axlowdimlem6 - Metamath Proof Explorer

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Theorem axlowdimlem6 29020

Description: Lemma for axlowdim 29034. Show that three points are non-colinear. (Contributed by Scott Fenton, 29-Jun-2013.)

**Hypotheses**
Ref	Expression
axlowdimlem6.1	⊢ 𝐴 = ({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))
axlowdimlem6.2	⊢ 𝐵 = ({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))
axlowdimlem6.3	⊢ 𝐶 = ({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))

**Assertion**
Ref	Expression
axlowdimlem6	⊢ (𝑁 ∈ (ℤ_≥‘2) → ¬ (𝐴 Btwn ⟨𝐵, 𝐶⟩ ∨ 𝐵 Btwn ⟨𝐶, 𝐴⟩ ∨ 𝐶 Btwn ⟨𝐴, 𝐵⟩))

Proof of Theorem axlowdimlem6 Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1zzd 12522	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 1 ∈ ℤ)
2		eluzelz 12761	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℤ)
3		2nn 12218	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ
4		uznnssnn 12808	. . . . . . . . . . 11 ⊢ (2 ∈ ℕ → (ℤ_≥‘2) ⊆ ℕ)
5	3, 4	ax-mp 5	. . . . . . . . . 10 ⊢ (ℤ_≥‘2) ⊆ ℕ
6		nnuz 12790	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
7	5, 6	sseqtri 3982	. . . . . . . . 9 ⊢ (ℤ_≥‘2) ⊆ (ℤ_≥‘1)
8	7	sseli 3929	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ (ℤ_≥‘1))
9		eluzle 12764	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 1 ≤ 𝑁)
10	8, 9	syl 17	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 1 ≤ 𝑁)
11		1re 11132	. . . . . . . 8 ⊢ 1 ∈ ℝ
12	11	leidi 11671	. . . . . . 7 ⊢ 1 ≤ 1
13	10, 12	jctil 519	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (1 ≤ 1 ∧ 1 ≤ 𝑁))
14		elfz4 13433	. . . . . 6 ⊢ (((1 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 1 ∈ ℤ) ∧ (1 ≤ 1 ∧ 1 ≤ 𝑁)) → 1 ∈ (1...𝑁))
15	1, 2, 1, 13, 14	syl31anc 1375	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 1 ∈ (1...𝑁))
16		eluzel2 12756	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ∈ ℤ)
17		eluzle 12764	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ≤ 𝑁)
18		1le2 12349	. . . . . . 7 ⊢ 1 ≤ 2
19	17, 18	jctil 519	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (1 ≤ 2 ∧ 2 ≤ 𝑁))
20		elfz4 13433	. . . . . 6 ⊢ (((1 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 2 ∈ ℤ) ∧ (1 ≤ 2 ∧ 2 ≤ 𝑁)) → 2 ∈ (1...𝑁))
21	1, 2, 16, 19, 20	syl31anc 1375	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ∈ (1...𝑁))
22		ax-1ne0 11095	. . . . . . 7 ⊢ 1 ≠ 0
23		1t1e1 12302	. . . . . . . 8 ⊢ (1 · 1) = 1
24		0cn 11124	. . . . . . . . 9 ⊢ 0 ∈ ℂ
25	24	mul01i 11323	. . . . . . . 8 ⊢ (0 · 0) = 0
26	23, 25	neeq12i 2998	. . . . . . 7 ⊢ ((1 · 1) ≠ (0 · 0) ↔ 1 ≠ 0)
27	22, 26	mpbir 231	. . . . . 6 ⊢ (1 · 1) ≠ (0 · 0)
28		fveq2 6834	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) = (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘1))
29		0re 11134	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
30	11, 29	axlowdimlem4 29018	. . . . . . . . . . . . . . 15 ⊢ {⟨1, 1⟩, ⟨2, 0⟩}:(1...2)⟶ℝ
31		ffn 6662	. . . . . . . . . . . . . . 15 ⊢ ({⟨1, 1⟩, ⟨2, 0⟩}:(1...2)⟶ℝ → {⟨1, 1⟩, ⟨2, 0⟩} Fn (1...2))
32	30, 31	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ {⟨1, 1⟩, ⟨2, 0⟩} Fn (1...2)
33		axlowdimlem1 29015	. . . . . . . . . . . . . . 15 ⊢ ((3...𝑁) × {0}):(3...𝑁)⟶ℝ
34		ffn 6662	. . . . . . . . . . . . . . 15 ⊢ (((3...𝑁) × {0}):(3...𝑁)⟶ℝ → ((3...𝑁) × {0}) Fn (3...𝑁))
35	33, 34	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((3...𝑁) × {0}) Fn (3...𝑁)
36		axlowdimlem2 29016	. . . . . . . . . . . . . . 15 ⊢ ((1...2) ∩ (3...𝑁)) = ∅
37		1z 12521	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℤ
38		2z 12523	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℤ
39	37, 38, 37	3pm3.2i 1340	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ ℤ ∧ 2 ∈ ℤ ∧ 1 ∈ ℤ)
