axlowdimlem16 - Metamath Proof Explorer

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Theorem axlowdimlem16 29288

Description: Lemma for axlowdim 29292. 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 13564	. . . . . 6 ⊢ (𝐼 ∈ (2...2) → 𝐼 = 2)
2		3z 12628	. . . . . . . 8 ⊢ 3 ∈ ℤ
3		ax-1cn 11159	. . . . . . . . 9 ⊢ 1 ∈ ℂ
4	3	sqcli 14219	. . . . . . . 8 ⊢ (1↑2) ∈ ℂ
5		fveq2 6883	. . . . . . . . . . . 12 ⊢ (𝑖 = 3 → (𝑃‘𝑖) = (𝑃‘3))
6		axlowdimlem16.1	. . . . . . . . . . . . 13 ⊢ 𝑃 = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))
7	6	axlowdimlem8 29280	. . . . . . . . . . . 12 ⊢ (𝑃‘3) = -1
8	5, 7	eqtrdi 2814	. . . . . . . . . . 11 ⊢ (𝑖 = 3 → (𝑃‘𝑖) = -1)
9	8	oveq1d 7427	. . . . . . . . . 10 ⊢ (𝑖 = 3 → ((𝑃‘𝑖)↑2) = (-1↑2))
10		sqneg 14153	. . . . . . . . . . 11 ⊢ (1 ∈ ℂ → (-1↑2) = (1↑2))
11	3, 10	ax-mp 5	. . . . . . . . . 10 ⊢ (-1↑2) = (1↑2)
12	9, 11	eqtrdi 2814	. . . . . . . . 9 ⊢ (𝑖 = 3 → ((𝑃‘𝑖)↑2) = (1↑2))
13	12	fsum1 15800	. . . . . . . 8 ⊢ ((3 ∈ ℤ ∧ (1↑2) ∈ ℂ) → Σ𝑖 ∈ (3...3)((𝑃‘𝑖)↑2) = (1↑2))
14	2, 4, 13	mp2an 704	. . . . . . 7 ⊢ Σ𝑖 ∈ (3...3)((𝑃‘𝑖)↑2) = (1↑2)
15		df-3 12305	. . . . . . . . . . 11 ⊢ 3 = (2 + 1)
16		oveq1 7419	. . . . . . . . . . 11 ⊢ (𝐼 = 2 → (𝐼 + 1) = (2 + 1))
17	15, 16	eqtr4id 2817	. . . . . . . . . 10 ⊢ (𝐼 = 2 → 3 = (𝐼 + 1))
18	17, 17	oveq12d 7430	. . . . . . . . 9 ⊢ (𝐼 = 2 → (3...3) = ((𝐼 + 1)...(𝐼 + 1)))
19	18	sumeq1d 15753	. . . . . . . 8 ⊢ (𝐼 = 2 → Σ𝑖 ∈ (3...3)((𝑄‘𝑖)↑2) = Σ𝑖 ∈ ((𝐼 + 1)...(𝐼 + 1))((𝑄‘𝑖)↑2))
20	16, 15	eqtr4di 2816	. . . . . . . . . 10 ⊢ (𝐼 = 2 → (𝐼 + 1) = 3)
21	20, 2	eqeltrdi 2871	. . . . . . . . 9 ⊢ (𝐼 = 2 → (𝐼 + 1) ∈ ℤ)
22		fveq2 6883	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝐼 + 1) → (𝑄‘𝑖) = (𝑄‘(𝐼 + 1)))
23		axlowdimlem16.2	. . . . . . . . . . . . 13 ⊢ 𝑄 = ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}))
24	23	axlowdimlem11 29283	. . . . . . . . . . . 12 ⊢ (𝑄‘(𝐼 + 1)) = 1
25	22, 24	eqtrdi 2814	. . . . . . . . . . 11 ⊢ (𝑖 = (𝐼 + 1) → (𝑄‘𝑖) = 1)
26	25	oveq1d 7427	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼 + 1) → ((𝑄‘𝑖)↑2) = (1↑2))
27	26	fsum1 15800	. . . . . . . . 9 ⊢ (((𝐼 + 1) ∈ ℤ ∧ (1↑2) ∈ ℂ) → Σ𝑖 ∈ ((𝐼 + 1)...(𝐼 + 1))((𝑄‘𝑖)↑2) = (1↑2))
28	21, 4, 27	sylancl 597	. . . . . . . 8 ⊢ (𝐼 = 2 → Σ𝑖 ∈ ((𝐼 + 1)...(𝐼 + 1))((𝑄‘𝑖)↑2) = (1↑2))
29	19, 28	eqtrd 2798	. . . . . . 7 ⊢ (𝐼 = 2 → Σ𝑖 ∈ (3...3)((𝑄‘𝑖)↑2) = (1↑2))
30	14, 29	eqtr4id 2817	. . . . . 6 ⊢ (𝐼 = 2 → Σ𝑖 ∈ (3...3)((𝑃‘𝑖)↑2) = Σ𝑖 ∈ (3...3)((𝑄‘𝑖)↑2))
31	1, 30	syl 18	. . . . 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 7419	. . . . . . 7 ⊢ (𝑁 = 3 → (𝑁 − 1) = (3 − 1))
34		3m1e2 12369	. . . . . . 7 ⊢ (3 − 1) = 2
35	33, 34	eqtrdi 2814	. . . . . 6 ⊢ (𝑁 = 3 → (𝑁 − 1) = 2)
36	35	oveq2d 7428	. . . . 5 ⊢ (𝑁 = 3 → (2...(𝑁 − 1)) = (2...2))
37	36	eleq2d 2849	. . . 4 ⊢ (𝑁 = 3 → (𝐼 ∈ (2...(𝑁 − 1)) ↔ 𝐼 ∈ (2...2)))
38		oveq2 7420	. . . . . 6 ⊢ (𝑁 = 3 → (3...𝑁) = (3...3))
39	38	sumeq1d 15753	. . . . 5 ⊢ (𝑁 = 3 → Σ𝑖 ∈ (3...𝑁)((𝑃‘𝑖)↑2) = Σ𝑖 ∈ (3...3)((𝑃‘𝑖)↑2))
40	38	sumeq1d 15753	. . . . 5 ⊢ (𝑁 = 3 → Σ𝑖 ∈ (3...𝑁)((𝑄‘𝑖)↑2) = Σ𝑖 ∈ (3...3)((𝑄‘𝑖)↑2))
41	39, 40	eqeq12d 2779	. . . 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 297	. . 3 ⊢ (𝑁 = 3 → (𝐼 ∈ (2...(𝑁 − 1)) → Σ𝑖 ∈ (3...𝑁)((𝑃‘𝑖)↑2) = Σ𝑖 ∈ (3...𝑁)((𝑄‘𝑖)↑2)))
43	42	adantld 495	. 2 ⊢ (𝑁 = 3 → ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (3...𝑁)((𝑃‘𝑖)↑2) = Σ𝑖 ∈ (3...𝑁)((𝑄‘𝑖)↑2)))
44		simprl 782	. . . 4 ⊢ ((𝑁 ≠ 3 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 𝑁 ∈ (ℤ_≥‘3))
45		eluzle 12876	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
46	45	adantl 486	. . . . . 6 ⊢ ((𝑁 ≠ 3 ∧ 𝑁 ∈ (ℤ_≥‘3)) → 3 ≤ 𝑁)
47		simpl 487	. . . . . 6 ⊢ ((𝑁 ≠ 3 ∧ 𝑁 ∈ (ℤ_≥‘3)) → 𝑁 ≠ 3)
48		3re 12322	. . . . . . 7 ⊢ 3 ∈ ℝ
49		eluzelre 12874	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℝ)
