axlowdimlem16 - Metamath Proof Explorer

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Theorem axlowdimlem16 26745

Description: Lemma for axlowdim 26749. Set up a summation that will help establish distance. (Contributed by Scott Fenton, 21-Apr-2013.)

**Hypotheses**
Ref	Expression
axlowdimlem16.1	⊢ 𝑃 = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))
axlowdimlem16.2	⊢ 𝑄 = ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}))

**Assertion**
Ref	Expression
axlowdimlem16	⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (3...𝑁)((𝑃‘𝑖)↑2) = Σ𝑖 ∈ (3...𝑁)((𝑄‘𝑖)↑2))

Distinct variable groups: 𝑃,𝑖 𝑖,𝐼 𝑖,𝑁 𝑄,𝑖

**Proof of Theorem axlowdimlem16**
Step	Hyp	Ref	Expression
1		elfz1eq 12921	. . . . . 6 ⊢ (𝐼 ∈ (2...2) → 𝐼 = 2)
2		oveq1 7165	. . . . . . . . . . 11 ⊢ (𝐼 = 2 → (𝐼 + 1) = (2 + 1))
3		df-3 11704	. . . . . . . . . . 11 ⊢ 3 = (2 + 1)
4	2, 3	syl6reqr 2877	. . . . . . . . . 10 ⊢ (𝐼 = 2 → 3 = (𝐼 + 1))
5	4, 4	oveq12d 7176	. . . . . . . . 9 ⊢ (𝐼 = 2 → (3...3) = ((𝐼 + 1)...(𝐼 + 1)))
6	5	sumeq1d 15060	. . . . . . . 8 ⊢ (𝐼 = 2 → Σ𝑖 ∈ (3...3)((𝑄‘𝑖)↑2) = Σ𝑖 ∈ ((𝐼 + 1)...(𝐼 + 1))((𝑄‘𝑖)↑2))
7	2, 3	syl6eqr 2876	. . . . . . . . . 10 ⊢ (𝐼 = 2 → (𝐼 + 1) = 3)
8		3z 12018	. . . . . . . . . 10 ⊢ 3 ∈ ℤ
9	7, 8	eqeltrdi 2923	. . . . . . . . 9 ⊢ (𝐼 = 2 → (𝐼 + 1) ∈ ℤ)
10		ax-1cn 10597	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
11	10	sqcli 13547	. . . . . . . . 9 ⊢ (1↑2) ∈ ℂ
12		fveq2 6672	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝐼 + 1) → (𝑄‘𝑖) = (𝑄‘(𝐼 + 1)))
13		axlowdimlem16.2	. . . . . . . . . . . . 13 ⊢ 𝑄 = ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}))
14	13	axlowdimlem11 26740	. . . . . . . . . . . 12 ⊢ (𝑄‘(𝐼 + 1)) = 1
15	12, 14	syl6eq 2874	. . . . . . . . . . 11 ⊢ (𝑖 = (𝐼 + 1) → (𝑄‘𝑖) = 1)
16	15	oveq1d 7173	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼 + 1) → ((𝑄‘𝑖)↑2) = (1↑2))
17	16	fsum1 15104	. . . . . . . . 9 ⊢ (((𝐼 + 1) ∈ ℤ ∧ (1↑2) ∈ ℂ) → Σ𝑖 ∈ ((𝐼 + 1)...(𝐼 + 1))((𝑄‘𝑖)↑2) = (1↑2))
18	9, 11, 17	sylancl 588	. . . . . . . 8 ⊢ (𝐼 = 2 → Σ𝑖 ∈ ((𝐼 + 1)...(𝐼 + 1))((𝑄‘𝑖)↑2) = (1↑2))
19	6, 18	eqtrd 2858	. . . . . . 7 ⊢ (𝐼 = 2 → Σ𝑖 ∈ (3...3)((𝑄‘𝑖)↑2) = (1↑2))
20		fveq2 6672	. . . . . . . . . . . 12 ⊢ (𝑖 = 3 → (𝑃‘𝑖) = (𝑃‘3))
21		axlowdimlem16.1	. . . . . . . . . . . . 13 ⊢ 𝑃 = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))
22	21	axlowdimlem8 26737	. . . . . . . . . . . 12 ⊢ (𝑃‘3) = -1
23	20, 22	syl6eq 2874	. . . . . . . . . . 11 ⊢ (𝑖 = 3 → (𝑃‘𝑖) = -1)
24	23	oveq1d 7173	. . . . . . . . . 10 ⊢ (𝑖 = 3 → ((𝑃‘𝑖)↑2) = (-1↑2))
25		sqneg 13485	. . . . . . . . . . 11 ⊢ (1 ∈ ℂ → (-1↑2) = (1↑2))
26	10, 25	ax-mp 5	. . . . . . . . . 10 ⊢ (-1↑2) = (1↑2)
27	24, 26	syl6eq 2874	. . . . . . . . 9 ⊢ (𝑖 = 3 → ((𝑃‘𝑖)↑2) = (1↑2))
28	27	fsum1 15104	. . . . . . . 8 ⊢ ((3 ∈ ℤ ∧ (1↑2) ∈ ℂ) → Σ𝑖 ∈ (3...3)((𝑃‘𝑖)↑2) = (1↑2))
29	8, 11, 28	mp2an 690	. . . . . . 7 ⊢ Σ𝑖 ∈ (3...3)((𝑃‘𝑖)↑2) = (1↑2)
30	19, 29	syl6reqr 2877	. . . . . 6 ⊢ (𝐼 = 2 → Σ𝑖 ∈ (3...3)((𝑃‘𝑖)↑2) = Σ𝑖 ∈ (3...3)((𝑄‘𝑖)↑2))
31	1, 30	syl 17	. . . . 5 ⊢ (𝐼 ∈ (2...2) → Σ𝑖 ∈ (3...3)((𝑃‘𝑖)↑2) = Σ𝑖 ∈ (3...3)((𝑄‘𝑖)↑2))
32	31	a1i 11	. . . 4 ⊢ (𝑁 = 3 → (𝐼 ∈ (2...2) → Σ𝑖 ∈ (3...3)((𝑃‘𝑖)↑2) = Σ𝑖 ∈ (3...3)((𝑄‘𝑖)↑2)))
33		oveq1 7165	. . . . . . 7 ⊢ (𝑁 = 3 → (𝑁 − 1) = (3 − 1))
34		3m1e2 11768	. . . . . . 7 ⊢ (3 − 1) = 2
35	33, 34	syl6eq 2874	. . . . . 6 ⊢ (𝑁 = 3 → (𝑁 − 1) = 2)
36	35	oveq2d 7174	. . . . 5 ⊢ (𝑁 = 3 → (2...(𝑁 − 1)) = (2...2))
37	36	eleq2d 2900	. . . 4 ⊢ (𝑁 = 3 → (𝐼 ∈ (2...(𝑁 − 1)) ↔ 𝐼 ∈ (2...2)))
38		oveq2 7166	. . . . . 6 ⊢ (𝑁 = 3 → (3...𝑁) = (3...3))
39	38	sumeq1d 15060	. . . . 5 ⊢ (𝑁 = 3 → Σ𝑖 ∈ (3...𝑁)((𝑃‘𝑖)↑2) = Σ𝑖 ∈ (3...3)((𝑃‘𝑖)↑2))
40	38	sumeq1d 15060	. . . . 5 ⊢ (𝑁 = 3 → Σ𝑖 ∈ (3...𝑁)((𝑄‘𝑖)↑2) = Σ𝑖 ∈ (3...3)((𝑄‘𝑖)↑2))
41	39, 40	eqeq12d 2839	. . . 4 ⊢ (𝑁 = 3 → (Σ𝑖 ∈ (3...𝑁)((𝑃‘𝑖)↑2) = Σ𝑖 ∈ (3...𝑁)((𝑄‘𝑖)↑2) ↔ Σ𝑖 ∈ (3...3)((𝑃‘𝑖)↑2) = Σ𝑖 ∈ (3...3)((𝑄‘𝑖)↑2)))
42	32, 37, 41	3imtr4d 296	. . 3 ⊢ (𝑁 = 3 → (𝐼 ∈ (2...(𝑁 − 1)) → Σ𝑖 ∈ (3...𝑁)((𝑃‘𝑖)↑2) = Σ𝑖 ∈ (3...𝑁)((𝑄‘𝑖)↑2)))
43	42	adantld 493	. 2 ⊢ (𝑁 = 3 → ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (3...𝑁)((𝑃‘𝑖)↑2) = Σ𝑖 ∈ (3...𝑁)((𝑄‘𝑖)↑2)))