40	12, 18	pm3.2i 470	. . . . . . . . . . . . . . . 16 ⊢ (1 ≤ 1 ∧ 1 ≤ 2)
41		elfz4 13433	. . . . . . . . . . . . . . . 16 ⊢ (((1 ∈ ℤ ∧ 2 ∈ ℤ ∧ 1 ∈ ℤ) ∧ (1 ≤ 1 ∧ 1 ≤ 2)) → 1 ∈ (1...2))
42	39, 40, 41	mp2an 692	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ (1...2)
43	36, 42	pm3.2i 470	. . . . . . . . . . . . . 14 ⊢ (((1...2) ∩ (3...𝑁)) = ∅ ∧ 1 ∈ (1...2))
44		fvun1 6925	. . . . . . . . . . . . . 14 ⊢ (({⟨1, 1⟩, ⟨2, 0⟩} Fn (1...2) ∧ ((3...𝑁) × {0}) Fn (3...𝑁) ∧ (((1...2) ∩ (3...𝑁)) = ∅ ∧ 1 ∈ (1...2))) → (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘1) = ({⟨1, 1⟩, ⟨2, 0⟩}‘1))
45	32, 35, 43, 44	mp3an 1463	. . . . . . . . . . . . 13 ⊢ (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘1) = ({⟨1, 1⟩, ⟨2, 0⟩}‘1)
46		1ne2 12348	. . . . . . . . . . . . . 14 ⊢ 1 ≠ 2
47		1ex 11128	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ V
48	47, 47	fvpr1 7138	. . . . . . . . . . . . . 14 ⊢ (1 ≠ 2 → ({⟨1, 1⟩, ⟨2, 0⟩}‘1) = 1)
49	46, 48	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ({⟨1, 1⟩, ⟨2, 0⟩}‘1) = 1
50	45, 49	eqtri 2759	. . . . . . . . . . . 12 ⊢ (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘1) = 1
51	28, 50	eqtrdi 2787	. . . . . . . . . . 11 ⊢ (𝑖 = 1 → (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) = 1)
52		fveq2 6834	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) = (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘1))
53	29, 29	axlowdimlem4 29018	. . . . . . . . . . . . . . 15 ⊢ {⟨1, 0⟩, ⟨2, 0⟩}:(1...2)⟶ℝ
54		ffn 6662	. . . . . . . . . . . . . . 15 ⊢ ({⟨1, 0⟩, ⟨2, 0⟩}:(1...2)⟶ℝ → {⟨1, 0⟩, ⟨2, 0⟩} Fn (1...2))
55	53, 54	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ {⟨1, 0⟩, ⟨2, 0⟩} Fn (1...2)
56		fvun1 6925	. . . . . . . . . . . . . 14 ⊢ (({⟨1, 0⟩, ⟨2, 0⟩} Fn (1...2) ∧ ((3...𝑁) × {0}) Fn (3...𝑁) ∧ (((1...2) ∩ (3...𝑁)) = ∅ ∧ 1 ∈ (1...2))) → (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘1) = ({⟨1, 0⟩, ⟨2, 0⟩}‘1))
57	55, 35, 43, 56	mp3an 1463	. . . . . . . . . . . . 13 ⊢ (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘1) = ({⟨1, 0⟩, ⟨2, 0⟩}‘1)
58	29	elexi 3463	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
59	47, 58	fvpr1 7138	. . . . . . . . . . . . . 14 ⊢ (1 ≠ 2 → ({⟨1, 0⟩, ⟨2, 0⟩}‘1) = 0)
60	46, 59	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ({⟨1, 0⟩, ⟨2, 0⟩}‘1) = 0
61	57, 60	eqtri 2759	. . . . . . . . . . . 12 ⊢ (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘1) = 0
62	52, 61	eqtrdi 2787	. . . . . . . . . . 11 ⊢ (𝑖 = 1 → (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) = 0)
63	51, 62	oveq12d 7376	. . . . . . . . . 10 ⊢ (𝑖 = 1 → ((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) = (1 − 0))
64		1m0e1 12261	. . . . . . . . . 10 ⊢ (1 − 0) = 1
65	63, 64	eqtrdi 2787	. . . . . . . . 9 ⊢ (𝑖 = 1 → ((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) = 1)
66	65	oveq1d 7373	. . . . . . . 8 ⊢ (𝑖 = 1 → (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))) = (1 · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))))
67		fveq2 6834	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) = (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘1))
68	29, 11	axlowdimlem4 29018	. . . . . . . . . . . . . . 15 ⊢ {⟨1, 0⟩, ⟨2, 1⟩}:(1...2)⟶ℝ
69		ffn 6662	. . . . . . . . . . . . . . 15 ⊢ ({⟨1, 0⟩, ⟨2, 1⟩}:(1...2)⟶ℝ → {⟨1, 0⟩, ⟨2, 1⟩} Fn (1...2))
70	68, 69	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ {⟨1, 0⟩, ⟨2, 1⟩} Fn (1...2)
71		fvun1 6925	. . . . . . . . . . . . . 14 ⊢ (({⟨1, 0⟩, ⟨2, 1⟩} Fn (1...2) ∧ ((3...𝑁) × {0}) Fn (3...𝑁) ∧ (((1...2) ∩ (3...𝑁)) = ∅ ∧ 1 ∈ (1...2))) → (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘1) = ({⟨1, 0⟩, ⟨2, 1⟩}‘1))
