axlowdimlem6 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > axlowdimlem6		Structured version Visualization version GIF version

Theorem axlowdimlem6 26733

Description: Lemma for axlowdim 26747. 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 12014	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 1 ∈ ℤ)
2		eluzelz 12254	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℤ)
3		2nn 11711	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ
4		uznnssnn 12296	. . . . . . . . . . 11 ⊢ (2 ∈ ℕ → (ℤ_≥‘2) ⊆ ℕ)
5	3, 4	ax-mp 5	. . . . . . . . . 10 ⊢ (ℤ_≥‘2) ⊆ ℕ
6		nnuz 12282	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
7	5, 6	sseqtri 4003	. . . . . . . . 9 ⊢ (ℤ_≥‘2) ⊆ (ℤ_≥‘1)
8	7	sseli 3963	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ (ℤ_≥‘1))
9		eluzle 12257	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 1 ≤ 𝑁)
10	8, 9	syl 17	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 1 ≤ 𝑁)
11		1re 10641	. . . . . . . 8 ⊢ 1 ∈ ℝ
12	11	leidi 11174	. . . . . . 7 ⊢ 1 ≤ 1
13	10, 12	jctil 522	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (1 ≤ 1 ∧ 1 ≤ 𝑁))
14		elfz4 12902	. . . . . 6 ⊢ (((1 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 1 ∈ ℤ) ∧ (1 ≤ 1 ∧ 1 ≤ 𝑁)) → 1 ∈ (1...𝑁))
15	1, 2, 1, 13, 14	syl31anc 1369	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 1 ∈ (1...𝑁))
16		eluzel2 12249	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ∈ ℤ)
17		eluzle 12257	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ≤ 𝑁)
18		1le2 11847	. . . . . . 7 ⊢ 1 ≤ 2
19	17, 18	jctil 522	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (1 ≤ 2 ∧ 2 ≤ 𝑁))
20		elfz4 12902	. . . . . 6 ⊢ (((1 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 2 ∈ ℤ) ∧ (1 ≤ 2 ∧ 2 ≤ 𝑁)) → 2 ∈ (1...𝑁))
21	1, 2, 16, 19, 20	syl31anc 1369	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ∈ (1...𝑁))
22		ax-1ne0 10606	. . . . . . 7 ⊢ 1 ≠ 0
23		1t1e1 11800	. . . . . . . 8 ⊢ (1 · 1) = 1
24		0cn 10633	. . . . . . . . 9 ⊢ 0 ∈ ℂ
25	24	mul01i 10830	. . . . . . . 8 ⊢ (0 · 0) = 0
26	23, 25	neeq12i 3082	. . . . . . 7 ⊢ ((1 · 1) ≠ (0 · 0) ↔ 1 ≠ 0)
27	22, 26	mpbir 233	. . . . . 6 ⊢ (1 · 1) ≠ (0 · 0)
28		fveq2 6670	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) = (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘1))
29		0re 10643	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
30	11, 29	axlowdimlem4 26731	. . . . . . . . . . . . . . 15 ⊢ {⟨1, 1⟩, ⟨2, 0⟩}:(1...2)⟶ℝ
31		ffn 6514	. . . . . . . . . . . . . . 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 26728	. . . . . . . . . . . . . . 15 ⊢ ((3...𝑁) × {0}):(3...𝑁)⟶ℝ
34		ffn 6514	. . . . . . . . . . . . . . 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 26729	. . . . . . . . . . . . . . 15 ⊢ ((1...2) ∩ (3...𝑁)) = ∅
37		1z 12013	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℤ
38		2z 12015	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℤ
39	37, 38, 37	3pm3.2i 1335	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ ℤ ∧ 2 ∈ ℤ ∧ 1 ∈ ℤ)
40	12, 18	pm3.2i 473	. . . . . . . . . . . . . . . 16 ⊢ (1 ≤ 1 ∧ 1 ≤ 2)
41		elfz4 12902	. . . . . . . . . . . . . . . 16 ⊢ (((1 ∈ ℤ ∧ 2 ∈ ℤ ∧ 1 ∈ ℤ) ∧ (1 ≤ 1 ∧ 1 ≤ 2)) → 1 ∈ (1...2))
42	39, 40, 41	mp2an 690	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ (1...2)
43	36, 42	pm3.2i 473	. . . . . . . . . . . . . 14 ⊢ (((1...2) ∩ (3...𝑁)) = ∅ ∧ 1 ∈ (1...2))
44		fvun1 6754	. . . . . . . . . . . . . 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 1457	. . . . . . . . . . . . 13 ⊢ (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘1) = ({⟨1, 1⟩, ⟨2, 0⟩}‘1)
46		1ne2 11846	. . . . . . . . . . . . . 14 ⊢ 1 ≠ 2
47		1ex 10637	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ V
48	47, 47	fvpr1 6952	. . . . . . . . . . . . . 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 2844	. . . . . . . . . . . 12 ⊢ (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘1) = 1
51	28, 50	syl6eq 2872	. . . . . . . . . . 11 ⊢ (𝑖 = 1 → (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) = 1)
52		fveq2 6670	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) = (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘1))
53	29, 29	axlowdimlem4 26731	. . . . . . . . . . . . . . 15 ⊢ {⟨1, 0⟩, ⟨2, 0⟩}:(1...2)⟶ℝ