44		simprl 769	. . . 4 ⊢ ((𝑁 ≠ 3 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 𝑁 ∈ (ℤ_≥‘3))
45		eluzle 12259	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
46	45	adantl 484	. . . . . 6 ⊢ ((𝑁 ≠ 3 ∧ 𝑁 ∈ (ℤ_≥‘3)) → 3 ≤ 𝑁)
47		simpl 485	. . . . . 6 ⊢ ((𝑁 ≠ 3 ∧ 𝑁 ∈ (ℤ_≥‘3)) → 𝑁 ≠ 3)
48		3re 11720	. . . . . . 7 ⊢ 3 ∈ ℝ
49		eluzelre 12257	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℝ)
50	49	adantl 484	. . . . . . 7 ⊢ ((𝑁 ≠ 3 ∧ 𝑁 ∈ (ℤ_≥‘3)) → 𝑁 ∈ ℝ)
51		ltlen 10743	. . . . . . 7 ⊢ ((3 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (3 < 𝑁 ↔ (3 ≤ 𝑁 ∧ 𝑁 ≠ 3)))
52	48, 50, 51	sylancr 589	. . . . . 6 ⊢ ((𝑁 ≠ 3 ∧ 𝑁 ∈ (ℤ_≥‘3)) → (3 < 𝑁 ↔ (3 ≤ 𝑁 ∧ 𝑁 ≠ 3)))
53	46, 47, 52	mpbir2and 711	. . . . 5 ⊢ ((𝑁 ≠ 3 ∧ 𝑁 ∈ (ℤ_≥‘3)) → 3 < 𝑁)
54	53	adantrr 715	. . . 4 ⊢ ((𝑁 ≠ 3 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 3 < 𝑁)
55		simprr 771	. . . 4 ⊢ ((𝑁 ≠ 3 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 𝐼 ∈ (2...(𝑁 − 1)))
56		fzssp1 12953	. . . . . . . . . . . . 13 ⊢ (2...(𝑁 − 1)) ⊆ (2...((𝑁 − 1) + 1))
57		simp3 1134	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝐼 ∈ (2...(𝑁 − 1)))
58	56, 57	sseldi 3967	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝐼 ∈ (2...((𝑁 − 1) + 1)))
59		eluzelz 12256	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
60	59	3ad2ant1 1129	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝑁 ∈ ℤ)
61	60	zcnd 12091	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝑁 ∈ ℂ)
62		npcan 10897	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
63	61, 10, 62	sylancl 588	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁)
64	63	oveq2d 7174	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (2...((𝑁 − 1) + 1)) = (2...𝑁))
65	58, 64	eleqtrd 2917	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝐼 ∈ (2...𝑁))
66		elfzelz 12911	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (2...𝑁) → 𝐼 ∈ ℤ)
67	65, 66	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝐼 ∈ ℤ)
68	67	zred 12090	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝐼 ∈ ℝ)
69	68	ltp1d 11572	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝐼 < (𝐼 + 1))
70		fzdisj 12937	. . . . . . . 8 ⊢ (𝐼 < (𝐼 + 1) → ((2...𝐼) ∩ ((𝐼 + 1)...𝑁)) = ∅)
71	69, 70	syl 17	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → ((2...𝐼) ∩ ((𝐼 + 1)...𝑁)) = ∅)
72		fzsplit 12936	. . . . . . . 8 ⊢ (𝐼 ∈ (2...𝑁) → (2...𝑁) = ((2...𝐼) ∪ ((𝐼 + 1)...𝑁)))
73	65, 72	syl 17	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (2...𝑁) = ((2...𝐼) ∪ ((𝐼 + 1)...𝑁)))
74		fzfid 13344	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (2...𝑁) ∈ Fin)
75		eluzge3nn 12293	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℕ)
76		2eluzge1 12297	. . . . . . . . . . . . 13 ⊢ 2 ∈ (ℤ_≥‘1)
77		fzss1 12949	. . . . . . . . . . . . 13 ⊢ (2 ∈ (ℤ_≥‘1) → (2...(𝑁 − 1)) ⊆ (1...(𝑁 − 1)))
78	76, 77	ax-mp 5	. . . . . . . . . . . 12 ⊢ (2...(𝑁 − 1)) ⊆ (1...(𝑁 − 1))
79	78	sseli 3965	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (2...(𝑁 − 1)) → 𝐼 ∈ (1...(𝑁 − 1)))
80	13	axlowdimlem10 26739	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → 𝑄 ∈ (𝔼‘𝑁))
81	75, 79, 80	syl2an 597	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝑄 ∈ (𝔼‘𝑁))
82		fzss1 12949	. . . . . . . . . . . 12 ⊢ (2 ∈ (ℤ_≥‘1) → (2...𝑁) ⊆ (1...𝑁))
83	76, 82	ax-mp 5	. . . . . . . . . . 11 ⊢ (2...𝑁) ⊆ (1...𝑁)
84	83	sseli 3965	. . . . . . . . . 10 ⊢ (𝑖 ∈ (2...𝑁) → 𝑖 ∈ (1...𝑁))
85		fveecn 26690	. . . . . . . . . 10 ⊢ ((𝑄 ∈ (𝔼‘𝑁) ∧ 𝑖 ∈ (1...𝑁)) → (𝑄‘𝑖) ∈ ℂ)
86	81, 84, 85	syl2an 597	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (2...𝑁)) → (𝑄‘𝑖) ∈ ℂ)
87	86	sqcld 13511	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (2...𝑁)) → ((𝑄‘𝑖)↑2) ∈ ℂ)
88	87	3adantl2 1163	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (2...𝑁)) → ((𝑄‘𝑖)↑2) ∈ ℂ)
89	71, 73, 74, 88	fsumsplit 15099	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (2...𝑁)((𝑄‘𝑖)↑2) = (Σ𝑖 ∈ (2...𝐼)((𝑄‘𝑖)↑2) + Σ𝑖 ∈ ((𝐼 + 1)...𝑁)((𝑄‘𝑖)↑2)))
90		elfzelz 12911	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ (2...(𝑁 − 1)) → 𝐼 ∈ ℤ)
91	90	zred 12090	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐼 ∈ (2...(𝑁 − 1)) → 𝐼 ∈ ℝ)