72	70, 35, 43, 71	mp3an 1463	. . . . . . . . . . . . 13 ⊢ (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘1) = ({⟨1, 0⟩, ⟨2, 1⟩}‘1)
73	47, 58	fvpr1 7138	. . . . . . . . . . . . . 14 ⊢ (1 ≠ 2 → ({⟨1, 0⟩, ⟨2, 1⟩}‘1) = 0)
74	46, 73	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ({⟨1, 0⟩, ⟨2, 1⟩}‘1) = 0
75	72, 74	eqtri 2759	. . . . . . . . . . . 12 ⊢ (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘1) = 0
76	67, 75	eqtrdi 2787	. . . . . . . . . . 11 ⊢ (𝑖 = 1 → (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) = 0)
77	76, 62	oveq12d 7376	. . . . . . . . . 10 ⊢ (𝑖 = 1 → ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) = (0 − 0))
78		0m0e0 12260	. . . . . . . . . 10 ⊢ (0 − 0) = 0
79	77, 78	eqtrdi 2787	. . . . . . . . 9 ⊢ (𝑖 = 1 → ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) = 0)
80	79	oveq2d 7374	. . . . . . . 8 ⊢ (𝑖 = 1 → (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖))) = (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) · 0))
81	66, 80	neeq12d 2993	. . . . . . 7 ⊢ (𝑖 = 1 → ((((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))) ≠ (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖))) ↔ (1 · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))) ≠ (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) · 0)))
82		fveq2 6834	. . . . . . . . . . . 12 ⊢ (𝑗 = 2 → (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) = (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘2))
83	37, 38, 38	3pm3.2i 1340	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ ℤ ∧ 2 ∈ ℤ ∧ 2 ∈ ℤ)
84		2re 12219	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℝ
85	84	leidi 11671	. . . . . . . . . . . . . . . . 17 ⊢ 2 ≤ 2
86	18, 85	pm3.2i 470	. . . . . . . . . . . . . . . 16 ⊢ (1 ≤ 2 ∧ 2 ≤ 2)
87		elfz4 13433	. . . . . . . . . . . . . . . 16 ⊢ (((1 ∈ ℤ ∧ 2 ∈ ℤ ∧ 2 ∈ ℤ) ∧ (1 ≤ 2 ∧ 2 ≤ 2)) → 2 ∈ (1...2))
88	83, 86, 87	mp2an 692	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ (1...2)
89	36, 88	pm3.2i 470	. . . . . . . . . . . . . 14 ⊢ (((1...2) ∩ (3...𝑁)) = ∅ ∧ 2 ∈ (1...2))
90		fvun1 6925	. . . . . . . . . . . . . 14 ⊢ (({⟨1, 0⟩, ⟨2, 1⟩} Fn (1...2) ∧ ((3...𝑁) × {0}) Fn (3...𝑁) ∧ (((1...2) ∩ (3...𝑁)) = ∅ ∧ 2 ∈ (1...2))) → (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘2) = ({⟨1, 0⟩, ⟨2, 1⟩}‘2))
91	70, 35, 89, 90	mp3an 1463	. . . . . . . . . . . . 13 ⊢ (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘2) = ({⟨1, 0⟩, ⟨2, 1⟩}‘2)
92	38	elexi 3463	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ V
93	92, 47	fvpr2 7139	. . . . . . . . . . . . . 14 ⊢ (1 ≠ 2 → ({⟨1, 0⟩, ⟨2, 1⟩}‘2) = 1)
94	46, 93	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ({⟨1, 0⟩, ⟨2, 1⟩}‘2) = 1
95	91, 94	eqtri 2759	. . . . . . . . . . . 12 ⊢ (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘2) = 1
96	82, 95	eqtrdi 2787	. . . . . . . . . . 11 ⊢ (𝑗 = 2 → (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) = 1)
97		fveq2 6834	. . . . . . . . . . . 12 ⊢ (𝑗 = 2 → (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) = (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘2))
98		fvun1 6925	. . . . . . . . . . . . . 14 ⊢ (({⟨1, 0⟩, ⟨2, 0⟩} Fn (1...2) ∧ ((3...𝑁) × {0}) Fn (3...𝑁) ∧ (((1...2) ∩ (3...𝑁)) = ∅ ∧ 2 ∈ (1...2))) → (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘2) = ({⟨1, 0⟩, ⟨2, 0⟩}‘2))
99	55, 35, 89, 98	mp3an 1463	. . . . . . . . . . . . 13 ⊢ (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘2) = ({⟨1, 0⟩, ⟨2, 0⟩}‘2)
100	92, 58	fvpr2 7139	. . . . . . . . . . . . . 14 ⊢ (1 ≠ 2 → ({⟨1, 0⟩, ⟨2, 0⟩}‘2) = 0)
101	46, 100	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ({⟨1, 0⟩, ⟨2, 0⟩}‘2) = 0