50	49	adantl 486	. . . . . . 7 ⊢ ((𝑁 ≠ 3 ∧ 𝑁 ∈ (ℤ_≥‘3)) → 𝑁 ∈ ℝ)
51		ltlen 11312	. . . . . . 7 ⊢ ((3 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (3 < 𝑁 ↔ (3 ≤ 𝑁 ∧ 𝑁 ≠ 3)))
52	48, 50, 51	sylancr 598	. . . . . 6 ⊢ ((𝑁 ≠ 3 ∧ 𝑁 ∈ (ℤ_≥‘3)) → (3 < 𝑁 ↔ (3 ≤ 𝑁 ∧ 𝑁 ≠ 3)))
53	46, 47, 52	mpbir2and 725	. . . . 5 ⊢ ((𝑁 ≠ 3 ∧ 𝑁 ∈ (ℤ_≥‘3)) → 3 < 𝑁)
54	53	adantrr 729	. . . 4 ⊢ ((𝑁 ≠ 3 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 3 < 𝑁)
55		simprr 784	. . . 4 ⊢ ((𝑁 ≠ 3 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 𝐼 ∈ (2...(𝑁 − 1)))
56		fzssp1 13597	. . . . . . . . . . . . 13 ⊢ (2...(𝑁 − 1)) ⊆ (2...((𝑁 − 1) + 1))
57		simp3 1156	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝐼 ∈ (2...(𝑁 − 1)))
58	56, 57	sselid 3936	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝐼 ∈ (2...((𝑁 − 1) + 1)))
59		eluzelz 12873	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
60	59	3ad2ant1 1151	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝑁 ∈ ℤ)
61	60	zcnd 12702	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝑁 ∈ ℂ)
62		npcan 11467	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
63	61, 3, 62	sylancl 597	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁)
64	63	oveq2d 7428	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (2...((𝑁 − 1) + 1)) = (2...𝑁))
65	58, 64	eleqtrd 2865	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝐼 ∈ (2...𝑁))
66		elfzelz 13553	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (2...𝑁) → 𝐼 ∈ ℤ)
67	65, 66	syl 18	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝐼 ∈ ℤ)
68	67	zred 12701	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝐼 ∈ ℝ)
69	68	ltp1d 12146	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝐼 < (𝐼 + 1))
70		fzdisj 13581	. . . . . . . 8 ⊢ (𝐼 < (𝐼 + 1) → ((2...𝐼) ∩ ((𝐼 + 1)...𝑁)) = ∅)
71	69, 70	syl 18	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → ((2...𝐼) ∩ ((𝐼 + 1)...𝑁)) = ∅)
72		fzsplit 13580	. . . . . . . 8 ⊢ (𝐼 ∈ (2...𝑁) → (2...𝑁) = ((2...𝐼) ∪ ((𝐼 + 1)...𝑁)))
73	65, 72	syl 18	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (2...𝑁) = ((2...𝐼) ∪ ((𝐼 + 1)...𝑁)))
74		fzfid 14011	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (2...𝑁) ∈ Fin)
75		eluz3nn 12914	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℕ)
76		2eluzge1 12907	. . . . . . . . . . . . 13 ⊢ 2 ∈ (ℤ_≥‘1)
77		fzss1 13593	. . . . . . . . . . . . 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 3934	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (2...(𝑁 − 1)) → 𝐼 ∈ (1...(𝑁 − 1)))
80	23	axlowdimlem10 29282	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → 𝑄 ∈ (𝔼‘𝑁))
81	75, 79, 80	syl2an 607	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝑄 ∈ (𝔼‘𝑁))
82		fzss1 13593	. . . . . . . . . . . 12 ⊢ (2 ∈ (ℤ_≥‘1) → (2...𝑁) ⊆ (1...𝑁))
83	76, 82	ax-mp 5	. . . . . . . . . . 11 ⊢ (2...𝑁) ⊆ (1...𝑁)
84	83	sseli 3934	. . . . . . . . . 10 ⊢ (𝑖 ∈ (2...𝑁) → 𝑖 ∈ (1...𝑁))
85		fveecn 29233	. . . . . . . . . 10 ⊢ ((𝑄 ∈ (𝔼‘𝑁) ∧ 𝑖 ∈ (1...𝑁)) → (𝑄‘𝑖) ∈ ℂ)
86	81, 84, 85	syl2an 607	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (2...𝑁)) → (𝑄‘𝑖) ∈ ℂ)
87	86	sqcld 14182	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (2...𝑁)) → ((𝑄‘𝑖)↑2) ∈ ℂ)
88	87	3adantl2 1186	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (2...𝑁)) → ((𝑄‘𝑖)↑2) ∈ ℂ)
89	71, 73, 74, 88	fsumsplit 15794	. . . . . 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 13553	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ (2...(𝑁 − 1)) → 𝐼 ∈ ℤ)
91	90	zred 12701	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐼 ∈ (2...(𝑁 − 1)) → 𝐼 ∈ ℝ)
92	91	3ad2ant3 1153	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝐼 ∈ ℝ)
93	49	3ad2ant1 1151	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝑁 ∈ ℝ)
94		peano2rem 11526	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℝ → (𝑁 − 1) ∈ ℝ)