92	91	3ad2ant3 1131	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝐼 ∈ ℝ)
93	49	3ad2ant1 1129	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝑁 ∈ ℝ)
94		peano2rem 10955	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℝ → (𝑁 − 1) ∈ ℝ)
95	93, 94	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (𝑁 − 1) ∈ ℝ)
96		elfzle2 12914	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ (2...(𝑁 − 1)) → 𝐼 ≤ (𝑁 − 1))
97	96	3ad2ant3 1131	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝐼 ≤ (𝑁 − 1))
98	93	ltm1d 11574	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (𝑁 − 1) < 𝑁)
99	92, 95, 93, 97, 98	lelttrd 10800	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝐼 < 𝑁)
100	92, 93, 99	ltled 10790	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝐼 ≤ 𝑁)
101	90	3ad2ant3 1131	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝐼 ∈ ℤ)
102		eluz 12260	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ_≥‘𝐼) ↔ 𝐼 ≤ 𝑁))
103	101, 60, 102	syl2anc 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (𝑁 ∈ (ℤ_≥‘𝐼) ↔ 𝐼 ≤ 𝑁))
104	100, 103	mpbird 259	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝑁 ∈ (ℤ_≥‘𝐼))
105		fzss2 12950	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘𝐼) → (1...𝐼) ⊆ (1...𝑁))
106	104, 105	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (1...𝐼) ⊆ (1...𝑁))
107	106	sseld 3968	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (𝑖 ∈ (1...𝐼) → 𝑖 ∈ (1...𝑁)))
108		fzss1 12949	. . . . . . . . . . . . . . 15 ⊢ (2 ∈ (ℤ_≥‘1) → (2...𝐼) ⊆ (1...𝐼))
109	76, 108	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (2...𝐼) ⊆ (1...𝐼)
110	109	sseli 3965	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (2...𝐼) → 𝑖 ∈ (1...𝐼))
111	107, 110	impel 508	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (2...𝐼)) → 𝑖 ∈ (1...𝑁))
112		elfzelz 12911	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (2...𝐼) → 𝑖 ∈ ℤ)
113	112	zred 12090	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (2...𝐼) → 𝑖 ∈ ℝ)
114	113	adantl 484	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (2...𝐼)) → 𝑖 ∈ ℝ)
115	92	adantr 483	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (2...𝐼)) → 𝐼 ∈ ℝ)
116		peano2re 10815	. . . . . . . . . . . . . . . . 17 ⊢ (𝐼 ∈ ℝ → (𝐼 + 1) ∈ ℝ)
117	91, 116	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ (2...(𝑁 − 1)) → (𝐼 + 1) ∈ ℝ)
118	117	3ad2ant3 1131	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (𝐼 + 1) ∈ ℝ)
119	118	adantr 483	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (2...𝐼)) → (𝐼 + 1) ∈ ℝ)
120		elfzle2 12914	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (2...𝐼) → 𝑖 ≤ 𝐼)
121	120	adantl 484	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (2...𝐼)) → 𝑖 ≤ 𝐼)
122	115	ltp1d 11572	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (2...𝐼)) → 𝐼 < (𝐼 + 1))
123	114, 115, 119, 121, 122	lelttrd 10800	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (2...𝐼)) → 𝑖 < (𝐼 + 1))
124	114, 123	ltned 10778	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (2...𝐼)) → 𝑖 ≠ (𝐼 + 1))
125	13	axlowdimlem12 26741	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (1...𝑁) ∧ 𝑖 ≠ (𝐼 + 1)) → (𝑄‘𝑖) = 0)
126	111, 124, 125	syl2anc 586	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (2...𝐼)) → (𝑄‘𝑖) = 0)
127	126	sq0id 13560	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (2...𝐼)) → ((𝑄‘𝑖)↑2) = 0)
128	127	sumeq2dv 15062	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (2...𝐼)((𝑄‘𝑖)↑2) = Σ𝑖 ∈ (2...𝐼)0)
129		fzfi 13343	. . . . . . . . . . 11 ⊢ (2...𝐼) ∈ Fin
130	129	olci 862	. . . . . . . . . 10 ⊢ ((2...𝐼) ⊆ (ℤ_≥‘1) ∨ (2...𝐼) ∈ Fin)
131		sumz 15081	. . . . . . . . . 10 ⊢ (((2...𝐼) ⊆ (ℤ_≥‘1) ∨ (2...𝐼) ∈ Fin) → Σ𝑖 ∈ (2...𝐼)0 = 0)
132	130, 131	ax-mp 5	. . . . . . . . 9 ⊢ Σ𝑖 ∈ (2...𝐼)0 = 0
133	128, 132	syl6eq 2874	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (2...𝐼)((𝑄‘𝑖)↑2) = 0)
134	101	peano2zd 12093	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (𝐼 + 1) ∈ ℤ)
135		sq1 13561	. . . . . . . . . . . . 13 ⊢ (1↑2) = 1
136	16, 135	syl6eq 2874	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝐼 + 1) → ((𝑄‘𝑖)↑2) = 1)
