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Theorem sumcubes 42347
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 7439 . . . . 5 (𝑥 = 0 → (1...𝑥) = (1...0))
21sumeq1d 15736 . . . 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 15736 . . . . . 6 (𝑥 = 0 → Σ𝑘 ∈ (1...𝑥)𝑘 = Σ𝑘 ∈ (1...0)𝑘)
43oveq2d 7447 . . . . 5 (𝑥 = 0 → (1...Σ𝑘 ∈ (1...𝑥)𝑘) = (1...Σ𝑘 ∈ (1...0)𝑘))
54sumeq1d 15736 . . . 4 (𝑥 = 0 → Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑥)𝑘)((2 · 𝑚) − 1) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...0)𝑘)((2 · 𝑚) − 1))
62, 5eqeq12d 2753 . . 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 7439 . . . . 5 (𝑥 = 𝑦 → (1...𝑥) = (1...𝑦))
87sumeq1d 15736 . . . 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 15736 . . . . . 6 (𝑥 = 𝑦 → Σ𝑘 ∈ (1...𝑥)𝑘 = Σ𝑘 ∈ (1...𝑦)𝑘)
109oveq2d 7447 . . . . 5 (𝑥 = 𝑦 → (1...Σ𝑘 ∈ (1...𝑥)𝑘) = (1...Σ𝑘 ∈ (1...𝑦)𝑘))
1110sumeq1d 15736 . . . 4 (𝑥 = 𝑦 → Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑥)𝑘)((2 · 𝑚) − 1) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑦)𝑘)((2 · 𝑚) − 1))
128, 11eqeq12d 2753 . . 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 7439 . . . . 5 (𝑥 = (𝑦 + 1) → (1...𝑥) = (1...(𝑦 + 1)))
1413sumeq1d 15736 . . . 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 15736 . . . . . 6 (𝑥 = (𝑦 + 1) → Σ𝑘 ∈ (1...𝑥)𝑘 = Σ𝑘 ∈ (1...(𝑦 + 1))𝑘)
1615oveq2d 7447 . . . . 5 (𝑥 = (𝑦 + 1) → (1...Σ𝑘 ∈ (1...𝑥)𝑘) = (1...Σ𝑘 ∈ (1...(𝑦 + 1))𝑘))
1716sumeq1d 15736 . . . 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 2753 . . 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 7439 . . . . 5 (𝑥 = 𝑁 → (1...𝑥) = (1...𝑁))
2019sumeq1d 15736 . . . 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 15736 . . . . . 6 (𝑥 = 𝑁 → Σ𝑘 ∈ (1...𝑥)𝑘 = Σ𝑘 ∈ (1...𝑁)𝑘)
2221oveq2d 7447 . . . . 5 (𝑥 = 𝑁 → (1...Σ𝑘 ∈ (1...𝑥)𝑘) = (1...Σ𝑘 ∈ (1...𝑁)𝑘))
2322sumeq1d 15736 . . . 4 (𝑥 = 𝑁 → Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑥)𝑘)((2 · 𝑚) − 1) = Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...𝑁)𝑘)((2 · 𝑚) − 1))
2420, 23eqeq12d 2753 . . 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 15757 . . . . 5 Σ𝑘 ∈ ∅ Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = 0
26 sum0 15757 . . . . 5 Σ𝑚 ∈ ∅ ((2 · 𝑚) − 1) = 0
2725, 26eqtr4i 2768 . . . 4 Σ𝑘 ∈ ∅ Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑚 ∈ ∅ ((2 · 𝑚) − 1)
28 fz10 13585 . . . . 5 (1...0) = ∅
2928sumeq1i 15733 . . . 4 Σ𝑘 ∈ (1...0)Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑘 ∈ ∅ Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1))
3028sumeq1i 15733 . . . . . . . 8 Σ𝑘 ∈ (1...0)𝑘 = Σ𝑘 ∈ ∅ 𝑘
31 sum0 15757 . . . . . . . 8 Σ𝑘 ∈ ∅ 𝑘 = 0
3230, 31eqtri 2765 . . . . . . 7 Σ𝑘 ∈ (1...0)𝑘 = 0
3332oveq2i 7442 . . . . . 6 (1...Σ𝑘 ∈ (1...0)𝑘) = (1...0)
3433, 28eqtri 2765 . . . . 5 (1...Σ𝑘 ∈ (1...0)𝑘) = ∅
3534sumeq1i 15733 . . . 4 Σ𝑚 ∈ (1...Σ𝑘 ∈ (1...0)𝑘)((2 · 𝑚) − 1) = Σ𝑚 ∈ ∅ ((2 · 𝑚) − 1)
3627, 29, 353eqtr4i 2775 . . 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 14014 . . . . . . . . . . 11 (𝑦 ∈ ℕ0 → (1...𝑦) ∈ Fin)
39 elfznn 13593 . . . . . . . . . . . . 13 (𝑘 ∈ (1...𝑦) → 𝑘 ∈ ℕ)