95	93, 94	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (𝑁 − 1) ∈ ℝ)
96		elfzle2 13557	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ (2...(𝑁 − 1)) → 𝐼 ≤ (𝑁 − 1))
97	96	3ad2ant3 1153	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝐼 ≤ (𝑁 − 1))
98	93	ltm1d 12148	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (𝑁 − 1) < 𝑁)
99	92, 95, 93, 97, 98	lelttrd 11369	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝐼 < 𝑁)
100	92, 93, 99	ltled 11359	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝐼 ≤ 𝑁)
101	90	3ad2ant3 1153	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝐼 ∈ ℤ)
102		eluz 12877	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ_≥‘𝐼) ↔ 𝐼 ≤ 𝑁))
103	101, 60, 102	syl2anc 595	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (𝑁 ∈ (ℤ_≥‘𝐼) ↔ 𝐼 ≤ 𝑁))
104	100, 103	mpbird 260	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝑁 ∈ (ℤ_≥‘𝐼))
105		fzss2 13594	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘𝐼) → (1...𝐼) ⊆ (1...𝑁))
106	104, 105	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (1...𝐼) ⊆ (1...𝑁))
107	106	sseld 3937	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (𝑖 ∈ (1...𝐼) → 𝑖 ∈ (1...𝑁)))
108		fzss1 13593	. . . . . . . . . . . . . . 15 ⊢ (2 ∈ (ℤ_≥‘1) → (2...𝐼) ⊆ (1...𝐼))
109	76, 108	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (2...𝐼) ⊆ (1...𝐼)
110	109	sseli 3934	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (2...𝐼) → 𝑖 ∈ (1...𝐼))
111	107, 110	impel 514	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (2...𝐼)) → 𝑖 ∈ (1...𝑁))
112		elfzelz 13553	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (2...𝐼) → 𝑖 ∈ ℤ)
113	112	zred 12701	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (2...𝐼) → 𝑖 ∈ ℝ)
114	113	adantl 486	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (2...𝐼)) → 𝑖 ∈ ℝ)
115	92	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (2...𝐼)) → 𝐼 ∈ ℝ)
116		peano2re 11384	. . . . . . . . . . . . . . . . 17 ⊢ (𝐼 ∈ ℝ → (𝐼 + 1) ∈ ℝ)
117	91, 116	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ (2...(𝑁 − 1)) → (𝐼 + 1) ∈ ℝ)
118	117	3ad2ant3 1153	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (𝐼 + 1) ∈ ℝ)
119	118	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (2...𝐼)) → (𝐼 + 1) ∈ ℝ)
120		elfzle2 13557	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (2...𝐼) → 𝑖 ≤ 𝐼)
121	120	adantl 486	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (2...𝐼)) → 𝑖 ≤ 𝐼)
122	115	ltp1d 12146	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (2...𝐼)) → 𝐼 < (𝐼 + 1))
123	114, 115, 119, 121, 122	lelttrd 11369	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (2...𝐼)) → 𝑖 < (𝐼 + 1))
124	114, 123	ltned 11347	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (2...𝐼)) → 𝑖 ≠ (𝐼 + 1))
125	23	axlowdimlem12 29284	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (1...𝑁) ∧ 𝑖 ≠ (𝐼 + 1)) → (𝑄‘𝑖) = 0)
126	111, 124, 125	syl2anc 595	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (2...𝐼)) → (𝑄‘𝑖) = 0)
127	126	sq0id 14232	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (2...𝐼)) → ((𝑄‘𝑖)↑2) = 0)
128	127	sumeq2dv 15755	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (2...𝐼)((𝑄‘𝑖)↑2) = Σ𝑖 ∈ (2...𝐼)0)
129		fzfi 14010	. . . . . . . . . . 11 ⊢ (2...𝐼) ∈ Fin
130	129	olci 879	. . . . . . . . . 10 ⊢ ((2...𝐼) ⊆ (ℤ_≥‘1) ∨ (2...𝐼) ∈ Fin)
131		sumz 15775	. . . . . . . . . 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	eqtrdi 2814	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (2...𝐼)((𝑄‘𝑖)↑2) = 0)
134	101	peano2zd 12704	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (𝐼 + 1) ∈ ℤ)
135		sq1 14233	. . . . . . . . . . . . 13 ⊢ (1↑2) = 1
136	26, 135	eqtrdi 2814	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝐼 + 1) → ((𝑄‘𝑖)↑2) = 1)