54		ffn 6514	. . . . . . . . . . . . . . 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 6754	. . . . . . . . . . . . . 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 1457	. . . . . . . . . . . . 13 ⊢ (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘1) = ({⟨1, 0⟩, ⟨2, 0⟩}‘1)
58	29	elexi 3513	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
59	47, 58	fvpr1 6952	. . . . . . . . . . . . . 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 2844	. . . . . . . . . . . 12 ⊢ (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘1) = 0
62	52, 61	syl6eq 2872	. . . . . . . . . . 11 ⊢ (𝑖 = 1 → (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) = 0)
63	51, 62	oveq12d 7174	. . . . . . . . . 10 ⊢ (𝑖 = 1 → ((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) = (1 − 0))
64		1m0e1 11759	. . . . . . . . . 10 ⊢ (1 − 0) = 1
65	63, 64	syl6eq 2872	. . . . . . . . 9 ⊢ (𝑖 = 1 → ((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) = 1)
66	65	oveq1d 7171	. . . . . . . 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 6670	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) = (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘1))
68	29, 11	axlowdimlem4 26731	. . . . . . . . . . . . . . 15 ⊢ {⟨1, 0⟩, ⟨2, 1⟩}:(1...2)⟶ℝ
69		ffn 6514	. . . . . . . . . . . . . . 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 6754	. . . . . . . . . . . . . 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 1457	. . . . . . . . . . . . 13 ⊢ (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘1) = ({⟨1, 0⟩, ⟨2, 1⟩}‘1)
73	47, 58	fvpr1 6952	. . . . . . . . . . . . . 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 2844	. . . . . . . . . . . 12 ⊢ (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘1) = 0
76	67, 75	syl6eq 2872	. . . . . . . . . . 11 ⊢ (𝑖 = 1 → (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) = 0)
77	76, 62	oveq12d 7174	. . . . . . . . . 10 ⊢ (𝑖 = 1 → ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) = (0 − 0))
78		0m0e0 11758	. . . . . . . . . 10 ⊢ (0 − 0) = 0
79	77, 78	syl6eq 2872	. . . . . . . . 9 ⊢ (𝑖 = 1 → ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) = 0)
80	79	oveq2d 7172	. . . . . . . 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 3077	. . . . . . 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 6670	. . . . . . . . . . . 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 1335	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ ℤ ∧ 2 ∈ ℤ ∧ 2 ∈ ℤ)
84		2re 11712	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℝ
85	84	leidi 11174	. . . . . . . . . . . . . . . . 17 ⊢ 2 ≤ 2
86	18, 85	pm3.2i 473	. . . . . . . . . . . . . . . 16 ⊢ (1 ≤ 2 ∧ 2 ≤ 2)
87		elfz4 12902	. . . . . . . . . . . . . . . 16 ⊢ (((1 ∈ ℤ ∧ 2 ∈ ℤ ∧ 2 ∈ ℤ) ∧ (1 ≤ 2 ∧ 2 ≤ 2)) → 2 ∈ (1...2))
88	83, 86, 87	mp2an 690	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ (1...2)
89	36, 88	pm3.2i 473	. . . . . . . . . . . . . 14 ⊢ (((1...2) ∩ (3...𝑁)) = ∅ ∧ 2 ∈ (1...2))
90		fvun1 6754	. . . . . . . . . . . . . 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 1457	. . . . . . . . . . . . 13 ⊢ (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘2) = ({⟨1, 0⟩, ⟨2, 1⟩}‘2)
92	38	elexi 3513	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ V
93	92, 47	fvpr2 6953	. . . . . . . . . . . . . 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 2844	. . . . . . . . . . . 12 ⊢ (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘2) = 1
96	82, 95	syl6eq 2872	. . . . . . . . . . 11 ⊢ (𝑗 = 2 → (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) = 1)
97		fveq2 6670	. . . . . . . . . . . 12 ⊢ (𝑗 = 2 → (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) = (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘2))
98		fvun1 6754	. . . . . . . . . . . . . 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 1457	. . . . . . . . . . . . 13 ⊢ (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘2) = ({⟨1, 0⟩, ⟨2, 0⟩}‘2)
100	92, 58	fvpr2 6953	. . . . . . . . . . . . . 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 2844	. . . . . . . . . . . 12 ⊢ (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘2) = 0
103	97, 102	syl6eq 2872	. . . . . . . . . . 11 ⊢ (𝑗 = 2 → (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) = 0)