102	99, 101	eqtri 2759	. . . . . . . . . . . 12 ⊢ (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘2) = 0
103	97, 102	eqtrdi 2787	. . . . . . . . . . 11 ⊢ (𝑗 = 2 → (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) = 0)
104	96, 103	oveq12d 7376	. . . . . . . . . 10 ⊢ (𝑗 = 2 → ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) = (1 − 0))
105	104, 64	eqtrdi 2787	. . . . . . . . 9 ⊢ (𝑗 = 2 → ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) = 1)
106	105	oveq2d 7374	. . . . . . . 8 ⊢ (𝑗 = 2 → (1 · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))) = (1 · 1))
107		fveq2 6834	. . . . . . . . . . . 12 ⊢ (𝑗 = 2 → (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) = (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘2))
108		fvun1 6925	. . . . . . . . . . . . . 14 ⊢ (({⟨1, 1⟩, ⟨2, 0⟩} Fn (1...2) ∧ ((3...𝑁) × {0}) Fn (3...𝑁) ∧ (((1...2) ∩ (3...𝑁)) = ∅ ∧ 2 ∈ (1...2))) → (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘2) = ({⟨1, 1⟩, ⟨2, 0⟩}‘2))
109	32, 35, 89, 108	mp3an 1463	. . . . . . . . . . . . 13 ⊢ (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘2) = ({⟨1, 1⟩, ⟨2, 0⟩}‘2)
110	92, 58	fvpr2 7139	. . . . . . . . . . . . . 14 ⊢ (1 ≠ 2 → ({⟨1, 1⟩, ⟨2, 0⟩}‘2) = 0)
111	46, 110	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ({⟨1, 1⟩, ⟨2, 0⟩}‘2) = 0
112	109, 111	eqtri 2759	. . . . . . . . . . . 12 ⊢ (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘2) = 0
113	107, 112	eqtrdi 2787	. . . . . . . . . . 11 ⊢ (𝑗 = 2 → (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) = 0)
114	113, 103	oveq12d 7376	. . . . . . . . . 10 ⊢ (𝑗 = 2 → ((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) = (0 − 0))
115	114, 78	eqtrdi 2787	. . . . . . . . 9 ⊢ (𝑗 = 2 → ((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) = 0)
116	115	oveq1d 7373	. . . . . . . 8 ⊢ (𝑗 = 2 → (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) · 0) = (0 · 0))
117	106, 116	neeq12d 2993	. . . . . . 7 ⊢ (𝑗 = 2 → ((1 · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))) ≠ (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) · 0) ↔ (1 · 1) ≠ (0 · 0)))
118	81, 117	rspc2ev 3589	. . . . . 6 ⊢ ((1 ∈ (1...𝑁) ∧ 2 ∈ (1...𝑁) ∧ (1 · 1) ≠ (0 · 0)) → ∃𝑖 ∈ (1...𝑁)∃𝑗 ∈ (1...𝑁)(((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))) ≠ (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖))))
119	27, 118	mp3an3 1452	. . . . 5 ⊢ ((1 ∈ (1...𝑁) ∧ 2 ∈ (1...𝑁)) → ∃𝑖 ∈ (1...𝑁)∃𝑗 ∈ (1...𝑁)(((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))) ≠ (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖))))
120	15, 21, 119	syl2anc 584	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ∃𝑖 ∈ (1...𝑁)∃𝑗 ∈ (1...𝑁)(((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))) ≠ (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖))))
121		df-ne 2933	. . . . . . . 8 ⊢ ((((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))) ≠ (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖))) ↔ ¬ (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))) = (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖))))
122	121	rexbii 3083	. . . . . . 7 ⊢ (∃𝑗 ∈ (1...𝑁)(((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))) ≠ (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖))) ↔ ∃𝑗 ∈ (1...𝑁) ¬ (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))) = (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖))))
123		rexnal 3088	. . . . . . 7 ⊢ (∃𝑗 ∈ (1...𝑁) ¬ (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))) = (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖))) ↔ ¬ ∀𝑗 ∈ (1...𝑁)(((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))) = (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖))))