4039adantl 481 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0𝑘 ∈ (1...𝑦)) → 𝑘 ∈ ℕ)
4140nnnn0d 12587 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0𝑘 ∈ (1...𝑦)) → 𝑘 ∈ ℕ0)
4238, 41fsumnn0cl 15772 . . . . . . . . . 10 (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℕ0)
4342nn0zd 12639 . . . . . . . . 9 (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℤ)
44 nn0p1nn 12565 . . . . . . . . . . 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 12640 . . . . . . . . 9 (𝑦 ∈ ℕ0 → (Σ𝑘 ∈ (1...𝑦)𝑘 + 1) ∈ ℤ)
47 peano2nn0 12566 . . . . . . . . . . 11 (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ0)
4847nn0zd 12639 . . . . . . . . . 10 (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℤ)
4943, 48zaddcld 12726 . . . . . . . . 9 (𝑦 ∈ ℕ0 → (Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)) ∈ ℤ)
50 2cnd 12344 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 2 ∈ ℂ)
51 elfzelz 13564 . . . . . . . . . . . . 13 (𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1))) → 𝑚 ∈ ℤ)
5251zcnd 12723 . . . . . . . . . . . 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 11281 . . . . . . . . . 10 ((𝑦 ∈ ℕ0𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → (2 · 𝑚) ∈ ℂ)
55 1cnd 11256 . . . . . . . . . 10 ((𝑦 ∈ ℕ0𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 1 ∈ ℂ)
5654, 55subcld 11620 . . . . . . . . 9 ((𝑦 ∈ ℕ0𝑚 ∈ ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1)...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → ((2 · 𝑚) − 1) ∈ ℂ)
57 oveq2 7439 . . . . . . . . . 10 (𝑚 = (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘) → (2 · 𝑚) = (2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)))
5857oveq1d 7446 . . . . . . . . 9 (𝑚 = (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘) → ((2 · 𝑚) − 1) = ((2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)) − 1))
5943, 46, 49, 56, 58fsumshftm 15817 . . . . . . . 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 13564 . . . . . . . . . . . . . . 15 (𝑘 ∈ (1...𝑦) → 𝑘 ∈ ℤ)
6160adantl 481 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℕ0𝑘 ∈ (1...𝑦)) → 𝑘 ∈ ℤ)
6261zred 12722 . . . . . . . . . . . . 13 ((𝑦 ∈ ℕ0𝑘 ∈ (1...𝑦)) → 𝑘 ∈ ℝ)
6338, 62fsumrecl 15770 . . . . . . . . . . . 12 (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℝ)
6463recnd 11289 . . . . . . . . . . 11 (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ ℂ)
65 1cnd 11256 . . . . . . . . . . 11 (𝑦 ∈ ℕ0 → 1 ∈ ℂ)
6664, 65pncan2d 11622 . . . . . . . . . 10 (𝑦 ∈ ℕ0 → ((Σ𝑘 ∈ (1...𝑦)𝑘 + 1) − Σ𝑘 ∈ (1...𝑦)𝑘) = 1)
6747nn0cnd 12589 . . . . . . . . . . 11 (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℂ)
6864, 67pncan2d 11622 . . . . . . . . . 10 (𝑦 ∈ ℕ0 → ((Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)) − Σ𝑘 ∈ (1...𝑦)𝑘) = (𝑦 + 1))
6966, 68oveq12d 7449 . . . . . . . . 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 13564 . . . . . . . . . . 11 (𝑙 ∈ (((Σ𝑘 ∈ (1...𝑦)𝑘 + 1) − Σ𝑘 ∈ (1...𝑦)𝑘)...((Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)) − Σ𝑘 ∈ (1...𝑦)𝑘)) → 𝑙 ∈ ℤ)
