘 ∈ (1...0)𝑘 = Σ𝑘 ∈ ∅ 𝑘
31		sum0 15694	. . . . . . . 8 ⊢ Σ𝑘 ∈ ∅ 𝑘 = 0
32	30, 31	eqtri 2753	. . . . . . 7 ⊢ Σ𝑘 ∈ (1...0)𝑘 = 0
33	32	oveq2i 7401	. . . . . 6 ⊢ (1...Σ𝑘 ∈ (1...0)𝑘) = (1...0)
34	33, 28	eqtri 2753	. . . . 5 ⊢ (1...Σ𝑘 ∈ (1...0)𝑘) = ∅
35	34	sumeq1i 15670	. . . 4 ⊢ Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...0)𝑘)((2 · 𝑚) − 1) = Σ𝑚 ∈ ∅ ((2 · 𝑚) − 1)
36	27, 29, 35	3eqtr4i 2763	. . 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 13945	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 → (1...𝑦) ∈ Fin)
39		elfznn 13521	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝑦) → 𝑘 ∈ ℕ)
40	39	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑦)) → 𝑘 ∈ ℕ)
41	40	nnnn0d 12510	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑦)) → 𝑘 ∈ ℕ0)
42	38, 41	fsumnn0cl 15709	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℕ0)
43	42	nn0zd 12562	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℤ)
44		nn0p1nn 12488	. . . . . . . . . . 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 12563	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0 → (Σ𝑘 ∈ (1...𝑦)𝑘 + 1) ∈ ℤ)
47		peano2nn0 12489	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ0)
48	47	nn0zd 12562	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℤ)
49	43, 48	zaddcld 12649	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0 → (Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)) ∈ ℤ)
50		2cnd 12271	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 2 ∈ ℂ)
51		elfzelz 13492	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1))) → 𝑚 ∈ ℤ)
52	51	zcnd 12646	. . . . . . . . . . . 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 11201	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → (2 · 𝑚) ∈ ℂ)
55		1cnd 11176	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 1 ∈ ℂ)
56	54, 55	subcld 11540	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → ((2 · 𝑚) − 1) ∈ ℂ)
57		oveq2 7398	. . . . . . . . . 10 ⊢ (𝑚 = (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘) → (2 · 𝑚) = (2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)))
58	57	oveq1d 7405	. . . . . . . . 9 ⊢ (𝑚 = (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘) → ((2 · 𝑚) − 1) = ((2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)) − 1))
59	43, 46, 49, 56, 58	fsumshftm 15754	. . . . . . . 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 13492	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑦) → 𝑘 ∈ ℤ)
61	60	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑦)) → 𝑘 ∈ ℤ)
62	61	zred 12645	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑦)) → 𝑘 ∈ ℝ)
63	38, 62	fsumrecl 15707	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℝ)
64	63	recnd 11209	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℂ)
65		1cnd 11176	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 → 1 ∈ ℂ)
66	64, 65	pncan2d 11542	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0 → ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1) − Σ𝑘 ∈ (1...𝑦)𝑘) = 1)
67	47	nn0cnd 12512	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℂ)
68	64, 67	pncan2d 11542	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0 → ((Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)) − Σ𝑘 ∈ (1...𝑦)𝑘) = (𝑦 + 1))
69	66, 68	oveq12d 7408	. . . . . . . . 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 13492	. . . . . . . . . . 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 12646	. . . . . . . . . 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 12271	. . . . . . . . . . . . 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 11205	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ) → (2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)) = ((2 · 𝑙) + (2 · Σ𝑘 ∈ (1...𝑦)𝑘)))
76	75	oveq1d 7405	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ) → ((2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)) − 1) = (((2 · 𝑙) + (2 · Σ𝑘 ∈ (1...𝑦)𝑘)) − 1))
77	72, 73	mulcld 11201	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ) → (2 · 𝑙) ∈ ℂ)
78	72, 74	mulcld 11201	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ) → (2 · Σ𝑘 ∈ (1...𝑦)𝑘) ∈ ℂ)
