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Theorem sumcubes 41526
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.
StepHypRef Expression
1 oveq2 7420 . . . . 5 (𝑥 = 0 → (1...𝑥) = (1...0))
21sumeq1d 15654 . . . 4 (𝑥 = 0 → Σ𝑘 ∈ (1...𝑥𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑘 ∈ (1...0)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)))
31sumeq1d 15654 . . . . . 6 (𝑥 = 0 → Σ𝑘 ∈ (1...𝑥)𝑘 = Σ𝑘 ∈ (1...0)𝑘)
43oveq2d 7428 . . . . 5 (𝑥 = 0 → (1...Σ𝑘 ∈ (1...𝑥)𝑘) = (1...Σ𝑘 ∈ (1...0)𝑘))
54sumeq1d 15654 . . . 4 (𝑥 = 0 → Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑥)𝑘)((2 · 𝑚) − 1) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...0)𝑘)((2 · 𝑚) − 1))
62, 5eqeq12d 2747 . . 3 (𝑥 = 0 → (Σ𝑘 ∈ (1...𝑥𝑙 ∈ (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 7420 . . . . 5 (𝑥 = 𝑦 → (1...𝑥) = (1...𝑦))
87sumeq1d 15654 . . . 4 (𝑥 = 𝑦 → Σ𝑘 ∈ (1...𝑥𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑘 ∈ (1...𝑦𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)))
97sumeq1d 15654 . . . . . 6 (𝑥 = 𝑦 → Σ𝑘 ∈ (1...𝑥)𝑘 = Σ𝑘 ∈ (1...𝑦)𝑘)
109oveq2d 7428 . . . . 5 (𝑥 = 𝑦 → (1...Σ𝑘 ∈ (1...𝑥)𝑘) = (1...Σ𝑘 ∈ (1...𝑦)𝑘))
1110sumeq1d 15654 . . . 4 (𝑥 = 𝑦 → Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑥)𝑘)((2 · 𝑚) − 1) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑦)𝑘)((2 · 𝑚) − 1))
128, 11eqeq12d 2747 . . 3 (𝑥 = 𝑦 → (Σ𝑘 ∈ (1...𝑥𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑥)𝑘)((2 · 𝑚) − 1) ↔ Σ𝑘 ∈ (1...𝑦𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑦)𝑘)((2 · 𝑚) − 1)))
13 oveq2 7420 . . . . 5 (𝑥 = (𝑦 + 1) → (1...𝑥) = (1...(𝑦 + 1)))
1413sumeq1d 15654 . . . 4 (𝑥 = (𝑦 + 1) → Σ𝑘 ∈ (1...𝑥𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑘 ∈ (1...(𝑦 + 1))Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)))
1513sumeq1d 15654 . . . . . 6 (𝑥 = (𝑦 + 1) → Σ𝑘 ∈ (1...𝑥)𝑘 = Σ𝑘 ∈ (1...(𝑦 + 1))𝑘)
1615oveq2d 7428 . . . . 5 (𝑥 = (𝑦 + 1) → (1...Σ𝑘 ∈ (1...𝑥)𝑘) = (1...Σ𝑘 ∈ (1...(𝑦 + 1))𝑘))
1716sumeq1d 15654 . . . 4 (𝑥 = (𝑦 + 1) → Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑥)𝑘)((2 · 𝑚) − 1) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...(𝑦 + 1))𝑘)((2 · 𝑚) − 1))
1814, 17eqeq12d 2747 . . 3 (𝑥 = (𝑦 + 1) → (Σ𝑘 ∈ (1...𝑥𝑙 ∈ (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 7420 . . . . 5 (𝑥 = 𝑁 → (1...𝑥) = (1...𝑁))
2019sumeq1d 15654 . . . 4 (𝑥 = 𝑁 → Σ𝑘 ∈ (1...𝑥𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑘 ∈ (1...𝑁𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)))
2119sumeq1d 15654 . . . . . 6 (𝑥 = 𝑁 → Σ𝑘 ∈ (1...𝑥)𝑘 = Σ𝑘 ∈ (1...𝑁)𝑘)
2221oveq2d 7428 . . . . 5 (𝑥 = 𝑁 → (1...Σ𝑘 ∈ (1...𝑥)𝑘) = (1...Σ𝑘 ∈ (1...𝑁)𝑘))
2322sumeq1d 15654 . . . 4 (𝑥 = 𝑁 → Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑥)𝑘)((2 · 𝑚) − 1) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑁)𝑘)((2 · 𝑚) − 1))