137	136	fsum1 15104	. . . . . . . . . . 11 ⊢ (((𝐼 + 1) ∈ ℤ ∧ 1 ∈ ℂ) → Σ𝑖 ∈ ((𝐼 + 1)...(𝐼 + 1))((𝑄‘𝑖)↑2) = 1)
138	134, 10, 137	sylancl 588	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ ((𝐼 + 1)...(𝐼 + 1))((𝑄‘𝑖)↑2) = 1)
139		oveq2 7166	. . . . . . . . . . . 12 ⊢ ((𝐼 + 1) = 𝑁 → ((𝐼 + 1)...(𝐼 + 1)) = ((𝐼 + 1)...𝑁))
140	139	sumeq1d 15060	. . . . . . . . . . 11 ⊢ ((𝐼 + 1) = 𝑁 → Σ𝑖 ∈ ((𝐼 + 1)...(𝐼 + 1))((𝑄‘𝑖)↑2) = Σ𝑖 ∈ ((𝐼 + 1)...𝑁)((𝑄‘𝑖)↑2))
141	140	eqeq1d 2825	. . . . . . . . . 10 ⊢ ((𝐼 + 1) = 𝑁 → (Σ𝑖 ∈ ((𝐼 + 1)...(𝐼 + 1))((𝑄‘𝑖)↑2) = 1 ↔ Σ𝑖 ∈ ((𝐼 + 1)...𝑁)((𝑄‘𝑖)↑2) = 1))
142	138, 141	syl5ib 246	. . . . . . . . 9 ⊢ ((𝐼 + 1) = 𝑁 → ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ ((𝐼 + 1)...𝑁)((𝑄‘𝑖)↑2) = 1))
143	101	adantl 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 𝐼 ∈ ℤ)
144	143	zred 12090	. . . . . . . . . . . . . . 15 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 𝐼 ∈ ℝ)
145	60	adantl 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 𝑁 ∈ ℤ)
146	145	zred 12090	. . . . . . . . . . . . . . . 16 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 𝑁 ∈ ℝ)
147	146, 94	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → (𝑁 − 1) ∈ ℝ)
148	97	adantl 484	. . . . . . . . . . . . . . 15 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 𝐼 ≤ (𝑁 − 1))
149	146	ltm1d 11574	. . . . . . . . . . . . . . 15 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → (𝑁 − 1) < 𝑁)
150	144, 147, 146, 148, 149	lelttrd 10800	. . . . . . . . . . . . . 14 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 𝐼 < 𝑁)
151		1red 10644	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ (2...(𝑁 − 1)) → 1 ∈ ℝ)
152		2re 11714	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℝ
153	152	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ (2...(𝑁 − 1)) → 2 ∈ ℝ)
154		1le2 11849	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ≤ 2
155	154	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ (2...(𝑁 − 1)) → 1 ≤ 2)
156		elfzle1 12913	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ (2...(𝑁 − 1)) → 2 ≤ 𝐼)
157	151, 153, 91, 155, 156	letrd 10799	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐼 ∈ (2...(𝑁 − 1)) → 1 ≤ 𝐼)
158	157	3ad2ant3 1131	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 1 ≤ 𝐼)
159	158	adantl 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 1 ≤ 𝐼)
160		elnnz1 12011	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ ℕ ↔ (𝐼 ∈ ℤ ∧ 1 ≤ 𝐼))
161	143, 159, 160	sylanbrc 585	. . . . . . . . . . . . . . 15 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 𝐼 ∈ ℕ)
162	75	3ad2ant1 1129	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝑁 ∈ ℕ)
163	162	adantl 484	. . . . . . . . . . . . . . 15 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 𝑁 ∈ ℕ)
164		nnltp1le 12041	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐼 < 𝑁 ↔ (𝐼 + 1) ≤ 𝑁))
165	161, 163, 164	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → (𝐼 < 𝑁 ↔ (𝐼 + 1) ≤ 𝑁))
166	150, 165	mpbid 234	. . . . . . . . . . . . 13 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → (𝐼 + 1) ≤ 𝑁)
167		eluz 12260	. . . . . . . . . . . . . 14 ⊢ (((𝐼 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ_≥‘(𝐼 + 1)) ↔ (𝐼 + 1) ≤ 𝑁))
168	134, 145, 167	syl2an2 684	. . . . . . . . . . . . 13 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → (𝑁 ∈ (ℤ_≥‘(𝐼 + 1)) ↔ (𝐼 + 1) ≤ 𝑁))
169	166, 168	mpbird 259	. . . . . . . . . . . 12 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 𝑁 ∈ (ℤ_≥‘(𝐼 + 1)))
170		simpr1 1190	. . . . . . . . . . . . . . . 16 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 𝑁 ∈ (ℤ_≥‘3))
171		simpr3 1192	. . . . . . . . . . . . . . . 16 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 𝐼 ∈ (2...(𝑁 − 1)))
172	170, 171, 81	syl2anc 586	. . . . . . . . . . . . . . 15 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 𝑄 ∈ (𝔼‘𝑁))
173	172	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) ∧ 𝑖 ∈ ((𝐼 + 1)...𝑁)) → 𝑄 ∈ (𝔼‘𝑁))
174	161	peano2nnd 11657	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → (𝐼 + 1) ∈ ℕ)
175		nnuz 12284	. . . . . . . . . . . . . . . . 17 ⊢ ℕ = (ℤ_≥‘1)
