axlowdimlem17 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > axlowdimlem17		Structured version Visualization version GIF version

Theorem axlowdimlem17 28685

Description: Lemma for axlowdim 28688. Establish a congruence result. (Contributed by Scott Fenton, 22-Apr-2013.) (Proof shortened by Mario Carneiro, 22-May-2014.)

**Hypotheses**
Ref	Expression
axlowdimlem16.1	⊢ 𝑃 = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))
axlowdimlem16.2	⊢ 𝑄 = ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}))
axlowdimlem17.3	⊢ 𝐴 = ({⟨1, 𝑋⟩, ⟨2, 𝑌⟩} ∪ ((3...𝑁) × {0}))
axlowdimlem17.4	⊢ 𝑋 ∈ ℝ
axlowdimlem17.5	⊢ 𝑌 ∈ ℝ

**Assertion**
Ref	Expression
axlowdimlem17	⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → ⟨𝑃, 𝐴⟩Cgr⟨𝑄, 𝐴⟩)

Proof of Theorem axlowdimlem17 Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uzuzle23 12870	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ (ℤ_≥‘2))
2	1	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...2)) → 𝑁 ∈ (ℤ_≥‘2))
3		fzss2 13538	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (1...2) ⊆ (1...𝑁))
4	2, 3	syl 17	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...2)) → (1...2) ⊆ (1...𝑁))
5		simpr 484	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...2)) → 𝑖 ∈ (1...2))
6	4, 5	sseldd 3975	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...2)) → 𝑖 ∈ (1...𝑁))
7		fznuz 13580	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...2) → ¬ 𝑖 ∈ (ℤ_≥‘(2 + 1)))
8	7	adantl 481	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...2)) → ¬ 𝑖 ∈ (ℤ_≥‘(2 + 1)))
9		3z 12592	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℤ
10		uzid 12834	. . . . . . . . . . . . . 14 ⊢ (3 ∈ ℤ → 3 ∈ (ℤ_≥‘3))
11	9, 10	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 3 ∈ (ℤ_≥‘3)
12		df-3 12273	. . . . . . . . . . . . . 14 ⊢ 3 = (2 + 1)
13	12	fveq2i 6884	. . . . . . . . . . . . 13 ⊢ (ℤ_≥‘3) = (ℤ_≥‘(2 + 1))
14	11, 13	eleqtri 2823	. . . . . . . . . . . 12 ⊢ 3 ∈ (ℤ_≥‘(2 + 1))
15		eleq1 2813	. . . . . . . . . . . 12 ⊢ (𝑖 = 3 → (𝑖 ∈ (ℤ_≥‘(2 + 1)) ↔ 3 ∈ (ℤ_≥‘(2 + 1))))
16	14, 15	mpbiri 258	. . . . . . . . . . 11 ⊢ (𝑖 = 3 → 𝑖 ∈ (ℤ_≥‘(2 + 1)))
17	16	necon3bi 2959	. . . . . . . . . 10 ⊢ (¬ 𝑖 ∈ (ℤ_≥‘(2 + 1)) → 𝑖 ≠ 3)
18	8, 17	syl 17	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...2)) → 𝑖 ≠ 3)
19		axlowdimlem16.1	. . . . . . . . . 10 ⊢ 𝑃 = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))
20	19	axlowdimlem9 28677	. . . . . . . . 9 ⊢ ((𝑖 ∈ (1...𝑁) ∧ 𝑖 ≠ 3) → (𝑃‘𝑖) = 0)
21	6, 18, 20	syl2anc 583	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...2)) → (𝑃‘𝑖) = 0)
22		elfzuz 13494	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ (2...(𝑁 − 1)) → 𝐼 ∈ (ℤ_≥‘2))