137	136	fsum1 15800	. . . . . . . . . . 11 ⊢ (((𝐼 + 1) ∈ ℤ ∧ 1 ∈ ℂ) → Σ𝑖 ∈ ((𝐼 + 1)...(𝐼 + 1))((𝑄‘𝑖)↑2) = 1)
138	134, 3, 137	sylancl 597	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ ((𝐼 + 1)...(𝐼 + 1))((𝑄‘𝑖)↑2) = 1)
139		oveq2 7420	. . . . . . . . . . . 12 ⊢ ((𝐼 + 1) = 𝑁 → ((𝐼 + 1)...(𝐼 + 1)) = ((𝐼 + 1)...𝑁))
140	139	sumeq1d 15753	. . . . . . . . . . 11 ⊢ ((𝐼 + 1) = 𝑁 → Σ𝑖 ∈ ((𝐼 + 1)...(𝐼 + 1))((𝑄‘𝑖)↑2) = Σ𝑖 ∈ ((𝐼 + 1)...𝑁)((𝑄‘𝑖)↑2))
141	140	eqeq1d 2765	. . . . . . . . . 10 ⊢ ((𝐼 + 1) = 𝑁 → (Σ𝑖 ∈ ((𝐼 + 1)...(𝐼 + 1))((𝑄‘𝑖)↑2) = 1 ↔ Σ𝑖 ∈ ((𝐼 + 1)...𝑁)((𝑄‘𝑖)↑2) = 1))
142	138, 141	imbitrid 247	. . . . . . . . 9 ⊢ ((𝐼 + 1) = 𝑁 → ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ ((𝐼 + 1)...𝑁)((𝑄‘𝑖)↑2) = 1))
143	101	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 𝐼 ∈ ℤ)
144	143	zred 12701	. . . . . . . . . . . . . . 15 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 𝐼 ∈ ℝ)
145	60	adantl 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 𝑁 ∈ ℤ)
146	145	zred 12701	. . . . . . . . . . . . . . . 16 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 𝑁 ∈ ℝ)
147	146, 94	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → (𝑁 − 1) ∈ ℝ)
148	97	adantl 486	. . . . . . . . . . . . . . 15 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 𝐼 ≤ (𝑁 − 1))
149	146	ltm1d 12148	. . . . . . . . . . . . . . 15 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → (𝑁 − 1) < 𝑁)
150	144, 147, 146, 148, 149	lelttrd 11369	. . . . . . . . . . . . . 14 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 𝐼 < 𝑁)
151		1red 11210	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ (2...(𝑁 − 1)) → 1 ∈ ℝ)
152		2re 12316	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℝ
153	152	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ (2...(𝑁 − 1)) → 2 ∈ ℝ)
154		1le2 12453	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ≤ 2
155	154	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ (2...(𝑁 − 1)) → 1 ≤ 2)
156		elfzle1 13556	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ (2...(𝑁 − 1)) → 2 ≤ 𝐼)
157	151, 153, 91, 155, 156	letrd 11368	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐼 ∈ (2...(𝑁 − 1)) → 1 ≤ 𝐼)
158	157	3ad2ant3 1153	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 1 ≤ 𝐼)
159	158	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 1 ≤ 𝐼)
160		elnnz1 12621	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ ℕ ↔ (𝐼 ∈ ℤ ∧ 1 ≤ 𝐼))
161	143, 159, 160	sylanbrc 594	. . . . . . . . . . . . . . 15 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 𝐼 ∈ ℕ)
162	75	3ad2ant1 1151	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝑁 ∈ ℕ)
163	162	adantl 486	. . . . . . . . . . . . . . 15 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 𝑁 ∈ ℕ)
164		nnltp1le 12653	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐼 < 𝑁 ↔ (𝐼 + 1) ≤ 𝑁))
165	161, 163, 164	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → (𝐼 < 𝑁 ↔ (𝐼 + 1) ≤ 𝑁))
166	150, 165	mpbid 235	. . . . . . . . . . . . 13 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → (𝐼 + 1) ≤ 𝑁)
167		eluz 12877	. . . . . . . . . . . . . 14 ⊢ (((𝐼 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ_≥‘(𝐼 + 1)) ↔ (𝐼 + 1) ≤ 𝑁))
168	134, 145, 167	syl2an2 698	. . . . . . . . . . . . 13 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → (𝑁 ∈ (ℤ_≥‘(𝐼 + 1)) ↔ (𝐼 + 1) ≤ 𝑁))
169	166, 168	mpbird 260	. . . . . . . . . . . 12 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 𝑁 ∈ (ℤ_≥‘(𝐼 + 1)))
170		simpr1 1213	. . . . . . . . . . . . . . . 16 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 𝑁 ∈ (ℤ_≥‘3))
171		simpr3 1215	. . . . . . . . . . . . . . . 16 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 𝐼 ∈ (2...(𝑁 − 1)))
172	170, 171, 81	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → 𝑄 ∈ (𝔼‘𝑁))