176	174, 175	eleqtrdi 2925	. . . . . . . . . . . . . . . 16 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → (𝐼 + 1) ∈ (ℤ_≥‘1))
177		fzss1 12949	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 + 1) ∈ (ℤ_≥‘1) → ((𝐼 + 1)...𝑁) ⊆ (1...𝑁))
178	176, 177	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → ((𝐼 + 1)...𝑁) ⊆ (1...𝑁))
179	178	sselda 3969	. . . . . . . . . . . . . 14 ⊢ ((((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) ∧ 𝑖 ∈ ((𝐼 + 1)...𝑁)) → 𝑖 ∈ (1...𝑁))
180	173, 179, 85	syl2anc 586	. . . . . . . . . . . . 13 ⊢ ((((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) ∧ 𝑖 ∈ ((𝐼 + 1)...𝑁)) → (𝑄‘𝑖) ∈ ℂ)
181	180	sqcld 13511	. . . . . . . . . . . 12 ⊢ ((((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) ∧ 𝑖 ∈ ((𝐼 + 1)...𝑁)) → ((𝑄‘𝑖)↑2) ∈ ℂ)
182	12	oveq1d 7173	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝐼 + 1) → ((𝑄‘𝑖)↑2) = ((𝑄‘(𝐼 + 1))↑2))
183	14	oveq1i 7168	. . . . . . . . . . . . . 14 ⊢ ((𝑄‘(𝐼 + 1))↑2) = (1↑2)
184	183, 135	eqtri 2846	. . . . . . . . . . . . 13 ⊢ ((𝑄‘(𝐼 + 1))↑2) = 1
185	182, 184	syl6eq 2874	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝐼 + 1) → ((𝑄‘𝑖)↑2) = 1)
186	169, 181, 185	fsum1p 15110	. . . . . . . . . . 11 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → Σ𝑖 ∈ ((𝐼 + 1)...𝑁)((𝑄‘𝑖)↑2) = (1 + Σ𝑖 ∈ (((𝐼 + 1) + 1)...𝑁)((𝑄‘𝑖)↑2)))
187	174	peano2nnd 11657	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → ((𝐼 + 1) + 1) ∈ ℕ)
188	187, 175	eleqtrdi 2925	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → ((𝐼 + 1) + 1) ∈ (ℤ_≥‘1))
189		fzss1 12949	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐼 + 1) + 1) ∈ (ℤ_≥‘1) → (((𝐼 + 1) + 1)...𝑁) ⊆ (1...𝑁))
190	188, 189	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → (((𝐼 + 1) + 1)...𝑁) ⊆ (1...𝑁))
191	190	sselda 3969	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) ∧ 𝑖 ∈ (((𝐼 + 1) + 1)...𝑁)) → 𝑖 ∈ (1...𝑁))
192	144, 116	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → (𝐼 + 1) ∈ ℝ)
193	192	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) ∧ 𝑖 ∈ (((𝐼 + 1) + 1)...𝑁)) → (𝐼 + 1) ∈ ℝ)
194		peano2re 10815	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐼 + 1) ∈ ℝ → ((𝐼 + 1) + 1) ∈ ℝ)
195	193, 194	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) ∧ 𝑖 ∈ (((𝐼 + 1) + 1)...𝑁)) → ((𝐼 + 1) + 1) ∈ ℝ)
196		elfzelz 12911	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (((𝐼 + 1) + 1)...𝑁) → 𝑖 ∈ ℤ)
197	196	zred 12090	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (((𝐼 + 1) + 1)...𝑁) → 𝑖 ∈ ℝ)
198	197	adantl 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) ∧ 𝑖 ∈ (((𝐼 + 1) + 1)...𝑁)) → 𝑖 ∈ ℝ)
199	193	ltp1d 11572	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) ∧ 𝑖 ∈ (((𝐼 + 1) + 1)...𝑁)) → (𝐼 + 1) < ((𝐼 + 1) + 1))
200		elfzle1 12913	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (((𝐼 + 1) + 1)...𝑁) → ((𝐼 + 1) + 1) ≤ 𝑖)
201	200	adantl 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) ∧ 𝑖 ∈ (((𝐼 + 1) + 1)...𝑁)) → ((𝐼 + 1) + 1) ≤ 𝑖)
202	193, 195, 198, 199, 201	ltletrd 10802	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) ∧ 𝑖 ∈ (((𝐼 + 1) + 1)...𝑁)) → (𝐼 + 1) < 𝑖)
203	193, 202	gtned 10777	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) ∧ 𝑖 ∈ (((𝐼 + 1) + 1)...𝑁)) → 𝑖 ≠ (𝐼 + 1))
204	191, 203, 125	syl2anc 586	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) ∧ 𝑖 ∈ (((𝐼 + 1) + 1)...𝑁)) → (𝑄‘𝑖) = 0)
205	204	sq0id 13560	. . . . . . . . . . . . . . 15 ⊢ ((((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) ∧ 𝑖 ∈ (((𝐼 + 1) + 1)...𝑁)) → ((𝑄‘𝑖)↑2) = 0)
206	205	sumeq2dv 15062	. . . . . . . . . . . . . 14 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → Σ𝑖 ∈ (((𝐼 + 1) + 1)...𝑁)((𝑄‘𝑖)↑2) = Σ𝑖 ∈ (((𝐼 + 1) + 1)...𝑁)0)
207		fzfi 13343	. . . . . . . . . . . . . . . 16 ⊢ (((𝐼 + 1) + 1)...𝑁) ∈ Fin
208	207	olci 862	. . . . . . . . . . . . . . 15 ⊢ ((((𝐼 + 1) + 1)...𝑁) ⊆ (ℤ_≥‘1) ∨ (((𝐼 + 1) + 1)...𝑁) ∈ Fin)
209		sumz 15081	. . . . . . . . . . . . . . 15 ⊢ (((((𝐼 + 1) + 1)...𝑁) ⊆ (ℤ_≥‘1) ∨ (((𝐼 + 1) + 1)...𝑁) ∈ Fin) → Σ𝑖 ∈ (((𝐼 + 1) + 1)...𝑁)0 = 0)
210	208, 209	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ Σ𝑖 ∈ (((𝐼 + 1) + 1)...𝑁)0 = 0