23	22	ad2antlr 724	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...2)) → 𝐼 ∈ (ℤ_≥‘2))
24		eluzp1p1 12847	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ (ℤ_≥‘2) → (𝐼 + 1) ∈ (ℤ_≥‘(2 + 1)))
25	23, 24	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...2)) → (𝐼 + 1) ∈ (ℤ_≥‘(2 + 1)))
26		uzss 12842	. . . . . . . . . . . 12 ⊢ ((𝐼 + 1) ∈ (ℤ_≥‘(2 + 1)) → (ℤ_≥‘(𝐼 + 1)) ⊆ (ℤ_≥‘(2 + 1)))
27	25, 26	syl 17	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...2)) → (ℤ_≥‘(𝐼 + 1)) ⊆ (ℤ_≥‘(2 + 1)))
28	27, 8	ssneldd 3977	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...2)) → ¬ 𝑖 ∈ (ℤ_≥‘(𝐼 + 1)))
29		eluzelz 12829	. . . . . . . . . . . . . 14 ⊢ ((𝐼 + 1) ∈ (ℤ_≥‘(2 + 1)) → (𝐼 + 1) ∈ ℤ)
30	25, 29	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...2)) → (𝐼 + 1) ∈ ℤ)
31		uzid 12834	. . . . . . . . . . . . 13 ⊢ ((𝐼 + 1) ∈ ℤ → (𝐼 + 1) ∈ (ℤ_≥‘(𝐼 + 1)))
32	30, 31	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...2)) → (𝐼 + 1) ∈ (ℤ_≥‘(𝐼 + 1)))
33		eleq1 2813	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝐼 + 1) → (𝑖 ∈ (ℤ_≥‘(𝐼 + 1)) ↔ (𝐼 + 1) ∈ (ℤ_≥‘(𝐼 + 1))))
34	32, 33	syl5ibrcom 246	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...2)) → (𝑖 = (𝐼 + 1) → 𝑖 ∈ (ℤ_≥‘(𝐼 + 1))))
35	34	necon3bd 2946	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...2)) → (¬ 𝑖 ∈ (ℤ_≥‘(𝐼 + 1)) → 𝑖 ≠ (𝐼 + 1)))
36	28, 35	mpd 15	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...2)) → 𝑖 ≠ (𝐼 + 1))
37		axlowdimlem16.2	. . . . . . . . . 10 ⊢ 𝑄 = ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}))
38	37	axlowdimlem12 28680	. . . . . . . . 9 ⊢ ((𝑖 ∈ (1...𝑁) ∧ 𝑖 ≠ (𝐼 + 1)) → (𝑄‘𝑖) = 0)
39	6, 36, 38	syl2anc 583	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...2)) → (𝑄‘𝑖) = 0)
40	21, 39	eqtr4d 2767	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...2)) → (𝑃‘𝑖) = (𝑄‘𝑖))
41	40	oveq1d 7416	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...2)) → ((𝑃‘𝑖) − (𝐴‘𝑖)) = ((𝑄‘𝑖) − (𝐴‘𝑖)))
42	41	oveq1d 7416	. . . . 5 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...2)) → (((𝑃‘𝑖) − (𝐴‘𝑖))↑2) = (((𝑄‘𝑖) − (𝐴‘𝑖))↑2))
43	42	sumeq2dv 15646	. . . 4 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (1...2)(((𝑃‘𝑖) − (𝐴‘𝑖))↑2) = Σ𝑖 ∈ (1...2)(((𝑄‘𝑖) − (𝐴‘𝑖))↑2))
44	19, 37	axlowdimlem16 28684	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (3...𝑁)((𝑃‘𝑖)↑2) = Σ𝑖 ∈ (3...𝑁)((𝑄‘𝑖)↑2))
45		axlowdimlem17.3	. . . . . . . . . . . . 13 ⊢ 𝐴 = ({⟨1, 𝑋⟩, ⟨2, 𝑌⟩} ∪ ((3...𝑁) × {0}))
46	45	fveq1i 6882	. . . . . . . . . . . 12 ⊢ (𝐴‘𝑖) = (({⟨1, 𝑋⟩, ⟨2, 𝑌⟩} ∪ ((3...𝑁) × {0}))‘𝑖)
47		axlowdimlem2 28670	. . . . . . . . . . . . 13 ⊢ ((1...2) ∩ (3...𝑁)) = ∅
48		axlowdimlem17.4	. . . . . . . . . . . . . . . 16 ⊢ 𝑋 ∈ ℝ