173	172	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) ∧ 𝑖 ∈ ((𝐼 + 1)...𝑁)) → 𝑄 ∈ (𝔼‘𝑁))
174	161	peano2nnd 12251	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → (𝐼 + 1) ∈ ℕ)
175		nnuz 12902	. . . . . . . . . . . . . . . . 17 ⊢ ℕ = (ℤ_≥‘1)
176	174, 175	eleqtrdi 2873	. . . . . . . . . . . . . . . 16 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → (𝐼 + 1) ∈ (ℤ_≥‘1))
177		fzss1 13593	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 + 1) ∈ (ℤ_≥‘1) → ((𝐼 + 1)...𝑁) ⊆ (1...𝑁))
178	176, 177	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → ((𝐼 + 1)...𝑁) ⊆ (1...𝑁))
179	178	sselda 3938	. . . . . . . . . . . . . 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 595	. . . . . . . . . . . . 13 ⊢ ((((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) ∧ 𝑖 ∈ ((𝐼 + 1)...𝑁)) → (𝑄‘𝑖) ∈ ℂ)
181	180	sqcld 14182	. . . . . . . . . . . 12 ⊢ ((((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) ∧ 𝑖 ∈ ((𝐼 + 1)...𝑁)) → ((𝑄‘𝑖)↑2) ∈ ℂ)
182	22	oveq1d 7427	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝐼 + 1) → ((𝑄‘𝑖)↑2) = ((𝑄‘(𝐼 + 1))↑2))
183	24	oveq1i 7422	. . . . . . . . . . . . . 14 ⊢ ((𝑄‘(𝐼 + 1))↑2) = (1↑2)
184	183, 135	eqtri 2786	. . . . . . . . . . . . 13 ⊢ ((𝑄‘(𝐼 + 1))↑2) = 1
185	182, 184	eqtrdi 2814	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝐼 + 1) → ((𝑄‘𝑖)↑2) = 1)
186	169, 181, 185	fsum1p 15806	. . . . . . . . . . 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 12251	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → ((𝐼 + 1) + 1) ∈ ℕ)
188	187, 175	eleqtrdi 2873	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → ((𝐼 + 1) + 1) ∈ (ℤ_≥‘1))
189		fzss1 13593	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐼 + 1) + 1) ∈ (ℤ_≥‘1) → (((𝐼 + 1) + 1)...𝑁) ⊆ (1...𝑁))
190	188, 189	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → (((𝐼 + 1) + 1)...𝑁) ⊆ (1...𝑁))
191	190	sselda 3938	. . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → (𝐼 + 1) ∈ ℝ)
193	192	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) ∧ 𝑖 ∈ (((𝐼 + 1) + 1)...𝑁)) → (𝐼 + 1) ∈ ℝ)
194		peano2re 11384	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐼 + 1) ∈ ℝ → ((𝐼 + 1) + 1) ∈ ℝ)
195	193, 194	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) ∧ 𝑖 ∈ (((𝐼 + 1) + 1)...𝑁)) → ((𝐼 + 1) + 1) ∈ ℝ)
196		elfzelz 13553	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (((𝐼 + 1) + 1)...𝑁) → 𝑖 ∈ ℤ)
197	196	zred 12701	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (((𝐼 + 1) + 1)...𝑁) → 𝑖 ∈ ℝ)
198	197	adantl 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) ∧ 𝑖 ∈ (((𝐼 + 1) + 1)...𝑁)) → 𝑖 ∈ ℝ)
199	193	ltp1d 12146	. . . . . . . . . . . . . . . . . . 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 13556	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (((𝐼 + 1) + 1)...𝑁) → ((𝐼 + 1) + 1) ≤ 𝑖)
201	200	adantl 486	. . . . . . . . . . . . . . . . . . 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 11371	. . . . . . . . . . . . . . . . . 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 11346	. . . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) ∧ 𝑖 ∈ (((𝐼 + 1) + 1)...𝑁)) → (𝑄‘𝑖) = 0)
205	204	sq0id 14232	. . . . . . . . . . . . . . 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 15755	. . . . . . . . . . . . . 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 14010	. . . . . . . . . . . . . . . 16 ⊢ (((𝐼 + 1) + 1)...𝑁) ∈ Fin
208	207	olci 879	. . . . . . . . . . . . . . 15 ⊢ ((((𝐼 + 1) + 1)...𝑁) ⊆ (ℤ_≥‘1) ∨ (((𝐼 + 1) + 1)...𝑁) ∈ Fin)
209		sumz 15775	. . . . . . . . . . . . . . 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	eqtrdi 2814	. . . . . . . . . . . . 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 7428	. . . . . . . . . . . 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 12364	. . . . . . . . . . . 12 ⊢ (1 + 0) = 1