211	206, 210	syl6eq 2874	. . . . . . . . . . . . 13 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → Σ𝑖 ∈ (((𝐼 + 1) + 1)...𝑁)((𝑄‘𝑖)↑2) = 0)
212	211	oveq2d 7174	. . . . . . . . . . . 12 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → (1 + Σ𝑖 ∈ (((𝐼 + 1) + 1)...𝑁)((𝑄‘𝑖)↑2)) = (1 + 0))
213		1p0e1 11764	. . . . . . . . . . . 12 ⊢ (1 + 0) = 1
214	212, 213	syl6eq 2874	. . . . . . . . . . 11 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → (1 + Σ𝑖 ∈ (((𝐼 + 1) + 1)...𝑁)((𝑄‘𝑖)↑2)) = 1)
215	186, 214	eqtrd 2858	. . . . . . . . . 10 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → Σ𝑖 ∈ ((𝐼 + 1)...𝑁)((𝑄‘𝑖)↑2) = 1)
216	215	ex 415	. . . . . . . . 9 ⊢ ((𝐼 + 1) ≠ 𝑁 → ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ ((𝐼 + 1)...𝑁)((𝑄‘𝑖)↑2) = 1))
217	142, 216	pm2.61ine 3102	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ ((𝐼 + 1)...𝑁)((𝑄‘𝑖)↑2) = 1)
218	133, 217	oveq12d 7176	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (Σ𝑖 ∈ (2...𝐼)((𝑄‘𝑖)↑2) + Σ𝑖 ∈ ((𝐼 + 1)...𝑁)((𝑄‘𝑖)↑2)) = (0 + 1))
219		0p1e1 11762	. . . . . . 7 ⊢ (0 + 1) = 1
220	218, 219	syl6eq 2874	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (Σ𝑖 ∈ (2...𝐼)((𝑄‘𝑖)↑2) + Σ𝑖 ∈ ((𝐼 + 1)...𝑁)((𝑄‘𝑖)↑2)) = 1)
221	89, 220	eqtrd 2858	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (2...𝑁)((𝑄‘𝑖)↑2) = 1)
222		simp1 1132	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝑁 ∈ (ℤ_≥‘3))
223		2lt3 11812	. . . . . . . . . 10 ⊢ 2 < 3
224	152, 48, 223	ltleii 10765	. . . . . . . . 9 ⊢ 2 ≤ 3
225		2z 12017	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
226	225	eluz1i 12254	. . . . . . . . 9 ⊢ (3 ∈ (ℤ_≥‘2) ↔ (3 ∈ ℤ ∧ 2 ≤ 3))
227	8, 224, 226	mpbir2an 709	. . . . . . . 8 ⊢ 3 ∈ (ℤ_≥‘2)
228		uztrn 12264	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 ∈ (ℤ_≥‘2)) → 𝑁 ∈ (ℤ_≥‘2))
229	222, 227, 228	sylancl 588	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝑁 ∈ (ℤ_≥‘2))
230		fveq2 6672	. . . . . . . 8 ⊢ (𝑖 = 2 → (𝑄‘𝑖) = (𝑄‘2))
231	230	oveq1d 7173	. . . . . . 7 ⊢ (𝑖 = 2 → ((𝑄‘𝑖)↑2) = ((𝑄‘2)↑2))
232	229, 88, 231	fsum1p 15110	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (2...𝑁)((𝑄‘𝑖)↑2) = (((𝑄‘2)↑2) + Σ𝑖 ∈ ((2 + 1)...𝑁)((𝑄‘𝑖)↑2)))
233	59	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → 𝑁 ∈ ℤ)
234	233	zred 12090	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → 𝑁 ∈ ℝ)
235		lttr 10719	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 ∈ ℝ ∧ 3 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((2 < 3 ∧ 3 < 𝑁) → 2 < 𝑁))
236	152, 48, 235	mp3an12 1447	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℝ → ((2 < 3 ∧ 3 < 𝑁) → 2 < 𝑁))
237	223, 236	mpani 694	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℝ → (3 < 𝑁 → 2 < 𝑁))
238	49, 237	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘3) → (3 < 𝑁 → 2 < 𝑁))
239	238	imp 409	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → 2 < 𝑁)
240		ltle 10731	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (2 < 𝑁 → 2 ≤ 𝑁))
241	152, 240	mpan 688	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℝ → (2 < 𝑁 → 2 ≤ 𝑁))
242	234, 239, 241	sylc 65	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → 2 ≤ 𝑁)
243	242, 154	jctil 522	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → (1 ≤ 2 ∧ 2 ≤ 𝑁))
244		1z 12015	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℤ
245		elfz 12901	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (2 ∈ (1...𝑁) ↔ (1 ≤ 2 ∧ 2 ≤ 𝑁)))
246	225, 244, 233, 245	mp3an12i 1461	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → (2 ∈ (1...𝑁) ↔ (1 ≤ 2 ∧ 2 ≤ 𝑁)))
247	243, 246	mpbird 259	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → 2 ∈ (1...𝑁))
248	247	3adant3 1128	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 2 ∈ (1...𝑁))
249	91	ltp1d 11572	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ (2...(𝑁 − 1)) → 𝐼 < (𝐼 + 1))
250	153, 91, 117, 156, 249	lelttrd 10800	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ (2...(𝑁 − 1)) → 2 < (𝐼 + 1))
251	250	3ad2ant3 1131	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 2 < (𝐼 + 1))
252		ltne 10739	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ ∧ 2 < (𝐼 + 1)) → (𝐼 + 1) ≠ 2)