49		axlowdimlem17.5	. . . . . . . . . . . . . . . 16 ⊢ 𝑌 ∈ ℝ
50	48, 49	axlowdimlem4 28672	. . . . . . . . . . . . . . 15 ⊢ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩}:(1...2)⟶ℝ
51		ffn 6707	. . . . . . . . . . . . . . 15 ⊢ ({⟨1, 𝑋⟩, ⟨2, 𝑌⟩}:(1...2)⟶ℝ → {⟨1, 𝑋⟩, ⟨2, 𝑌⟩} Fn (1...2))
52	50, 51	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩} Fn (1...2)
53		axlowdimlem1 28669	. . . . . . . . . . . . . . 15 ⊢ ((3...𝑁) × {0}):(3...𝑁)⟶ℝ
54		ffn 6707	. . . . . . . . . . . . . . 15 ⊢ (((3...𝑁) × {0}):(3...𝑁)⟶ℝ → ((3...𝑁) × {0}) Fn (3...𝑁))
55	53, 54	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((3...𝑁) × {0}) Fn (3...𝑁)
56		fvun2 6973	. . . . . . . . . . . . . 14 ⊢ (({⟨1, 𝑋⟩, ⟨2, 𝑌⟩} Fn (1...2) ∧ ((3...𝑁) × {0}) Fn (3...𝑁) ∧ (((1...2) ∩ (3...𝑁)) = ∅ ∧ 𝑖 ∈ (3...𝑁))) → (({⟨1, 𝑋⟩, ⟨2, 𝑌⟩} ∪ ((3...𝑁) × {0}))‘𝑖) = (((3...𝑁) × {0})‘𝑖))
57	52, 55, 56	mp3an12 1447	. . . . . . . . . . . . 13 ⊢ ((((1...2) ∩ (3...𝑁)) = ∅ ∧ 𝑖 ∈ (3...𝑁)) → (({⟨1, 𝑋⟩, ⟨2, 𝑌⟩} ∪ ((3...𝑁) × {0}))‘𝑖) = (((3...𝑁) × {0})‘𝑖))
58	47, 57	mpan 687	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (3...𝑁) → (({⟨1, 𝑋⟩, ⟨2, 𝑌⟩} ∪ ((3...𝑁) × {0}))‘𝑖) = (((3...𝑁) × {0})‘𝑖))
59	46, 58	eqtrid 2776	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (3...𝑁) → (𝐴‘𝑖) = (((3...𝑁) × {0})‘𝑖))
60		c0ex 11205	. . . . . . . . . . . 12 ⊢ 0 ∈ V
61	60	fvconst2 7197	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (3...𝑁) → (((3...𝑁) × {0})‘𝑖) = 0)
62	59, 61	eqtrd 2764	. . . . . . . . . 10 ⊢ (𝑖 ∈ (3...𝑁) → (𝐴‘𝑖) = 0)
63	62	adantl 481	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (3...𝑁)) → (𝐴‘𝑖) = 0)
64	63	oveq2d 7417	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (3...𝑁)) → ((𝑃‘𝑖) − (𝐴‘𝑖)) = ((𝑃‘𝑖) − 0))
65	19	axlowdimlem7 28675	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑃 ∈ (𝔼‘𝑁))
66	65	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (3...𝑁)) → 𝑃 ∈ (𝔼‘𝑁))
67		3nn 12288	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℕ
68		nnuz 12862	. . . . . . . . . . . . . 14 ⊢ ℕ = (ℤ_≥‘1)
69	67, 68	eleqtri 2823	. . . . . . . . . . . . 13 ⊢ 3 ∈ (ℤ_≥‘1)
70		fzss1 13537	. . . . . . . . . . . . 13 ⊢ (3 ∈ (ℤ_≥‘1) → (3...𝑁) ⊆ (1...𝑁))
71	69, 70	ax-mp 5	. . . . . . . . . . . 12 ⊢ (3...𝑁) ⊆ (1...𝑁)
72	71	sseli 3970	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (3...𝑁) → 𝑖 ∈ (1...𝑁))
73	72	adantl 481	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (3...𝑁)) → 𝑖 ∈ (1...𝑁))
74		fveecn 28629	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (𝔼‘𝑁) ∧ 𝑖 ∈ (1...𝑁)) → (𝑃‘𝑖) ∈ ℂ)
75	66, 73, 74	syl2anc 583	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (3...𝑁)) → (𝑃‘𝑖) ∈ ℂ)