2420, 23eqeq12d 2747 . . 3 (𝑥 = 𝑁 → (Σ𝑘 ∈ (1...𝑥𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑥)𝑘)((2 · 𝑚) − 1) ↔ Σ𝑘 ∈ (1...𝑁𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑁)𝑘)((2 · 𝑚) − 1)))
25 sum0 15674 . . . . 5 Σ𝑘 ∈ ∅ Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = 0
26 sum0 15674 . . . . 5 Σ𝑚 ∈ ∅ ((2 · 𝑚) − 1) = 0
2725, 26eqtr4i 2762 . . . 4 Σ𝑘 ∈ ∅ Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ ∅ ((2 · 𝑚) − 1)
28 fz10 13529 . . . . 5 (1...0) = ∅
2928sumeq1i 15651 . . . 4 Σ𝑘 ∈ (1...0)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑘 ∈ ∅ Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1))
3028sumeq1i 15651 . . . . . . . 8 Σ𝑘 ∈ (1...0)𝑘 = Σ𝑘 ∈ ∅ 𝑘
31 sum0 15674 . . . . . . . 8 Σ𝑘 ∈ ∅ 𝑘 = 0
3230, 31eqtri 2759 . . . . . . 7 Σ𝑘 ∈ (1...0)𝑘 = 0
3332oveq2i 7423 . . . . . 6 (1...Σ𝑘 ∈ (1...0)𝑘) = (1...0)
3433, 28eqtri 2759 . . . . 5 (1...Σ𝑘 ∈ (1...0)𝑘) = ∅
3534sumeq1i 15651 . . . 4 Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...0)𝑘)((2 · 𝑚) − 1) = Σ𝑚 ∈ ∅ ((2 · 𝑚) − 1)
3627, 29, 353eqtr4i 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...𝑦𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑦)𝑘)((2 · 𝑚) − 1)) → Σ𝑘 ∈ (1...𝑦𝑙 ∈ (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 13537 . . . . . . . . . . . . 13 (𝑘 ∈ (1...𝑦) → 𝑘 ∈ ℕ)
4039adantl 481 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0𝑘 ∈ (1...𝑦)) → 𝑘 ∈ ℕ)
4140nnnn0d 12539 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0𝑘 ∈ (1...𝑦)) → 𝑘 ∈ ℕ0)
4238, 41fsumnn0cl 15689 . . . . . . . . . 10 (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℕ0)
4342nn0zd 12591 . . . . . . . . 9 (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℤ)
44 nn0p1nn 12518 . . . . . . . . . . 11 𝑘 ∈ (1...𝑦)𝑘 ∈ ℕ0 → (Σ𝑘 ∈ (1...𝑦)𝑘 + 1) ∈ ℕ)
4542, 44syl 17 . . . . . . . . . 10 (𝑦 ∈ ℕ0 → (Σ𝑘 ∈ (1...𝑦)𝑘 + 1) ∈ ℕ)
4645nnzd 12592 . . . . . . . . 9 (𝑦 ∈ ℕ0 → (Σ𝑘 ∈ (1...𝑦)𝑘 + 1) ∈ ℤ)
47 peano2nn0 12519 . . . . . . . . . . 11 (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ0)
4847nn0zd 12591 . . . . . . . . . 10 (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℤ)
4943, 48zaddcld 12677 . . . . . . . . 9 (𝑦 ∈ ℕ0 → (Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)) ∈ ℤ)
50 2cnd 12297 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 2 ∈ ℂ)
51 elfzelz 13508 . . . . . . . . . . . . 13 (𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1))) → 𝑚 ∈ ℤ)
5251zcnd 12674 . . . . . . . . . . . 12 (𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1))) → 𝑚 ∈ ℂ)
5352adantl 481 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 𝑚 ∈ ℂ)
5450, 53mulcld 11241 . . . . . . . . . 10 ((𝑦 ∈ ℕ0𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → (2 · 𝑚) ∈ ℂ)
55 1cnd 11216 . . . . . . . . . 10 ((𝑦 ∈ ℕ0𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 1 ∈ ℂ)
5654, 55subcld 11578 . . . . . . . . 9 ((𝑦 ∈ ℕ0𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → ((2 · 𝑚) − 1) ∈ ℂ)