124	122, 123	bitri 275	. . . . . 6 ⊢ (∃𝑗 ∈ (1...𝑁)(((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))) ≠ (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖))) ↔ ¬ ∀𝑗 ∈ (1...𝑁)(((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))) = (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖))))
125	124	rexbii 3083	. . . . 5 ⊢ (∃𝑖 ∈ (1...𝑁)∃𝑗 ∈ (1...𝑁)(((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))) ≠ (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖))) ↔ ∃𝑖 ∈ (1...𝑁) ¬ ∀𝑗 ∈ (1...𝑁)(((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))) = (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖))))
126		rexnal 3088	. . . . 5 ⊢ (∃𝑖 ∈ (1...𝑁) ¬ ∀𝑗 ∈ (1...𝑁)(((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))) = (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖))) ↔ ¬ ∀𝑖 ∈ (1...𝑁)∀𝑗 ∈ (1...𝑁)(((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))) = (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖))))
127	125, 126	bitri 275	. . . 4 ⊢ (∃𝑖 ∈ (1...𝑁)∃𝑗 ∈ (1...𝑁)(((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))) ≠ (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖))) ↔ ¬ ∀𝑖 ∈ (1...𝑁)∀𝑗 ∈ (1...𝑁)(((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))) = (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖))))
128	120, 127	sylib 218	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ¬ ∀𝑖 ∈ (1...𝑁)∀𝑗 ∈ (1...𝑁)(((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))) = (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖))))
129	29, 29	axlowdimlem5 29019	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})) ∈ (𝔼‘𝑁))
130	11, 29	axlowdimlem5 29019	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})) ∈ (𝔼‘𝑁))
131	29, 11	axlowdimlem5 29019	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0})) ∈ (𝔼‘𝑁))
132		colinearalg 28983	. . . 4 ⊢ ((({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})) ∈ (𝔼‘𝑁) ∧ ({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})) ∈ (𝔼‘𝑁) ∧ ({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0})) ∈ (𝔼‘𝑁)) → ((({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})) Btwn ⟨({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})), ({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))⟩ ∨ ({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})) Btwn ⟨({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0})), ({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))⟩ ∨ ({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0})) Btwn ⟨({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})), ({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))⟩) ↔ ∀𝑖 ∈ (1...𝑁)∀𝑗 ∈ (1...𝑁)(((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))) = (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)))))
133	129, 130, 131, 132	syl3anc 1373	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ((({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})) Btwn ⟨({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})), ({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))⟩ ∨ ({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})) Btwn ⟨({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0})), ({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))⟩ ∨ ({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0})) Btwn ⟨({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})), ({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))⟩) ↔ ∀𝑖 ∈ (1...𝑁)∀𝑗 ∈ (1...𝑁)(((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))) = (((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)))))