7170zcnd 12723 . . . . . . . . . 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 12344 . . . . . . . . . . . . 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 11285 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0𝑙 ∈ ℂ) → (2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)) = ((2 · 𝑙) + (2 · Σ𝑘 ∈ (1...𝑦)𝑘)))
7675oveq1d 7446 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0𝑙 ∈ ℂ) → ((2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)) − 1) = (((2 · 𝑙) + (2 · Σ𝑘 ∈ (1...𝑦)𝑘)) − 1))
7772, 73mulcld 11281 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0𝑙 ∈ ℂ) → (2 · 𝑙) ∈ ℂ)
7872, 74mulcld 11281 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0𝑙 ∈ ℂ) → (2 · Σ𝑘 ∈ (1...𝑦)𝑘) ∈ ℂ)
79 1cnd 11256 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0𝑙 ∈ ℂ) → 1 ∈ ℂ)
8077, 78, 79addsubassd 11640 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0𝑙 ∈ ℂ) → (((2 · 𝑙) + (2 · Σ𝑘 ∈ (1...𝑦)𝑘)) − 1) = ((2 · 𝑙) + ((2 · Σ𝑘 ∈ (1...𝑦)𝑘) − 1)))
8177, 78, 79addsub12d 11643 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0𝑙 ∈ ℂ) → ((2 · 𝑙) + ((2 · Σ𝑘 ∈ (1...𝑦)𝑘) − 1)) = ((2 · Σ𝑘 ∈ (1...𝑦)𝑘) + ((2 · 𝑙) − 1)))
82 arisum 15896 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 = (((𝑦↑2) + 𝑦) / 2))
8382oveq2d 7447 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℕ0 → (2 · Σ𝑘 ∈ (1...𝑦)𝑘) = (2 · (((𝑦↑2) + 𝑦) / 2)))
84 nn0cn 12536 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℕ0𝑦 ∈ ℂ)
8584sqcld 14184 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℕ0 → (𝑦↑2) ∈ ℂ)
8685, 84addcld 11280 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ0 → ((𝑦↑2) + 𝑦) ∈ ℂ)
87 2cnd 12344 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ0 → 2 ∈ ℂ)
88 2ne0 12370 . . . . . . . . . . . . . . . . 17 2 ≠ 0
8988a1i 11 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ0 → 2 ≠ 0)
9086, 87, 89divcan2d 12045 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℕ0 → (2 · (((𝑦↑2) + 𝑦) / 2)) = ((𝑦↑2) + 𝑦))
91 binom21 14258 . . . . . . . . . . . . . . . . . 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 7446 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ0 → (((𝑦 + 1)↑2) − (𝑦 + 1)) = ((((𝑦↑2) + (2 · 𝑦)) + 1) − (𝑦 + 1)))
9487, 84mulcld 11281 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℕ0 → (2 · 𝑦) ∈ ℂ)
9585, 94addcld 11280 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℕ0 → ((𝑦↑2) + (2 · 𝑦)) ∈ ℂ)
9695, 84, 65pnpcan2d 11658 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ0 → ((((𝑦↑2) + (2 · 𝑦)) + 1) − (𝑦 + 1)) = (((𝑦↑2) + (2 · 𝑦)) − 𝑦))
9785, 94, 84addsubassd 11640 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℕ0 → (((𝑦↑2) + (2 · 𝑦)) − 𝑦) = ((𝑦↑2) + ((2 · 𝑦) − 𝑦)))
98842timesd 12509 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℕ0 → (2 · 𝑦) = (𝑦 + 𝑦))
9984, 84, 98mvrladdd 11676 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℕ0 → ((2 · 𝑦) − 𝑦) = 𝑦)
10099oveq2d 7447 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℕ0 → ((𝑦↑2) + ((2 · 𝑦) − 𝑦)) = ((𝑦↑2) + 𝑦))
10197, 100eqtrd 2777 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ0 → (((𝑦↑2) + (2 · 𝑦)) − 𝑦) = ((𝑦↑2) + 𝑦))