79		1cnd 11176	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ) → 1 ∈ ℂ)
80	77, 78, 79	addsubassd 11560	. . . . . . . . . . 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 11563	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ) → ((2 · 𝑙) + ((2 · Σ𝑘 ∈ (1...𝑦)𝑘) − 1)) = ((2 · Σ𝑘 ∈ (1...𝑦)𝑘) + ((2 · 𝑙) − 1)))
82		arisum 15833	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 = (((𝑦↑2) + 𝑦) / 2))
83	82	oveq2d 7406	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ0 → (2 · Σ𝑘 ∈ (1...𝑦)𝑘) = (2 · (((𝑦↑2) + 𝑦) / 2)))
84		nn0cn 12459	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℂ)
85	84	sqcld 14116	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℕ0 → (𝑦↑2) ∈ ℂ)
86	85, 84	addcld 11200	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ0 → ((𝑦↑2) + 𝑦) ∈ ℂ)
87		2cnd 12271	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ0 → 2 ∈ ℂ)
88		2ne0 12297	. . . . . . . . . . . . . . . . 17 ⊢ 2 ≠ 0
89	88	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ0 → 2 ≠ 0)
90	86, 87, 89	divcan2d 11967	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ0 → (2 · (((𝑦↑2) + 𝑦) / 2)) = ((𝑦↑2) + 𝑦))
91		binom21 14191	. . . . . . . . . . . . . . . . . 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 7405	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ0 → (((𝑦 + 1)↑2) − (𝑦 + 1)) = ((((𝑦↑2) + (2 · 𝑦)) + 1) − (𝑦 + 1)))
94	87, 84	mulcld 11201	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ0 → (2 · 𝑦) ∈ ℂ)
95	85, 94	addcld 11200	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℕ0 → ((𝑦↑2) + (2 · 𝑦)) ∈ ℂ)
96	95, 84, 65	pnpcan2d 11578	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ0 → ((((𝑦↑2) + (2 · 𝑦)) + 1) − (𝑦 + 1)) = (((𝑦↑2) + (2 · 𝑦)) − 𝑦))
97	85, 94, 84	addsubassd 11560	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℕ0 → (((𝑦↑2) + (2 · 𝑦)) − 𝑦) = ((𝑦↑2) + ((2 · 𝑦) − 𝑦)))
98	84	2timesd 12432	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℕ0 → (2 · 𝑦) = (𝑦 + 𝑦))
99	84, 84, 98	mvrladdd 11598	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ0 → ((2 · 𝑦) − 𝑦) = 𝑦)
100	99	oveq2d 7406	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℕ0 → ((𝑦↑2) + ((2 · 𝑦) − 𝑦)) = ((𝑦↑2) + 𝑦))
101	97, 100	eqtrd 2765	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ0 → (((𝑦↑2) + (2 · 𝑦)) − 𝑦) = ((𝑦↑2) + 𝑦))
102	93, 96, 101	3eqtrrd 2770	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ0 → ((𝑦↑2) + 𝑦) = (((𝑦 + 1)↑2) − (𝑦 + 1)))
103	83, 90, 102	3eqtrd 2769	. . . . . . . . . . . . . 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 7405	. . . . . . . . . . . 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 2765	. . . . . . . . . . 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 2769	. . . . . . . . . 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 15679	. . . . . . . 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 2766	. . . . . . 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 7408	. . . . 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 13945	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑘 ∈ (1...(𝑦 + 1))) → (1...𝑘) ∈ Fin)
115		elfzelz 13492	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...(𝑦 + 1)) → 𝑘 ∈ ℤ)
116	115	zcnd 12646	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...(𝑦 + 1)) → 𝑘 ∈ ℂ)
117	116	sqcld 14116	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...(𝑦 + 1)) → (𝑘↑2) ∈ ℂ)
118	117, 116	subcld 11540	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...(𝑦 + 1)) → ((𝑘↑2) − 𝑘) ∈ ℂ)
119		2cnd 12271	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ (1...𝑘) → 2 ∈ ℂ)
120		elfzelz 13492	. . . . . . . . . . . . 13 ⊢ (𝑙 ∈ (1...𝑘) → 𝑙 ∈ ℤ)
121	120	zcnd 12646	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ (1...𝑘) → 𝑙 ∈ ℂ)
122	119, 121	mulcld 11201	. . . . . . . . . . 11 ⊢ (𝑙 ∈ (1...𝑘) → (2 · 𝑙) ∈ ℂ)
123		1cnd 11176	. . . . . . . . . . 11 ⊢ (𝑙 ∈ (1...𝑘) → 1 ∈ ℂ)
124	122, 123	subcld 11540	. . . . . . . . . 10 ⊢ (𝑙 ∈ (1...𝑘) → ((2 · 𝑙) − 1) ∈ ℂ)
125		addcl 11157	. . . . . . . . . 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 714	. . . . . . . 8 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝑘 ∈ (1...(𝑦 + 1))) ∧ 𝑙 ∈ (1...𝑘)) → (((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) ∈ ℂ)