57 oveq2 7420 . . . . . . . . . 10 (𝑚 = (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘) → (2 · 𝑚) = (2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)))
5857oveq1d 7427 . . . . . . . . 9 (𝑚 = (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘) → ((2 · 𝑚) − 1) = ((2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)) − 1))
5943, 46, 49, 56, 58fsumshftm 15734 . . . . . . . 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 13508 . . . . . . . . . . . . . . 15 (𝑘 ∈ (1...𝑦) → 𝑘 ∈ ℤ)
6160adantl 481 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℕ0𝑘 ∈ (1...𝑦)) → 𝑘 ∈ ℤ)
6261zred 12673 . . . . . . . . . . . . 13 ((𝑦 ∈ ℕ0𝑘 ∈ (1...𝑦)) → 𝑘 ∈ ℝ)
6338, 62fsumrecl 15687 . . . . . . . . . . . 12 (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℝ)
6463recnd 11249 . . . . . . . . . . 11 (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℂ)
65 1cnd 11216 . . . . . . . . . . 11 (𝑦 ∈ ℕ0 → 1 ∈ ℂ)
6664, 65pncan2d 11580 . . . . . . . . . 10 (𝑦 ∈ ℕ0 → ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1) − Σ𝑘 ∈ (1...𝑦)𝑘) = 1)
6747nn0cnd 12541 . . . . . . . . . . 11 (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℂ)
6864, 67pncan2d 11580 . . . . . . . . . 10 (𝑦 ∈ ℕ0 → ((Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)) − Σ𝑘 ∈ (1...𝑦)𝑘) = (𝑦 + 1))
6966, 68oveq12d 7430 . . . . . . . . 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 13508 . . . . . . . . . . 11 (𝑙 ∈ (((Σ𝑘 ∈ (1...𝑦)𝑘 + 1) − Σ𝑘 ∈ (1...𝑦)𝑘)...((Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)) − Σ𝑘 ∈ (1...𝑦)𝑘)) → 𝑙 ∈ ℤ)
7170zcnd 12674 . . . . . . . . . 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 12297 . . . . . . . . . . . . 13 ((𝑦 ∈ ℕ0𝑙 ∈ ℂ) → 2 ∈ ℂ)
73 simpr 484 . . . . . . . . . . . . 13 ((𝑦 ∈ ℕ0𝑙 ∈ ℂ) → 𝑙 ∈ ℂ)
7464adantr 480 . . . . . . . . . . . . 13 ((𝑦 ∈ ℕ0𝑙 ∈ ℂ) → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℂ)
7572, 73, 74adddid 11245 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0𝑙 ∈ ℂ) → (2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)) = ((2 · 𝑙) + (2 · Σ𝑘 ∈ (1...𝑦)𝑘)))
7675oveq1d 7427 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0𝑙 ∈ ℂ) → ((2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)) − 1) = (((2 · 𝑙) + (2 · Σ𝑘 ∈ (1...𝑦)𝑘)) − 1))
7772, 73mulcld 11241 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0𝑙 ∈ ℂ) → (2 · 𝑙) ∈ ℂ)
7872, 74mulcld 11241 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0𝑙 ∈ ℂ) → (2 · Σ𝑘 ∈ (1...𝑦)𝑘) ∈ ℂ)
79 1cnd 11216 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0𝑙 ∈ ℂ) → 1 ∈ ℂ)
8077, 78, 79addsubassd 11598 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0𝑙 ∈ ℂ) → (((2 · 𝑙) + (2 · Σ𝑘 ∈ (1...𝑦)𝑘)) − 1) = ((2 · 𝑙) + ((2 · Σ𝑘 ∈ (1...𝑦)𝑘) − 1)))
8177, 78, 79addsub12d 11601 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0𝑙 ∈ ℂ) → ((2 · 𝑙) + ((2 · Σ𝑘 ∈ (1...𝑦)𝑘) − 1)) = ((2 · Σ𝑘 ∈ (1...𝑦)𝑘) + ((2 · 𝑙) − 1)))