104	96, 103	oveq12d 7174	. . . . . . . . . 10 ⊢ (𝑗 = 2 → ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) = (1 − 0))
105	104, 64	syl6eq 2872	. . . . . . . . 9 ⊢ (𝑗 = 2 → ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) = 1)
106	105	oveq2d 7172	. . . . . . . 8 ⊢ (𝑗 = 2 → (1 · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))) = (1 · 1))
107		fveq2 6670	. . . . . . . . . . . 12 ⊢ (𝑗 = 2 → (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) = (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘2))
108		fvun1 6754	. . . . . . . . . . . . . 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 1457	. . . . . . . . . . . . 13 ⊢ (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘2) = ({⟨1, 1⟩, ⟨2, 0⟩}‘2)
110	92, 58	fvpr2 6953	. . . . . . . . . . . . . 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 2844	. . . . . . . . . . . 12 ⊢ (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘2) = 0
113	107, 112	syl6eq 2872	. . . . . . . . . . 11 ⊢ (𝑗 = 2 → (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) = 0)
114	113, 103	oveq12d 7174	. . . . . . . . . 10 ⊢ (𝑗 = 2 → ((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) = (0 − 0))
115	114, 78	syl6eq 2872	. . . . . . . . 9 ⊢ (𝑗 = 2 → ((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) = 0)
116	115	oveq1d 7171	. . . . . . . 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 3077	. . . . . . 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 3635	. . . . . 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 1446	. . . . 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 586	. . . 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 3017	. . . . . . . 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 3247	. . . . . . 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 3238	. . . . . . 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 277	. . . . . 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 3247	. . . . 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 3238	. . . . 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 277	. . . 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 220	. . 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 26732	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})) ∈ (𝔼‘𝑁))
130	11, 29	axlowdimlem5 26732	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})) ∈ (𝔼‘𝑁))
131	29, 11	axlowdimlem5 26732	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0})) ∈ (𝔼‘𝑁))
132		colinearalg 26696	. . . 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 1367	. . 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 327	. 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 4808	. . . 4 ⊢ ⟨𝐵, 𝐶⟩ = ⟨({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})), ({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))⟩
139	135, 138	breq12i 5075	. . 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 4808	. . . 4 ⊢ ⟨𝐶, 𝐴⟩ = ⟨({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0})), ({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))⟩
141	136, 140	breq12i 5075	. . 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 4808	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ = ⟨({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})), ({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))⟩
143	137, 142	breq12i 5075	. . 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 1152	. 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 331	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 208 ∧ wa 398 ∨ w3o 1082 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 ∪ cun 3934 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 {csn 4567 {cpr 4569 ⟨cop 4573 class class class wbr 5066 × cxp 5553 Fn wfn 6350 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 0cc0 10537 1c1 10538 · cmul 10542 ≤ cle 10676 − cmin 10870 ℕcn 11638 2c2 11693 3c3 11694 ℤcz 11982 ℤ_≥cuz 12244 ...cfz 12893 𝔼cee 26674 Btwn cbtwn 26675

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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-icc 12746 df-fz 12894 df-seq 13371 df-exp 13431 df-ee 26677 df-btwn 26678

This theorem is referenced by: axlowdim2 26746 axlowdim 26747

W3C validator