128	114, 127	fsumcl 15706	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑘 ∈ (1...(𝑦 + 1))) → Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) ∈ ℂ)
129		oveq2 7398	. . . . . . . 8 ⊢ (𝑘 = (𝑦 + 1) → (1...𝑘) = (1...(𝑦 + 1)))
130		oveq1 7397	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑦 + 1) → (𝑘↑2) = ((𝑦 + 1)↑2))
131		id 22	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑦 + 1) → 𝑘 = (𝑦 + 1))
132	130, 131	oveq12d 7408	. . . . . . . . . 10 ⊢ (𝑘 = (𝑦 + 1) → ((𝑘↑2) − 𝑘) = (((𝑦 + 1)↑2) − (𝑦 + 1)))
133	132	oveq1d 7405	. . . . . . . . 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 15679	. . . . . . 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 42305	. . . . . 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 42305	. . . . . . . . 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 7406	. . . . . . 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 15673	. . . . . 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 12120	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 < (Σ𝑘 ∈ (1...𝑦)𝑘 + 1))
144		fzdisj 13519	. . . . . . . . 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 12843	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
147	45, 146	eleqtrdi 2839	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0 → (Σ𝑘 ∈ (1...𝑦)𝑘 + 1) ∈ (ℤ≥‘1))
148	43	uzidd 12816	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ (ℤ≥‘Σ𝑘 ∈ (1...𝑦)𝑘))
149		uzaddcl 12870	. . . . . . . . . 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 13517	. . . . . . . . 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 13945	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ0 → (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1))) ∈ Fin)
154		2cnd 12271	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 2 ∈ ℂ)
155		elfzelz 13492	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1))) → 𝑚 ∈ ℤ)
156	155	zcnd 12646	. . . . . . . . . . 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 11201	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → (2 · 𝑚) ∈ ℂ)
159		1cnd 11176	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 1 ∈ ℂ)
160	158, 159	subcld 11540	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → ((2 · 𝑚) − 1) ∈ ℂ)
161	145, 152, 153, 160	fsumsplit 15714	. . . . . . 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 2765	. . . . 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 2775	. . . 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 12636	. 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 13523	. . . . . . 7 ⊢ (1...𝑁) ⊆ ℕ
168		nnssnn0 12452	. . . . . . 7 ⊢ ℕ ⊆ ℕ0
169	167, 168	sstri 3959	. . . . . 6 ⊢ (1...𝑁) ⊆ ℕ0
170	169	a1i 11	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ⊆ ℕ0)
171	170	sselda 3949	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℕ0)
172		nicomachus 42307	. . . 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 15675	. 2 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑘 ∈ (1...𝑁)(𝑘↑3))
175		fzfid 13945	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ∈ Fin)
176	175, 171	fsumnn0cl 15709	. . 3 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)𝑘 ∈ ℕ0)
177		oddnumth 42306	. . 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 2773	1 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)(𝑘↑3) = (Σ𝑘 ∈ (1...𝑁)𝑘↑2))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926   ∪ cun 3915   ∩ cin 3916   ⊆ wss 3917  ∅c0 4299   class class class wbr 5110  ‘cfv 6514  (class class class)co 7390  ℂcc 11073  0cc0 11075  1c1 11076   + caddc 11078   · cmul 11080   < clt 11215   − cmin 11412   / cdiv 11842  ℕcn 12193  2c2 12248  3c3 12249  ℕ₀cn0 12449  ℤcz 12536  ℤ_≥cuz 12800  ...cfz 13475  ↑cexp 14033  Σcsu 15659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9400  df-oi 9470  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-n0 12450  df-z 12537  df-uz 12801  df-rp 12959  df-fz 13476  df-fzo 13623  df-seq 13974  df-exp 14034  df-fac 14246  df-bc 14275  df-hash 14303  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-clim 15461  df-sum 15660

This theorem is referenced by:  sum9cubes  42667