10293, 96, 1013eqtrrd 2782 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℕ0 → ((𝑦↑2) + 𝑦) = (((𝑦 + 1)↑2) − (𝑦 + 1)))
10383, 90, 1023eqtrd 2781 . . . . . . . . . . . . . 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 7446 . . . . . . . . . . . 12 ((𝑦 ∈ ℕ0𝑙 ∈ ℂ) → ((2 · Σ𝑘 ∈ (1...𝑦)𝑘) + ((2 · 𝑙) − 1)) = ((((𝑦 + 1)↑2) − (𝑦 + 1)) + ((2 · 𝑙) − 1)))
10681, 105eqtrd 2777 . . . . . . . . . . 11 ((𝑦 ∈ ℕ0𝑙 ∈ ℂ) → ((2 · 𝑙) + ((2 · Σ𝑘 ∈ (1...𝑦)𝑘) − 1)) = ((((𝑦 + 1)↑2) − (𝑦 + 1)) + ((2 · 𝑙) − 1)))
10776, 80, 1063eqtrd 2781 . . . . . . . . . 10 ((𝑦 ∈ ℕ0𝑙 ∈ ℂ) → ((2 · (𝑙 + Σ𝑘 ∈ (1...𝑦)𝑘)) − 1) = ((((𝑦 + 1)↑2) − (𝑦 + 1)) + ((2 · 𝑙) − 1)))
10871, 107sylan2 593 . . . . . . . . 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 15742 . . . . . . . 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 2778 . . . . . . 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 7449 . . . . 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 14014 . . . . . . . 8 ((𝑦 ∈ ℕ0𝑘 ∈ (1...(𝑦 + 1))) → (1...𝑘) ∈ Fin)
115 elfzelz 13564 . . . . . . . . . . . . 13 (𝑘 ∈ (1...(𝑦 + 1)) → 𝑘 ∈ ℤ)
116115zcnd 12723 . . . . . . . . . . . 12 (𝑘 ∈ (1...(𝑦 + 1)) → 𝑘 ∈ ℂ)
117116sqcld 14184 . . . . . . . . . . 11 (𝑘 ∈ (1...(𝑦 + 1)) → (𝑘↑2) ∈ ℂ)
118117, 116subcld 11620 . . . . . . . . . 10 (𝑘 ∈ (1...(𝑦 + 1)) → ((𝑘↑2) − 𝑘) ∈ ℂ)
119 2cnd 12344 . . . . . . . . . . . 12 (𝑙 ∈ (1...𝑘) → 2 ∈ ℂ)
120 elfzelz 13564 . . . . . . . . . . . . 13 (𝑙 ∈ (1...𝑘) → 𝑙 ∈ ℤ)
121120zcnd 12723 . . . . . . . . . . . 12 (𝑙 ∈ (1...𝑘) → 𝑙 ∈ ℂ)
122119, 121mulcld 11281 . . . . . . . . . . 11 (𝑙 ∈ (1...𝑘) → (2 · 𝑙) ∈ ℂ)
123 1cnd 11256 . . . . . . . . . . 11 (𝑙 ∈ (1...𝑘) → 1 ∈ ℂ)
124122, 123subcld 11620 . . . . . . . . . 10 (𝑙 ∈ (1...𝑘) → ((2 · 𝑙) − 1) ∈ ℂ)
125 addcl 11237 . . . . . . . . . 10 ((((𝑘↑2) − 𝑘) ∈ ℂ ∧ ((2 · 𝑙) − 1) ∈ ℂ) → (((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) ∈ ℂ)
126118, 124, 125syl2an 596 . . . . . . . . 9 ((𝑘 ∈ (1...(𝑦 + 1)) ∧ 𝑙 ∈ (1...𝑘)) → (((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) ∈ ℂ)
127126adantll 714 . . . . . . . 8 (((𝑦 ∈ ℕ0𝑘 ∈ (1...(𝑦 + 1))) ∧ 𝑙 ∈ (1...𝑘)) → (((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) ∈ ℂ)
128114, 127fsumcl 15769 . . . . . . 7 ((𝑦 ∈ ℕ0𝑘 ∈ (1...(𝑦 + 1))) → Σ𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) ∈ ℂ)
129 oveq2 7439 . . . . . . . 8 (𝑘 = (𝑦 + 1) → (1...𝑘) = (1...(𝑦 + 1)))
130 oveq1 7438 . . . . . . . . . . 11 (𝑘 = (𝑦 + 1) → (𝑘↑2) = ((𝑦 + 1)↑2))
131 id 22 . . . . . . . . . . 11 (𝑘 = (𝑦 + 1) → 𝑘 = (𝑦 + 1))
132130, 131oveq12d 7449 . . . . . . . . . 10 (𝑘 = (𝑦 + 1) → ((𝑘↑2) − 𝑘) = (((𝑦 + 1)↑2) − (𝑦 + 1)))
133132oveq1d 7446 . . . . . . . . 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 15742 . . . . . . 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 42344 . . . . . 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 42344 . . . . . . . . 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 7447 . . . . . . 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 15736 . . . . . 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 12198 . . . . . . . . 9 (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 < (Σ𝑘 ∈ (1...𝑦)𝑘 + 1))