82 arisum 15813 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 = (((𝑦↑2) + 𝑦) / 2))
8382oveq2d 7428 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℕ0 → (2 · Σ𝑘 ∈ (1...𝑦)𝑘) = (2 · (((𝑦↑2) + 𝑦) / 2)))
84 nn0cn 12489 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℕ0𝑦 ∈ ℂ)
8584sqcld 14116 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℕ0 → (𝑦↑2) ∈ ℂ)
8685, 84addcld 11240 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ0 → ((𝑦↑2) + 𝑦) ∈ ℂ)
87 2cnd 12297 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ0 → 2 ∈ ℂ)
88 2ne0 12323 . . . . . . . . . . . . . . . . 17 2 ≠ 0
8988a1i 11 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ0 → 2 ≠ 0)
9086, 87, 89divcan2d 11999 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℕ0 → (2 · (((𝑦↑2) + 𝑦) / 2)) = ((𝑦↑2) + 𝑦))
91 binom21 14189 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℂ → ((𝑦 + 1)↑2) = (((𝑦↑2) + (2 · 𝑦)) + 1))
9284, 91syl 17 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℕ0 → ((𝑦 + 1)↑2) = (((𝑦↑2) + (2 · 𝑦)) + 1))
9392oveq1d 7427 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ0 → (((𝑦 + 1)↑2) − (𝑦 + 1)) = ((((𝑦↑2) + (2 · 𝑦)) + 1) − (𝑦 + 1)))
9487, 84mulcld 11241 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℕ0 → (2 · 𝑦) ∈ ℂ)
9585, 94addcld 11240 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℕ0 → ((𝑦↑2) + (2 · 𝑦)) ∈ ℂ)
9695, 84, 65pnpcan2d 11616 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ0 → ((((𝑦↑2) + (2 · 𝑦)) + 1) − (𝑦 + 1)) = (((𝑦↑2) + (2 · 𝑦)) − 𝑦))
9785, 94, 84addsubassd 11598 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℕ0 → (((𝑦↑2) + (2 · 𝑦)) − 𝑦) = ((𝑦↑2) + ((2 · 𝑦) − 𝑦)))
98842timesd 12462 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℕ0 → (2 · 𝑦) = (𝑦 + 𝑦))
9984, 84, 98mvrladdd 11634 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℕ0 → ((2 · 𝑦) − 𝑦) = 𝑦)
10099oveq2d 7428 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℕ0 → ((𝑦↑2) + ((2 · 𝑦) − 𝑦)) = ((𝑦↑2) + 𝑦))
10197, 100eqtrd 2771 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ0 → (((𝑦↑2) + (2 · 𝑦)) − 𝑦) = ((𝑦↑2) + 𝑦))
10293, 96, 1013eqtrrd 2776 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℕ0 → ((𝑦↑2) + 𝑦) = (((𝑦 + 1)↑2) − (𝑦 + 1)))
10383, 90, 1023eqtrd 2775 . . . . . . . . . . . . . 14 (𝑦 ∈ ℕ0 → (2 · Σ𝑘 ∈ (1...𝑦)𝑘) = (((𝑦 + 1)↑2) − (𝑦 + 1)))
104103adantr 480 . . . . . . . . . . . . 13 ((𝑦 ∈ ℕ0𝑙 ∈ ℂ) → (2 · Σ𝑘 ∈ (1...𝑦)𝑘) = (((𝑦 + 1)↑2) − (𝑦 + 1)))
105104oveq1d 7427 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0𝑙 ∈ ℂ) → ((2 · Σ𝑘 ∈ (1...𝑦)𝑘) + ((2 · 𝑙) − 1)) = ((((𝑦 + 1)↑2) − (𝑦 + 1)) + ((2 · 𝑙) − 1)))
10681, 105eqtrd 2771 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0𝑙 ∈ ℂ) → ((2 · 𝑙) + ((2 · Σ𝑘 ∈ (1...𝑦)𝑘) − 1)) = ((((𝑦 + 1)↑2) − (𝑦 + 1)) + ((2 · 𝑙) − 1)))
10776, 80, 1063eqtrd 2775 . . . . . . . . . 10 ((𝑦 ∈ ℕ0𝑙 ∈ ℂ) → ((2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)) − 1) = ((((𝑦 + 1)↑2) − (𝑦 + 1)) + ((2 · 𝑙) − 1)))