253	152, 251, 252	sylancr 589	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (𝐼 + 1) ≠ 2)
254	253	necomd 3073	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 2 ≠ (𝐼 + 1))
255	13	axlowdimlem12 26741	. . . . . . . . . 10 ⊢ ((2 ∈ (1...𝑁) ∧ 2 ≠ (𝐼 + 1)) → (𝑄‘2) = 0)
256	248, 254, 255	syl2anc 586	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (𝑄‘2) = 0)
257	256	sq0id 13560	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → ((𝑄‘2)↑2) = 0)
258	257	oveq1d 7173	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (((𝑄‘2)↑2) + Σ𝑖 ∈ ((2 + 1)...𝑁)((𝑄‘𝑖)↑2)) = (0 + Σ𝑖 ∈ ((2 + 1)...𝑁)((𝑄‘𝑖)↑2)))
259	3	oveq1i 7168	. . . . . . . . 9 ⊢ (3...𝑁) = ((2 + 1)...𝑁)
260	259	sumeq1i 15057	. . . . . . . 8 ⊢ Σ𝑖 ∈ (3...𝑁)((𝑄‘𝑖)↑2) = Σ𝑖 ∈ ((2 + 1)...𝑁)((𝑄‘𝑖)↑2)
261	260	oveq2i 7169	. . . . . . 7 ⊢ (0 + Σ𝑖 ∈ (3...𝑁)((𝑄‘𝑖)↑2)) = (0 + Σ𝑖 ∈ ((2 + 1)...𝑁)((𝑄‘𝑖)↑2))
262	258, 261	syl6eqr 2876	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (((𝑄‘2)↑2) + Σ𝑖 ∈ ((2 + 1)...𝑁)((𝑄‘𝑖)↑2)) = (0 + Σ𝑖 ∈ (3...𝑁)((𝑄‘𝑖)↑2)))
263		fzfid 13344	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (3...𝑁) ∈ Fin)
264		3nn 11719	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℕ
265	264, 175	eleqtri 2913	. . . . . . . . . . . . 13 ⊢ 3 ∈ (ℤ_≥‘1)
266		fzss1 12949	. . . . . . . . . . . . 13 ⊢ (3 ∈ (ℤ_≥‘1) → (3...𝑁) ⊆ (1...𝑁))
267	265, 266	ax-mp 5	. . . . . . . . . . . 12 ⊢ (3...𝑁) ⊆ (1...𝑁)
268	267	sseli 3965	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (3...𝑁) → 𝑖 ∈ (1...𝑁))
269	81, 268, 85	syl2an 597	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (3...𝑁)) → (𝑄‘𝑖) ∈ ℂ)
270	269	sqcld 13511	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (3...𝑁)) → ((𝑄‘𝑖)↑2) ∈ ℂ)
271	270	3adantl2 1163	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (3...𝑁)) → ((𝑄‘𝑖)↑2) ∈ ℂ)
272	263, 271	fsumcl 15092	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (3...𝑁)((𝑄‘𝑖)↑2) ∈ ℂ)
273	272	addid2d 10843	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (0 + Σ𝑖 ∈ (3...𝑁)((𝑄‘𝑖)↑2)) = Σ𝑖 ∈ (3...𝑁)((𝑄‘𝑖)↑2))
274	232, 262, 273	3eqtrrd 2863	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (3...𝑁)((𝑄‘𝑖)↑2) = Σ𝑖 ∈ (2...𝑁)((𝑄‘𝑖)↑2))
275		simpl 485	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → 𝑁 ∈ (ℤ_≥‘3))
276	21	axlowdimlem7 26736	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑃 ∈ (𝔼‘𝑁))
277	276	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) ∧ 𝑖 ∈ (3...𝑁)) → 𝑃 ∈ (𝔼‘𝑁))
278	268	adantl 484	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) ∧ 𝑖 ∈ (3...𝑁)) → 𝑖 ∈ (1...𝑁))
279		fveecn 26690	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (𝔼‘𝑁) ∧ 𝑖 ∈ (1...𝑁)) → (𝑃‘𝑖) ∈ ℂ)
280	277, 278, 279	syl2anc 586	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) ∧ 𝑖 ∈ (3...𝑁)) → (𝑃‘𝑖) ∈ ℂ)
281	280	sqcld 13511	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) ∧ 𝑖 ∈ (3...𝑁)) → ((𝑃‘𝑖)↑2) ∈ ℂ)
282		neg1sqe1 13562	. . . . . . . . . 10 ⊢ (-1↑2) = 1
283	24, 282	syl6eq 2874	. . . . . . . . 9 ⊢ (𝑖 = 3 → ((𝑃‘𝑖)↑2) = 1)
284	275, 281, 283	fsum1p 15110	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → Σ𝑖 ∈ (3...𝑁)((𝑃‘𝑖)↑2) = (1 + Σ𝑖 ∈ ((3 + 1)...𝑁)((𝑃‘𝑖)↑2)))
285		1re 10643	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
286		zaddcl 12025	. . . . . . . . . . . . . . . . . . 19 ⊢ ((3 ∈ ℤ ∧ 1 ∈ ℤ) → (3 + 1) ∈ ℤ)
287	8, 244, 286	mp2an 690	. . . . . . . . . . . . . . . . . 18 ⊢ (3 + 1) ∈ ℤ
288	287	zrei 11990	. . . . . . . . . . . . . . . . 17 ⊢ (3 + 1) ∈ ℝ
289		1lt3 11813	. . . . . . . . . . . . . . . . . 18 ⊢ 1 < 3
290	48	ltp1i 11546	. . . . . . . . . . . . . . . . . 18 ⊢ 3 < (3 + 1)
291	285, 48, 288	lttri 10768	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 < 3 ∧ 3 < (3 + 1)) → 1 < (3 + 1))
292	289, 290, 291	mp2an 690	. . . . . . . . . . . . . . . . 17 ⊢ 1 < (3 + 1)
293	285, 288, 292	ltleii 10765	. . . . . . . . . . . . . . . 16 ⊢ 1 ≤ (3 + 1)
294		eluz 12260	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℤ ∧ (3 + 1) ∈ ℤ) → ((3 + 1) ∈ (ℤ_≥‘1) ↔ 1 ≤ (3 + 1)))
295	244, 287, 294	mp2an 690	. . . . . . . . . . . . . . . 16 ⊢ ((3 + 1) ∈ (ℤ_≥‘1) ↔ 1 ≤ (3 + 1))
296	293, 295	mpbir 233	. . . . . . . . . . . . . . 15 ⊢ (3 + 1) ∈ (ℤ_≥‘1)