76	75	subid1d 11557	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (3...𝑁)) → ((𝑃‘𝑖) − 0) = (𝑃‘𝑖))
77	64, 76	eqtrd 2764	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (3...𝑁)) → ((𝑃‘𝑖) − (𝐴‘𝑖)) = (𝑃‘𝑖))
78	77	oveq1d 7416	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (3...𝑁)) → (((𝑃‘𝑖) − (𝐴‘𝑖))↑2) = ((𝑃‘𝑖)↑2))
79	78	sumeq2dv 15646	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (3...𝑁)(((𝑃‘𝑖) − (𝐴‘𝑖))↑2) = Σ𝑖 ∈ (3...𝑁)((𝑃‘𝑖)↑2))
80	63	oveq2d 7417	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (3...𝑁)) → ((𝑄‘𝑖) − (𝐴‘𝑖)) = ((𝑄‘𝑖) − 0))
81		eluzge3nn 12871	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℕ)
82		2eluzge1 12875	. . . . . . . . . . . . 13 ⊢ 2 ∈ (ℤ_≥‘1)
83		fzss1 13537	. . . . . . . . . . . . 13 ⊢ (2 ∈ (ℤ_≥‘1) → (2...(𝑁 − 1)) ⊆ (1...(𝑁 − 1)))
84	82, 83	ax-mp 5	. . . . . . . . . . . 12 ⊢ (2...(𝑁 − 1)) ⊆ (1...(𝑁 − 1))
85	84	sseli 3970	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (2...(𝑁 − 1)) → 𝐼 ∈ (1...(𝑁 − 1)))
86	37	axlowdimlem10 28678	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → 𝑄 ∈ (𝔼‘𝑁))
87	81, 85, 86	syl2an 595	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝑄 ∈ (𝔼‘𝑁))
88		fveecn 28629	. . . . . . . . . 10 ⊢ ((𝑄 ∈ (𝔼‘𝑁) ∧ 𝑖 ∈ (1...𝑁)) → (𝑄‘𝑖) ∈ ℂ)
89	87, 72, 88	syl2an 595	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (3...𝑁)) → (𝑄‘𝑖) ∈ ℂ)
90	89	subid1d 11557	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (3...𝑁)) → ((𝑄‘𝑖) − 0) = (𝑄‘𝑖))
91	80, 90	eqtrd 2764	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (3...𝑁)) → ((𝑄‘𝑖) − (𝐴‘𝑖)) = (𝑄‘𝑖))
92	91	oveq1d 7416	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (3...𝑁)) → (((𝑄‘𝑖) − (𝐴‘𝑖))↑2) = ((𝑄‘𝑖)↑2))
93	92	sumeq2dv 15646	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (3...𝑁)(((𝑄‘𝑖) − (𝐴‘𝑖))↑2) = Σ𝑖 ∈ (3...𝑁)((𝑄‘𝑖)↑2))
94	44, 79, 93	3eqtr4d 2774	. . . 4 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (3...𝑁)(((𝑃‘𝑖) − (𝐴‘𝑖))↑2) = Σ𝑖 ∈ (3...𝑁)(((𝑄‘𝑖) − (𝐴‘𝑖))↑2))
95	43, 94	oveq12d 7419	. . 3 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (Σ𝑖 ∈ (1...2)(((𝑃‘𝑖) − (𝐴‘𝑖))↑2) + Σ𝑖 ∈ (3...𝑁)(((𝑃‘𝑖) − (𝐴‘𝑖))↑2)) = (Σ𝑖 ∈ (1...2)(((𝑄‘𝑖) − (𝐴‘𝑖))↑2) + Σ𝑖 ∈ (3...𝑁)(((𝑄‘𝑖) − (𝐴‘𝑖))↑2)))
96	47	a1i 11	. . . 4 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → ((1...2) ∩ (3...𝑁)) = ∅)
97		eluzelre 12830	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℝ)
98		eluzle 12832	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
99		2re 12283	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
100		3re 12289	. . . . . . . . . . . 12 ⊢ 3 ∈ ℝ
101		2lt3 12381	. . . . . . . . . . . 12 ⊢ 2 < 3
102	99, 100, 101	ltleii 11334	. . . . . . . . . . 11 ⊢ 2 ≤ 3