144 fzdisj 13591 . . . . . . . . 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 12921 . . . . . . . . . 10 ℕ = (ℤ‘1)
14745, 146eleqtrdi 2851 . . . . . . . . 9 (𝑦 ∈ ℕ0 → (Σ𝑘 ∈ (1...𝑦)𝑘 + 1) ∈ (ℤ‘1))
14843uzidd 12894 . . . . . . . . . 10 (𝑦 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑦)𝑘 ∈ (ℤ‘Σ𝑘 ∈ (1...𝑦)𝑘))
149 uzaddcl 12946 . . . . . . . . . 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 584 . . . . . . . . 9 (𝑦 ∈ ℕ0 → (Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)) ∈ (ℤ‘Σ𝑘 ∈ (1...𝑦)𝑘))
151 fzsplit2 13589 . . . . . . . . 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 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 14014 . . . . . . . 8 (𝑦 ∈ ℕ0 → (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1))) ∈ Fin)
154 2cnd 12344 . . . . . . . . . 10 ((𝑦 ∈ ℕ0𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 2 ∈ ℂ)
155 elfzelz 13564 . . . . . . . . . . . 12 (𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1))) → 𝑚 ∈ ℤ)
156155zcnd 12723 . . . . . . . . . . 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 11281 . . . . . . . . 9 ((𝑦 ∈ ℕ0𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → (2 · 𝑚) ∈ ℂ)
159 1cnd 11256 . . . . . . . . 9 ((𝑦 ∈ ℕ0𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → 1 ∈ ℂ)
160158, 159subcld 11620 . . . . . . . 8 ((𝑦 ∈ ℕ0𝑚 ∈ (1...(Σ𝑘 ∈ (1...𝑦)𝑘 + (𝑦 + 1)))) → ((2 · 𝑚) − 1) ∈ ℂ)
161145, 152, 153, 160fsumsplit 15777 . . . . . . 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 2777 . . . . 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 2787 . . . 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 12713 . 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 13595 . . . . . . 7 (1...𝑁) ⊆ ℕ
168 nnssnn0 12529 . . . . . . 7 ℕ ⊆ ℕ0
169167, 168sstri 3993 . . . . . 6 (1...𝑁) ⊆ ℕ0
170169a1i 11 . . . . 5 (𝑁 ∈ ℕ0 → (1...𝑁) ⊆ ℕ0)
171170sselda 3983 . . . 4 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℕ0)
172 nicomachus 42346 . . . 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 15738 . 2 (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁𝑙 ∈ (1...𝑘)(((𝑘↑2) − 𝑘) + ((2 · 𝑙) − 1)) = Σ𝑘 ∈ (1...𝑁)(𝑘↑3))
175 fzfid 14014 . . . 4 (𝑁 ∈ ℕ0 → (1...𝑁) ∈ Fin)
176175, 171fsumnn0cl 15772 . . 3 (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)𝑘 ∈ ℕ0)
177 oddnumth 42345 . . 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 2785 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 2108  wne 2940  cun 3949  cin 3950  wss 3951  c0 4333   class class class wbr 5143  cfv 6561  (class class class)co 7431  cc 11153  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160   < clt 11295  cmin 11492   / cdiv 11920  cn 12266  2c2 12321  3c3 12322  0cn0 12526  cz 12613  cuz 12878  ...cfz 13547  cexp 14102  Σcsu 15722
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-fac 14313  df-bc 14342  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723
This theorem is referenced by:  sum9cubes  42682
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