297		fzss1 12949	. . . . . . . . . . . . . . 15 ⊢ ((3 + 1) ∈ (ℤ_≥‘1) → ((3 + 1)...𝑁) ⊆ (1...𝑁))
298	296, 297	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((3 + 1)...𝑁) ⊆ (1...𝑁)
299	298	sseli 3965	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ ((3 + 1)...𝑁) → 𝑖 ∈ (1...𝑁))
300	48, 288	ltnlei 10763	. . . . . . . . . . . . . . . . . . 19 ⊢ (3 < (3 + 1) ↔ ¬ (3 + 1) ≤ 3)
301	290, 300	mpbi 232	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ (3 + 1) ≤ 3
302	301	intnanr 490	. . . . . . . . . . . . . . . . 17 ⊢ ¬ ((3 + 1) ≤ 3 ∧ 3 ≤ 𝑁)
303		elfz 12901	. . . . . . . . . . . . . . . . . 18 ⊢ ((3 ∈ ℤ ∧ (3 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (3 ∈ ((3 + 1)...𝑁) ↔ ((3 + 1) ≤ 3 ∧ 3 ≤ 𝑁)))
304	8, 287, 233, 303	mp3an12i 1461	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → (3 ∈ ((3 + 1)...𝑁) ↔ ((3 + 1) ≤ 3 ∧ 3 ≤ 𝑁)))
305	302, 304	mtbiri 329	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → ¬ 3 ∈ ((3 + 1)...𝑁))
306		eleq1 2902	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 3 → (𝑖 ∈ ((3 + 1)...𝑁) ↔ 3 ∈ ((3 + 1)...𝑁)))
307	306	notbid 320	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 3 → (¬ 𝑖 ∈ ((3 + 1)...𝑁) ↔ ¬ 3 ∈ ((3 + 1)...𝑁)))
308	305, 307	syl5ibrcom 249	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → (𝑖 = 3 → ¬ 𝑖 ∈ ((3 + 1)...𝑁)))
309	308	necon2ad 3033	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → (𝑖 ∈ ((3 + 1)...𝑁) → 𝑖 ≠ 3))
310	309	imp 409	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) ∧ 𝑖 ∈ ((3 + 1)...𝑁)) → 𝑖 ≠ 3)
311	21	axlowdimlem9 26738	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (1...𝑁) ∧ 𝑖 ≠ 3) → (𝑃‘𝑖) = 0)
312	299, 310, 311	syl2an2 684	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) ∧ 𝑖 ∈ ((3 + 1)...𝑁)) → (𝑃‘𝑖) = 0)
313	312	sq0id 13560	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) ∧ 𝑖 ∈ ((3 + 1)...𝑁)) → ((𝑃‘𝑖)↑2) = 0)
314	313	sumeq2dv 15062	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → Σ𝑖 ∈ ((3 + 1)...𝑁)((𝑃‘𝑖)↑2) = Σ𝑖 ∈ ((3 + 1)...𝑁)0)
315		fzfi 13343	. . . . . . . . . . . 12 ⊢ ((3 + 1)...𝑁) ∈ Fin
316	315	olci 862	. . . . . . . . . . 11 ⊢ (((3 + 1)...𝑁) ⊆ (ℤ_≥‘1) ∨ ((3 + 1)...𝑁) ∈ Fin)
317		sumz 15081	. . . . . . . . . . 11 ⊢ ((((3 + 1)...𝑁) ⊆ (ℤ_≥‘1) ∨ ((3 + 1)...𝑁) ∈ Fin) → Σ𝑖 ∈ ((3 + 1)...𝑁)0 = 0)
318	316, 317	ax-mp 5	. . . . . . . . . 10 ⊢ Σ𝑖 ∈ ((3 + 1)...𝑁)0 = 0
319	314, 318	syl6eq 2874	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → Σ𝑖 ∈ ((3 + 1)...𝑁)((𝑃‘𝑖)↑2) = 0)
320	319	oveq2d 7174	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → (1 + Σ𝑖 ∈ ((3 + 1)...𝑁)((𝑃‘𝑖)↑2)) = (1 + 0))
321	284, 320	eqtrd 2858	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → Σ𝑖 ∈ (3...𝑁)((𝑃‘𝑖)↑2) = (1 + 0))
322	321, 213	syl6eq 2874	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → Σ𝑖 ∈ (3...𝑁)((𝑃‘𝑖)↑2) = 1)
323	322	3adant3 1128	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (3...𝑁)((𝑃‘𝑖)↑2) = 1)
324	221, 274, 323	3eqtr4rd 2869	. . . 4 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (3...𝑁)((𝑃‘𝑖)↑2) = Σ𝑖 ∈ (3...𝑁)((𝑄‘𝑖)↑2))
325	44, 54, 55, 324	syl3anc 1367	. . 3 ⊢ ((𝑁 ≠ 3 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → Σ𝑖 ∈ (3...𝑁)((𝑃‘𝑖)↑2) = Σ𝑖 ∈ (3...𝑁)((𝑄‘𝑖)↑2))
326	325	ex 415	. 2 ⊢ (𝑁 ≠ 3 → ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (3...𝑁)((𝑃‘𝑖)↑2) = Σ𝑖 ∈ (3...𝑁)((𝑄‘𝑖)↑2)))
327	43, 326	pm2.61ine 3102	1 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (3...𝑁)((𝑃‘𝑖)↑2) = Σ𝑖 ∈ (3...𝑁)((𝑄‘𝑖)↑2))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∖ cdif 3935 ∪ cun 3936 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 {csn 4569 ⟨cop 4575 class class class wbr 5068 × cxp 5555 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 ℂcc 10537 ℝcr 10538 0cc0 10539 1c1 10540 + caddc 10542 < clt 10677 ≤ cle 10678 − cmin 10872 -cneg 10873 ℕcn 11640 2c2 11695 3c3 11696 ℤcz 11984 ℤ_≥cuz 12246 ...cfz 12895 ↑cexp 13432 Σcsu 15044 𝔼cee 26676

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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-sum 15045 df-ee 26679

This theorem is referenced by: axlowdimlem17 26746

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