103		letr 11305	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ ∧ 3 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((2 ≤ 3 ∧ 3 ≤ 𝑁) → 2 ≤ 𝑁))
104	99, 100, 103	mp3an12 1447	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℝ → ((2 ≤ 3 ∧ 3 ≤ 𝑁) → 2 ≤ 𝑁))
105	102, 104	mpani 693	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ → (3 ≤ 𝑁 → 2 ≤ 𝑁))
106	97, 98, 105	sylc 65	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 2 ≤ 𝑁)
107		1le2 12418	. . . . . . . . 9 ⊢ 1 ≤ 2
108	106, 107	jctil 519	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → (1 ≤ 2 ∧ 2 ≤ 𝑁))
109	108	adantr 480	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (1 ≤ 2 ∧ 2 ≤ 𝑁))
110		eluzelz 12829	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
111	110	adantr 480	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝑁 ∈ ℤ)
112		2z 12591	. . . . . . . . 9 ⊢ 2 ∈ ℤ
113		1z 12589	. . . . . . . . 9 ⊢ 1 ∈ ℤ
114		elfz 13487	. . . . . . . . 9 ⊢ ((2 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (2 ∈ (1...𝑁) ↔ (1 ≤ 2 ∧ 2 ≤ 𝑁)))
115	112, 113, 114	mp3an12 1447	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (2 ∈ (1...𝑁) ↔ (1 ≤ 2 ∧ 2 ≤ 𝑁)))
116	111, 115	syl 17	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (2 ∈ (1...𝑁) ↔ (1 ≤ 2 ∧ 2 ≤ 𝑁)))
117	109, 116	mpbird 257	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 2 ∈ (1...𝑁))
118		fzsplit 13524	. . . . . 6 ⊢ (2 ∈ (1...𝑁) → (1...𝑁) = ((1...2) ∪ ((2 + 1)...𝑁)))
119	117, 118	syl 17	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (1...𝑁) = ((1...2) ∪ ((2 + 1)...𝑁)))
120	12	oveq1i 7411	. . . . . 6 ⊢ (3...𝑁) = ((2 + 1)...𝑁)
121	120	uneq2i 4152	. . . . 5 ⊢ ((1...2) ∪ (3...𝑁)) = ((1...2) ∪ ((2 + 1)...𝑁))
122	119, 121	eqtr4di 2782	. . . 4 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (1...𝑁) = ((1...2) ∪ (3...𝑁)))
123		fzfid 13935	. . . 4 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (1...𝑁) ∈ Fin)
124	65	ad2antrr 723	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...𝑁)) → 𝑃 ∈ (𝔼‘𝑁))
125	124, 74	sylancom 587	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...𝑁)) → (𝑃‘𝑖) ∈ ℂ)
126	48, 49	axlowdimlem5 28673	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ({⟨1, 𝑋⟩, ⟨2, 𝑌⟩} ∪ ((3...𝑁) × {0})) ∈ (𝔼‘𝑁))
127	45, 126	eqeltrid 2829	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝐴 ∈ (𝔼‘𝑁))
128	1, 127	syl 17	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝐴 ∈ (𝔼‘𝑁))
129	128	ad2antrr 723	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...𝑁)) → 𝐴 ∈ (𝔼‘𝑁))
130		fveecn 28629	. . . . . . 7 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝑖 ∈ (1...𝑁)) → (𝐴‘𝑖) ∈ ℂ)
131	129, 130	sylancom 587	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...𝑁)) → (𝐴‘𝑖) ∈ ℂ)