10871, 107sylan2 592 . . . . . . . . 9 ((𝑦 ∈ ℕ0𝑙 ∈ (((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)))
10969, 108sumeq12dv 15659 . . . . . . . 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)))
11059, 109eqtr2d 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))
111110adantr 480 . . . . . 6 ((𝑦 ∈ ℕ0 ∧ Σ𝑘 ∈ (1...𝑦𝑙 ∈ (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))
11237, 111oveq12d 7430 . . . . 5 ((𝑦 ∈ ℕ0 ∧ Σ𝑘 ∈ (1...𝑦𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑦)𝑘)((2 · 𝑚) − 1)) → (Σ𝑘 ∈ (1...𝑦𝑙 ∈ (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 ∈ ℕ0𝑦 ∈ ℕ0)
114 fzfid 13945 . . . . . . . 8 ((𝑦 ∈ ℕ0𝑘 ∈ (1...(𝑦 + 1))) → (1...𝑘) ∈ Fin)
115 elfzelz 13508 . . . . . . . . . . . . 13 (𝑘 ∈ (1...(𝑦 + 1)) → 𝑘 ∈ ℤ)
116115zcnd 12674 . . . . . . . . . . . 12 (𝑘 ∈ (1...(𝑦 + 1)) → 𝑘 ∈ ℂ)
117116sqcld 14116 . . . . . . . . . . 11 (𝑘 ∈ (1...(𝑦 + 1)) → (𝑘↑2) ∈ ℂ)
118117, 116subcld 11578 . . . . . . . . . 10 (𝑘 ∈ (1...(𝑦 + 1)) → ((𝑘↑2) − 𝑘) ∈ ℂ)
119 2cnd 12297 . . . . . . . . . . . 12 (𝑙 ∈ (1...𝑘) → 2 ∈ ℂ)
120 elfzelz 13508 . . . . . . . . . . . . 13 (𝑙 ∈ (1...𝑘) → 𝑙 ∈ ℤ)
121120zcnd 12674 . . . . . . . . . . . 12 (𝑙 ∈ (1...𝑘) → 𝑙 ∈ ℂ)
122119, 121mulcld 11241 . . . . . . . . . . 11 (𝑙 ∈ (1...𝑘) → (2 · 𝑙) ∈ ℂ)
123 1cnd 11216 . . . . . . . . . . 11 (𝑙 ∈ (1...𝑘) → 1 ∈ ℂ)
124122, 123subcld 11578 . . . . . . . . . 10 (𝑙 ∈ (1...𝑘) → ((2 · 𝑙) − 1) ∈ ℂ)
125 addcl 11198 . . . . . . . . . 10 ((((𝑘↑2) − 𝑘) ∈ ℂ ∧ ((2 · 𝑙) − 1) ∈ ℂ) → (((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) ∈ ℂ)
126118, 124, 125syl2an 595 . . . . . . . . 9 ((𝑘 ∈ (1...(𝑦 + 1)) ∧ 𝑙 ∈ (1...𝑘)) → (((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) ∈ ℂ)
127126adantll 711 . . . . . . . 8 (((𝑦 ∈ ℕ0𝑘 ∈ (1...(𝑦 + 1))) ∧ 𝑙 ∈ (1...𝑘)) → (((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) ∈ ℂ)
128114, 127fsumcl 15686 . . . . . . 7 ((𝑦 ∈ ℕ0𝑘 ∈ (1...(𝑦 + 1))) → Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) ∈ ℂ)
129 oveq2 7420 . . . . . . . 8 (𝑘 = (𝑦 + 1) → (1...𝑘) = (1...(𝑦 + 1)))
130 oveq1 7419 . . . . . . . . . . 11 (𝑘 = (𝑦 + 1) → (𝑘↑2) = ((𝑦 + 1)↑2))
131 id 22 . . . . . . . . . . 11 (𝑘 = (𝑦 + 1) → 𝑘 = (𝑦 + 1))
132130, 131oveq12d 7430 . . . . . . . . . 10 (𝑘 = (𝑦 + 1) → ((𝑘↑2) − 𝑘) = (((𝑦 + 1)↑2) − (𝑦 + 1)))
133132oveq1d 7427 . . . . . . . . 9 (𝑘 = (𝑦 + 1) → (((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = ((((𝑦 + 1)↑2) − (𝑦 + 1)) + ((2 · 𝑙) − 1)))
134133adantr 480 . . . . . . . 8 ((𝑘 = (𝑦 + 1) ∧ 𝑙 ∈ (1...𝑘)) → (((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = ((((𝑦 + 1)↑2) − (𝑦 + 1)) + ((2 · 𝑙) − 1)))
135129, 134sumeq12dv 15659 . . . . . . 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)))
136113, 128, 135fz1sump1 41523 . . . . . 6 (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...(𝑦 + 1))Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = (Σ𝑘 ∈ (1...𝑦𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) + Σ𝑙 ∈ (1...(𝑦 + 1))((((𝑦 + 1)↑2) − (𝑦 + 1)) + ((2 · 𝑙) − 1))))