214	212, 213	eqtrdi 2814	. . . . . . . . . . 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 2798	. . . . . . . . . 10 ⊢ (((𝐼 + 1) ≠ 𝑁 ∧ (𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1)))) → Σ𝑖 ∈ ((𝐼 + 1)...𝑁)((𝑄‘𝑖)↑2) = 1)
216	215	ex 417	. . . . . . . . 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 3041	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ ((𝐼 + 1)...𝑁)((𝑄‘𝑖)↑2) = 1)
218	133, 217	oveq12d 7430	. . . . . . 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 12362	. . . . . . 7 ⊢ (0 + 1) = 1
220	218, 219	eqtrdi 2814	. . . . . 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 2798	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (2...𝑁)((𝑄‘𝑖)↑2) = 1)
222		simp1 1154	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝑁 ∈ (ℤ_≥‘3))
223		2lt3 12415	. . . . . . . . . 10 ⊢ 2 < 3
224	152, 48, 223	ltleii 11334	. . . . . . . . 9 ⊢ 2 ≤ 3
225		2z 12627	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
226	225	eluz1i 12871	. . . . . . . . 9 ⊢ (3 ∈ (ℤ_≥‘2) ↔ (3 ∈ ℤ ∧ 2 ≤ 3))
227	2, 224, 226	mpbir2an 723	. . . . . . . 8 ⊢ 3 ∈ (ℤ_≥‘2)
228		uztrn 12881	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 ∈ (ℤ_≥‘2)) → 𝑁 ∈ (ℤ_≥‘2))
229	222, 227, 228	sylancl 597	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝑁 ∈ (ℤ_≥‘2))
230		fveq2 6883	. . . . . . . 8 ⊢ (𝑖 = 2 → (𝑄‘𝑖) = (𝑄‘2))
231	230	oveq1d 7427	. . . . . . 7 ⊢ (𝑖 = 2 → ((𝑄‘𝑖)↑2) = ((𝑄‘2)↑2))
232	229, 88, 231	fsum1p 15806	. . . . . 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 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → 𝑁 ∈ ℤ)
234	233	zred 12701	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → 𝑁 ∈ ℝ)
235		lttr 11287	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 ∈ ℝ ∧ 3 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((2 < 3 ∧ 3 < 𝑁) → 2 < 𝑁))
236	152, 48, 235	mp3an12 1480	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℝ → ((2 < 3 ∧ 3 < 𝑁) → 2 < 𝑁))
237	223, 236	mpani 708	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℝ → (3 < 𝑁 → 2 < 𝑁))
238	49, 237	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘3) → (3 < 𝑁 → 2 < 𝑁))
239	238	imp 411	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → 2 < 𝑁)
240		ltle 11299	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (2 < 𝑁 → 2 ≤ 𝑁))
241	152, 240	mpan 702	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℝ → (2 < 𝑁 → 2 ≤ 𝑁))
242	234, 239, 241	sylc 66	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → 2 ≤ 𝑁)
243	242, 154	jctil 528	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → (1 ≤ 2 ∧ 2 ≤ 𝑁))
244		1z 12625	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℤ
245		elfz 13542	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (2 ∈ (1...𝑁) ↔ (1 ≤ 2 ∧ 2 ≤ 𝑁)))
246	225, 244, 233, 245	mp3an12i 1494	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → (2 ∈ (1...𝑁) ↔ (1 ≤ 2 ∧ 2 ≤ 𝑁)))
247	243, 246	mpbird 260	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → 2 ∈ (1...𝑁))
248	247	3adant3 1150	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 2 ∈ (1...𝑁))
249	91	ltp1d 12146	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ (2...(𝑁 − 1)) → 𝐼 < (𝐼 + 1))
250	153, 91, 117, 156, 249	lelttrd 11369	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ (2...(𝑁 − 1)) → 2 < (𝐼 + 1))
251	250	3ad2ant3 1153	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 2 < (𝐼 + 1))
252		ltne 11308	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ ∧ 2 < (𝐼 + 1)) → (𝐼 + 1) ≠ 2)
253	152, 251, 252	sylancr 598	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (𝐼 + 1) ≠ 2)
254	253	necomd 3013	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 2 ≠ (𝐼 + 1))
255	23	axlowdimlem12 29284	. . . . . . . . . 10 ⊢ ((2 ∈ (1...𝑁) ∧ 2 ≠ (𝐼 + 1)) → (𝑄‘2) = 0)
256	248, 254, 255	syl2anc 595	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (𝑄‘2) = 0)