134	128, 133	mtbird 325	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ¬ (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})) Btwn ⟨({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})), ({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))⟩ ∨ ({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})) Btwn ⟨({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0})), ({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))⟩ ∨ ({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0})) Btwn ⟨({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})), ({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))⟩))
135		axlowdimlem6.1	. . . 4 ⊢ 𝐴 = ({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))
136		axlowdimlem6.2	. . . . 5 ⊢ 𝐵 = ({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))
137		axlowdimlem6.3	. . . . 5 ⊢ 𝐶 = ({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))
138	136, 137	opeq12i 4834	. . . 4 ⊢ ⟨𝐵, 𝐶⟩ = ⟨({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})), ({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))⟩
139	135, 138	breq12i 5107	. . 3 ⊢ (𝐴 Btwn ⟨𝐵, 𝐶⟩ ↔ ({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})) Btwn ⟨({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})), ({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))⟩)
140	137, 135	opeq12i 4834	. . . 4 ⊢ ⟨𝐶, 𝐴⟩ = ⟨({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0})), ({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))⟩
141	136, 140	breq12i 5107	. . 3 ⊢ (𝐵 Btwn ⟨𝐶, 𝐴⟩ ↔ ({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})) Btwn ⟨({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0})), ({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))⟩)
142	135, 136	opeq12i 4834	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ = ⟨({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})), ({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))⟩
143	137, 142	breq12i 5107	. . 3 ⊢ (𝐶 Btwn ⟨𝐴, 𝐵⟩ ↔ ({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0})) Btwn ⟨({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})), ({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))⟩)
144	139, 141, 143	3orbi123i 1156	. 2 ⊢ ((𝐴 Btwn ⟨𝐵, 𝐶⟩ ∨ 𝐵 Btwn ⟨𝐶, 𝐴⟩ ∨ 𝐶 Btwn ⟨𝐴, 𝐵⟩) ↔ (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})) Btwn ⟨({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})), ({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))⟩ ∨ ({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})) Btwn ⟨({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0})), ({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))⟩ ∨ ({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0})) Btwn ⟨({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})), ({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))⟩))
145	134, 144	sylnibr 329	1 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ¬ (𝐴 Btwn ⟨𝐵, 𝐶⟩ ∨ 𝐵 Btwn ⟨𝐶, 𝐴⟩ ∨ 𝐶 Btwn ⟨𝐴, 𝐵⟩))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ w3o 1085 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 ∀wral 3051 ∃wrex 3060 ∪ cun 3899 ∩ cin 3900 ⊆ wss 3901 ∅c0 4285 {csn 4580 {cpr 4582 ⟨cop 4586 class class class wbr 5098 × cxp 5622 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 ℝcr 11025 0cc0 11026 1c1 11027 · cmul 11031 ≤ cle 11167 − cmin 11364 ℕcn 12145 2c2 12200 3c3 12201 ℤcz 12488 ℤ_≥cuz 12751 ...cfz 13423 𝔼cee 28960 Btwn cbtwn 28961

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-er 8635 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-n0 12402 df-z 12489 df-uz 12752 df-icc 13268 df-fz 13424 df-seq 13925 df-exp 13985 df-ee 28963 df-btwn 28964

This theorem is referenced by: axlowdim2 29033 axlowdim 29034

W3C validator