137136adantr 480 . . . . 5 ((𝑦 ∈ ℕ0 ∧ Σ𝑘 ∈ (1...𝑦𝑙 ∈ (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...𝑦𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) + Σ𝑙 ∈ (1...(𝑦 + 1))((((𝑦 + 1)↑2) − (𝑦 + 1)) + ((2 · 𝑙) − 1))))
138116adantl 481 . . . . . . . . . 10 ((𝑦 ∈ ℕ0𝑘 ∈ (1...(𝑦 + 1))) → 𝑘 ∈ ℂ)
139113, 138, 131fz1sump1 41523 . . . . . . . . 9 (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...(𝑦 + 1))𝑘 = (Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))
140139adantr 480 . . . . . . . 8 ((𝑦 ∈ ℕ0 ∧ Σ𝑘 ∈ (1...𝑦𝑙 ∈ (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)))
141140oveq2d 7428 . . . . . . 7 ((𝑦 ∈ ℕ0 ∧ Σ𝑘 ∈ (1...𝑦𝑙 ∈ (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))))
142141sumeq1d 15654 . . . . . 6 ((𝑦 ∈ ℕ0 ∧ Σ𝑘 ∈ (1...𝑦𝑙 ∈ (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))
14363ltp1d 12151 . . . . . . . . 9 (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 < (Σ𝑘 ∈ (1...𝑦)𝑘 + 1))
144 fzdisj 13535 . . . . . . . . 9 𝑘 ∈ (1...𝑦)𝑘 < (Σ𝑘 ∈ (1...𝑦)𝑘 + 1) → ((1...Σ𝑘 ∈ (1...𝑦)𝑘) ∩ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) = ∅)
145143, 144syl 17 . . . . . . . 8 (𝑦 ∈ ℕ0 → ((1...Σ𝑘 ∈ (1...𝑦)𝑘) ∩ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) = ∅)
146 nnuz 12872 . . . . . . . . . 10 ℕ = (ℤ‘1)
14745, 146eleqtrdi 2842 . . . . . . . . 9 (𝑦 ∈ ℕ0 → (Σ𝑘 ∈ (1...𝑦)𝑘 + 1) ∈ (ℤ‘1))
14843uzidd 12845 . . . . . . . . . 10 (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ (ℤ‘Σ𝑘 ∈ (1...𝑦)𝑘))
149 uzaddcl 12895 . . . . . . . . . 10 ((Σ𝑘 ∈ (1...𝑦)𝑘 ∈ (ℤ‘Σ𝑘 ∈ (1...𝑦)𝑘) ∧ (𝑦 + 1) ∈ ℕ0) → (Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)) ∈ (ℤ‘Σ𝑘 ∈ (1...𝑦)𝑘))
150148, 47, 149syl2anc 583 . . . . . . . . 9 (𝑦 ∈ ℕ0 → (Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)) ∈ (ℤ‘Σ𝑘 ∈ (1...𝑦)𝑘))
151 fzsplit2 13533 . . . . . . . . 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)))))
152147, 150, 151syl2anc 583 . . . . . . . 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 12297 . . . . . . . . . 10 ((𝑦 ∈ ℕ0𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 2 ∈ ℂ)
155 elfzelz 13508 . . . . . . . . . . . 12 (𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1))) → 𝑚 ∈ ℤ)
156155zcnd 12674 . . . . . . . . . . 11 (𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1))) → 𝑚 ∈ ℂ)
157156adantl 481 . . . . . . . . . 10 ((𝑦 ∈ ℕ0𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 𝑚 ∈ ℂ)
158154, 157mulcld 11241 . . . . . . . . 9 ((𝑦 ∈ ℕ0𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → (2 · 𝑚) ∈ ℂ)
159 1cnd 11216 . . . . . . . . 9 ((𝑦 ∈ ℕ0𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 1 ∈ ℂ)
160158, 159subcld 11578 . . . . . . . 8 ((𝑦 ∈ ℕ0𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → ((2 · 𝑚) − 1) ∈ ℂ)