257	256	sq0id 14232	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → ((𝑄‘2)↑2) = 0)
258	257	oveq1d 7427	. . . . . . 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	15	oveq1i 7422	. . . . . . . . 9 ⊢ (3...𝑁) = ((2 + 1)...𝑁)
260	259	sumeq1i 15750	. . . . . . . 8 ⊢ Σ𝑖 ∈ (3...𝑁)((𝑄‘𝑖)↑2) = Σ𝑖 ∈ ((2 + 1)...𝑁)((𝑄‘𝑖)↑2)
261	260	oveq2i 7423	. . . . . . 7 ⊢ (0 + Σ𝑖 ∈ (3...𝑁)((𝑄‘𝑖)↑2)) = (0 + Σ𝑖 ∈ ((2 + 1)...𝑁)((𝑄‘𝑖)↑2))
262	258, 261	eqtr4di 2816	. . . . . 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 14011	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (3...𝑁) ∈ Fin)
264		3nn 12321	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℕ
265	264, 175	eleqtri 2861	. . . . . . . . . . . . 13 ⊢ 3 ∈ (ℤ_≥‘1)
266		fzss1 13593	. . . . . . . . . . . . 13 ⊢ (3 ∈ (ℤ_≥‘1) → (3...𝑁) ⊆ (1...𝑁))
267	265, 266	ax-mp 5	. . . . . . . . . . . 12 ⊢ (3...𝑁) ⊆ (1...𝑁)
268	267	sseli 3934	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (3...𝑁) → 𝑖 ∈ (1...𝑁))
269	81, 268, 85	syl2an 607	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (3...𝑁)) → (𝑄‘𝑖) ∈ ℂ)
270	269	sqcld 14182	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (3...𝑁)) → ((𝑄‘𝑖)↑2) ∈ ℂ)
271	270	3adantl2 1186	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (3...𝑁)) → ((𝑄‘𝑖)↑2) ∈ ℂ)
272	263, 271	fsumcl 15786	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (3...𝑁)((𝑄‘𝑖)↑2) ∈ ℂ)
273	272	addlidd 11412	. . . . . 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 2803	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (3...𝑁)((𝑄‘𝑖)↑2) = Σ𝑖 ∈ (2...𝑁)((𝑄‘𝑖)↑2))
275		simpl 487	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → 𝑁 ∈ (ℤ_≥‘3))
276	6	axlowdimlem7 29279	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑃 ∈ (𝔼‘𝑁))
277	276	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) ∧ 𝑖 ∈ (3...𝑁)) → 𝑃 ∈ (𝔼‘𝑁))
278	268	adantl 486	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) ∧ 𝑖 ∈ (3...𝑁)) → 𝑖 ∈ (1...𝑁))
279		fveecn 29233	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (𝔼‘𝑁) ∧ 𝑖 ∈ (1...𝑁)) → (𝑃‘𝑖) ∈ ℂ)
280	277, 278, 279	syl2anc 595	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) ∧ 𝑖 ∈ (3...𝑁)) → (𝑃‘𝑖) ∈ ℂ)
281	280	sqcld 14182	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) ∧ 𝑖 ∈ (3...𝑁)) → ((𝑃‘𝑖)↑2) ∈ ℂ)
282		neg1sqe1 14234	. . . . . . . . . 10 ⊢ (-1↑2) = 1
283	9, 282	eqtrdi 2814	. . . . . . . . 9 ⊢ (𝑖 = 3 → ((𝑃‘𝑖)↑2) = 1)
284	275, 281, 283	fsum1p 15806	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → Σ𝑖 ∈ (3...𝑁)((𝑃‘𝑖)↑2) = (1 + Σ𝑖 ∈ ((3 + 1)...𝑁)((𝑃‘𝑖)↑2)))
285		1re 11209	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
286		zaddcl 12635	. . . . . . . . . . . . . . . . . . 19 ⊢ ((3 ∈ ℤ ∧ 1 ∈ ℤ) → (3 + 1) ∈ ℤ)
287	2, 244, 286	mp2an 704	. . . . . . . . . . . . . . . . . 18 ⊢ (3 + 1) ∈ ℤ
288	287	zrei 12598	. . . . . . . . . . . . . . . . 17 ⊢ (3 + 1) ∈ ℝ
289		1lt3 12417	. . . . . . . . . . . . . . . . . 18 ⊢ 1 < 3
290	48	ltp1i 12120	. . . . . . . . . . . . . . . . . 18 ⊢ 3 < (3 + 1)
291	285, 48, 288	lttri 11337	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 < 3 ∧ 3 < (3 + 1)) → 1 < (3 + 1))
292	289, 290, 291	mp2an 704	. . . . . . . . . . . . . . . . 17 ⊢ 1 < (3 + 1)
293	285, 288, 292	ltleii 11334	. . . . . . . . . . . . . . . 16 ⊢ 1 ≤ (3 + 1)
294		eluz 12877	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℤ ∧ (3 + 1) ∈ ℤ) → ((3 + 1) ∈ (ℤ_≥‘1) ↔ 1 ≤ (3 + 1)))
295	244, 287, 294	mp2an 704	. . . . . . . . . . . . . . . 16 ⊢ ((3 + 1) ∈ (ℤ_≥‘1) ↔ 1 ≤ (3 + 1))
296	293, 295	mpbir 234	. . . . . . . . . . . . . . 15 ⊢ (3 + 1) ∈ (ℤ_≥‘1)
297		fzss1 13593	. . . . . . . . . . . . . . 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 3934	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ ((3 + 1)...𝑁) → 𝑖 ∈ (1...𝑁))