132	125, 131	subcld 11568	. . . . 5 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑃‘𝑖) − (𝐴‘𝑖)) ∈ ℂ)
133	132	sqcld 14106	. . . 4 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑃‘𝑖) − (𝐴‘𝑖))↑2) ∈ ℂ)
134	96, 122, 123, 133	fsumsplit 15684	. . 3 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (1...𝑁)(((𝑃‘𝑖) − (𝐴‘𝑖))↑2) = (Σ𝑖 ∈ (1...2)(((𝑃‘𝑖) − (𝐴‘𝑖))↑2) + Σ𝑖 ∈ (3...𝑁)(((𝑃‘𝑖) − (𝐴‘𝑖))↑2)))
135	87, 88	sylan 579	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...𝑁)) → (𝑄‘𝑖) ∈ ℂ)
136	135, 131	subcld 11568	. . . . 5 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑄‘𝑖) − (𝐴‘𝑖)) ∈ ℂ)
137	136	sqcld 14106	. . . 4 ⊢ (((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑄‘𝑖) − (𝐴‘𝑖))↑2) ∈ ℂ)
138	96, 122, 123, 137	fsumsplit 15684	. . 3 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (1...𝑁)(((𝑄‘𝑖) − (𝐴‘𝑖))↑2) = (Σ𝑖 ∈ (1...2)(((𝑄‘𝑖) − (𝐴‘𝑖))↑2) + Σ𝑖 ∈ (3...𝑁)(((𝑄‘𝑖) − (𝐴‘𝑖))↑2)))
139	95, 134, 138	3eqtr4d 2774	. 2 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (1...𝑁)(((𝑃‘𝑖) − (𝐴‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝑄‘𝑖) − (𝐴‘𝑖))↑2))
140	65	adantr 480	. . 3 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝑃 ∈ (𝔼‘𝑁))
141	128	adantr 480	. . 3 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → 𝐴 ∈ (𝔼‘𝑁))
142		brcgr 28627	. . 3 ⊢ (((𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) ∧ (𝑄 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁))) → (⟨𝑃, 𝐴⟩Cgr⟨𝑄, 𝐴⟩ ↔ Σ𝑖 ∈ (1...𝑁)(((𝑃‘𝑖) − (𝐴‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝑄‘𝑖) − (𝐴‘𝑖))↑2)))
143	140, 141, 87, 141, 142	syl22anc 836	. 2 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → (⟨𝑃, 𝐴⟩Cgr⟨𝑄, 𝐴⟩ ↔ Σ𝑖 ∈ (1...𝑁)(((𝑃‘𝑖) − (𝐴‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝑄‘𝑖) − (𝐴‘𝑖))↑2)))
144	139, 143	mpbird 257	1 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → ⟨𝑃, 𝐴⟩Cgr⟨𝑄, 𝐴⟩)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∖ cdif 3937 ∪ cun 3938 ∩ cin 3939 ⊆ wss 3940 ∅c0 4314 {csn 4620 {cpr 4622 ⟨cop 4626 class class class wbr 5138 × cxp 5664 Fn wfn 6528 ⟶wf 6529 ‘cfv 6533 (class class class)co 7401 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 ≤ cle 11246 − cmin 11441 -cneg 11442 ℕcn 12209 2c2 12264 3c3 12265 ℤcz 12555 ℤ_≥cuz 12819 ...cfz 13481 ↑cexp 14024 Σcsu 15629 𝔼cee 28615 Cgrccgr 28617

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-z 12556 df-uz 12820 df-rp 12972 df-fz 13482 df-fzo 13625 df-seq 13964 df-exp 14025 df-hash 14288 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-clim 15429 df-sum 15630 df-ee 28618 df-cgr 28620

This theorem is referenced by: axlowdim 28688

W3C validator