161145, 152, 153, 160fsumsplit 15694 . . . . . . 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)))
162161adantr 480 . . . . . 6 ((𝑦 ∈ ℕ0 ∧ Σ𝑘 ∈ (1...𝑦𝑙 ∈ (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)))
163142, 162eqtrd 2771 . . . . 5 ((𝑦 ∈ ℕ0 ∧ Σ𝑘 ∈ (1...𝑦𝑙 ∈ (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)))
164112, 137, 1633eqtr4d 2781 . . . 4 ((𝑦 ∈ ℕ0 ∧ Σ𝑘 ∈ (1...𝑦𝑙 ∈ (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))
165164ex 412 . . 3 (𝑦 ∈ ℕ0 → (Σ𝑘 ∈ (1...𝑦𝑙 ∈ (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)))
1666, 12, 18, 24, 36, 165nn0ind 12664 . 2 (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑁)𝑘)((2 · 𝑚) − 1))
167 fz1ssnn 13539 . . . . . . 7 (1...𝑁) ⊆ ℕ
168 nnssnn0 12482 . . . . . . 7 ℕ ⊆ ℕ0
169167, 168sstri 3991 . . . . . 6 (1...𝑁) ⊆ ℕ0
170169a1i 11 . . . . 5 (𝑁 ∈ ℕ0 → (1...𝑁) ⊆ ℕ0)
171170sselda 3982 . . . 4 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℕ0)
172 nicomachus 41525 . . . 4 (𝑘 ∈ ℕ0 → Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = (𝑘↑3))
173171, 172syl 17 . . 3 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = (𝑘↑3))
174173sumeq2dv 15656 . 2 (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑘 ∈ (1...𝑁)(𝑘↑3))
175 fzfid 13945 . . . 4 (𝑁 ∈ ℕ0 → (1...𝑁) ∈ Fin)
176175, 171fsumnn0cl 15689 . . 3 (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)𝑘 ∈ ℕ0)
177 oddnumth 41524 . . 3 𝑘 ∈ (1...𝑁)𝑘 ∈ ℕ0 → Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑁)𝑘)((2 · 𝑚) − 1) = (Σ𝑘 ∈ (1...𝑁)𝑘↑2))
178176, 177syl 17 . 2 (𝑁 ∈ ℕ0 → Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑁)𝑘)((2 · 𝑚) − 1) = (Σ𝑘 ∈ (1...𝑁)𝑘↑2))
179166, 174, 1783eqtr3d 2779 1 (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)(𝑘↑3) = (Σ𝑘 ∈ (1...𝑁)𝑘↑2))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  wne 2939  cun 3946  cin 3947  wss 3948  c0 4322   class class class wbr 5148  cfv 6543  (class class class)co 7412  cc 11114  0cc0 11116  1c1 11117   + caddc 11119   · cmul 11121   < clt 11255  cmin 11451   / cdiv 11878  cn 12219  2c2 12274  3c3 12275  0cn0 12479  cz 12565  cuz 12829  ...cfz 13491  cexp 14034  Σcsu 15639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-inf2 9642  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-pre-sup 11194
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  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 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-sup 9443  df-oi 9511  df-card 9940  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-div 11879  df-nn 12220  df-2 12282  df-3 12283  df-n0 12480  df-z 12566  df-uz 12830  df-rp 12982  df-fz 13492  df-fzo 13635  df-seq 13974  df-exp 14035  df-fac 14241  df-bc 14270  df-hash 14298  df-cj 15053  df-re 15054  df-im 15055  df-sqrt 15189  df-abs 15190  df-clim 15439  df-sum 15640
This theorem is referenced by:  sum9cubes  41729
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