300	48, 288	ltnlei 11332	. . . . . . . . . . . . . . . . . . 19 ⊢ (3 < (3 + 1) ↔ ¬ (3 + 1) ≤ 3)
301	290, 300	mpbi 233	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ (3 + 1) ≤ 3
302	301	intnanr 492	. . . . . . . . . . . . . . . . 17 ⊢ ¬ ((3 + 1) ≤ 3 ∧ 3 ≤ 𝑁)
303		elfz 13542	. . . . . . . . . . . . . . . . . 18 ⊢ ((3 ∈ ℤ ∧ (3 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (3 ∈ ((3 + 1)...𝑁) ↔ ((3 + 1) ≤ 3 ∧ 3 ≤ 𝑁)))
304	2, 287, 233, 303	mp3an12i 1494	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → (3 ∈ ((3 + 1)...𝑁) ↔ ((3 + 1) ≤ 3 ∧ 3 ≤ 𝑁)))
305	302, 304	mtbiri 330	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → ¬ 3 ∈ ((3 + 1)...𝑁))
306		eleq1 2851	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 3 → (𝑖 ∈ ((3 + 1)...𝑁) ↔ 3 ∈ ((3 + 1)...𝑁)))
307	306	notbid 321	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 3 → (¬ 𝑖 ∈ ((3 + 1)...𝑁) ↔ ¬ 3 ∈ ((3 + 1)...𝑁)))
308	305, 307	syl5ibrcom 250	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → (𝑖 = 3 → ¬ 𝑖 ∈ ((3 + 1)...𝑁)))
309	308	necon2ad 2973	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → (𝑖 ∈ ((3 + 1)...𝑁) → 𝑖 ≠ 3))
310	309	imp 411	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) ∧ 𝑖 ∈ ((3 + 1)...𝑁)) → 𝑖 ≠ 3)
311	6	axlowdimlem9 29281	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (1...𝑁) ∧ 𝑖 ≠ 3) → (𝑃‘𝑖) = 0)
312	299, 310, 311	syl2an2 698	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) ∧ 𝑖 ∈ ((3 + 1)...𝑁)) → (𝑃‘𝑖) = 0)
313	312	sq0id 14232	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) ∧ 𝑖 ∈ ((3 + 1)...𝑁)) → ((𝑃‘𝑖)↑2) = 0)
314	313	sumeq2dv 15755	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → Σ𝑖 ∈ ((3 + 1)...𝑁)((𝑃‘𝑖)↑2) = Σ𝑖 ∈ ((3 + 1)...𝑁)0)
315		fzfi 14010	. . . . . . . . . . . 12 ⊢ ((3 + 1)...𝑁) ∈ Fin
316	315	olci 879	. . . . . . . . . . 11 ⊢ (((3 + 1)...𝑁) ⊆ (ℤ_≥‘1) ∨ ((3 + 1)...𝑁) ∈ Fin)
317		sumz 15775	. . . . . . . . . . 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	eqtrdi 2814	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → Σ𝑖 ∈ ((3 + 1)...𝑁)((𝑃‘𝑖)↑2) = 0)
320	319	oveq2d 7428	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → (1 + Σ𝑖 ∈ ((3 + 1)...𝑁)((𝑃‘𝑖)↑2)) = (1 + 0))
321	284, 320	eqtrd 2798	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → Σ𝑖 ∈ (3...𝑁)((𝑃‘𝑖)↑2) = (1 + 0))
322	321, 213	eqtrdi 2814	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁) → Σ𝑖 ∈ (3...𝑁)((𝑃‘𝑖)↑2) = 1)
323	322	3adant3 1150	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 3 < 𝑁 ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (3...𝑁)((𝑃‘𝑖)↑2) = 1)
324	221, 274, 323	3eqtr4rd 2809	. . . 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 1398	. . 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 417	. 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 3041	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 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∖ cdif 3903 ∪ cun 3904 ∩ cin 3905 ⊆ wss 3906 ∅c0 4287 {csn 4590 ⟨cop 4596 class class class wbr 5110 × cxp 5661 ‘cfv 6538 (class class class)co 7412 Fincfn 8944 ℂcc 11099 ℝcr 11100 0cc0 11101 1c1 11102 + caddc 11104 < clt 11244 ≤ cle 11245 − cmin 11442 -cneg 11443 ℕcn 12234 2c2 12296 3c3 12297 ℤcz 12592 ℤ_≥cuz 12863 ...cfz 13536 ↑cexp 14099 Σcsu 15739 𝔼cee 29218

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-inf2 9611 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-er 8695 df-map 8827 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-sup 9403 df-oi 9473 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-n0 12506 df-z 12593 df-uz 12864 df-rp 13018 df-fz 13537 df-fzo 13685 df-seq 14040 df-exp 14100 df-hash 14369 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-clim 15541 df-sum 15740 df-ee 29221

This theorem is referenced by: axlowdimlem17 29289

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