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 28931

Description: Lemma for axlowdim 28945. 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 12628	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 1 ∈ ℤ)
2		eluzelz 12867	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℤ)
3		2nn 12318	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ
4		uznnssnn 12916	. . . . . . . . . . 11 ⊢ (2 ∈ ℕ → (ℤ_≥‘2) ⊆ ℕ)
5	3, 4	ax-mp 5	. . . . . . . . . 10 ⊢ (ℤ_≥‘2) ⊆ ℕ
6		nnuz 12900	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
7	5, 6	sseqtri 4012	. . . . . . . . 9 ⊢ (ℤ_≥‘2) ⊆ (ℤ_≥‘1)
8	7	sseli 3959	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ (ℤ_≥‘1))
9		eluzle 12870	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 1 ≤ 𝑁)
10	8, 9	syl 17	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 1 ≤ 𝑁)
11		1re 11240	. . . . . . . 8 ⊢ 1 ∈ ℝ
12	11	leidi 11776	. . . . . . 7 ⊢ 1 ≤ 1
13	10, 12	jctil 519	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (1 ≤ 1 ∧ 1 ≤ 𝑁))
14		elfz4 13539	. . . . . 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 12862	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ∈ ℤ)
17		eluzle 12870	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ≤ 𝑁)
18		1le2 12454	. . . . . . 7 ⊢ 1 ≤ 2
19	17, 18	jctil 519	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (1 ≤ 2 ∧ 2 ≤ 𝑁))
20		elfz4 13539	. . . . . 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 11203	. . . . . . 7 ⊢ 1 ≠ 0
23		1t1e1 12407	. . . . . . . 8 ⊢ (1 · 1) = 1
24		0cn 11232	. . . . . . . . 9 ⊢ 0 ∈ ℂ
25	24	mul01i 11430	. . . . . . . 8 ⊢ (0 · 0) = 0
26	23, 25	neeq12i 2999	. . . . . . 7 ⊢ ((1 · 1) ≠ (0 · 0) ↔ 1 ≠ 0)
27	22, 26	mpbir 231	. . . . . 6 ⊢ (1 · 1) ≠ (0 · 0)
28		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) = (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘1))
29		0re 11242	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
30	11, 29	axlowdimlem4 28929	. . . . . . . . . . . . . . 15 ⊢ {⟨1, 1⟩, ⟨2, 0⟩}:(1...2)⟶ℝ
31		ffn 6711	. . . . . . . . . . . . . . 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 28926	. . . . . . . . . . . . . . 15 ⊢ ((3...𝑁) × {0}):(3...𝑁)⟶ℝ
34		ffn 6711	. . . . . . . . . . . . . . 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 28927	. . . . . . . . . . . . . . 15 ⊢ ((1...2) ∩ (3...𝑁)) = ∅
37		1z 12627	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℤ
38		2z 12629	. . . . . . . . . . . . . . . . 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 13539	. . . . . . . . . . . . . . . 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 6975	. . . . . . . . . . . . . 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 12453	. . . . . . . . . . . . . 14 ⊢ 1 ≠ 2
47		1ex 11236	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ V
48	47, 47	fvpr1 7189	. . . . . . . . . . . . . 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 6881	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) = (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘1))
53	29, 29	axlowdimlem4 28929	. . . . . . . . . . . . . . 15 ⊢ {⟨1, 0⟩, ⟨2, 0⟩}:(1...2)⟶ℝ
54		ffn 6711	. . . . . . . . . . . . . . 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 6975	. . . . . . . . . . . . . 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 3487	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
59	47, 58	fvpr1 7189	. . . . . . . . . . . . . 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 7428	. . . . . . . . . 10 ⊢ (𝑖 = 1 → ((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) = (1 − 0))
64		1m0e1 12366	. . . . . . . . . 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 7425	. . . . . . . 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 6881	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) = (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘1))
68	29, 11	axlowdimlem4 28929	. . . . . . . . . . . . . . 15 ⊢ {⟨1, 0⟩, ⟨2, 1⟩}:(1...2)⟶ℝ
69		ffn 6711	. . . . . . . . . . . . . . 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 6975	. . . . . . . . . . . . . 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 7189	. . . . . . . . . . . . . 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 7428	. . . . . . . . . 10 ⊢ (𝑖 = 1 → ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) = (0 − 0))
78		0m0e0 12365	. . . . . . . . . 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 7426	. . . . . . . 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 2994	. . . . . . 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 6881	. . . . . . . . . . . 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 12319	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℝ
85	84	leidi 11776	. . . . . . . . . . . . . . . . 17 ⊢ 2 ≤ 2
86	18, 85	pm3.2i 470	. . . . . . . . . . . . . . . 16 ⊢ (1 ≤ 2 ∧ 2 ≤ 2)
87		elfz4 13539	. . . . . . . . . . . . . . . 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 6975	. . . . . . . . . . . . . 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 3487	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ V
93	92, 47	fvpr2 7190	. . . . . . . . . . . . . 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 6881	. . . . . . . . . . . 12 ⊢ (𝑗 = 2 → (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) = (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘2))
98		fvun1 6975	. . . . . . . . . . . . . 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 7190	. . . . . . . . . . . . . 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 7428	. . . . . . . . . 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 7426	. . . . . . . 8 ⊢ (𝑗 = 2 → (1 · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))) = (1 · 1))
107		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑗 = 2 → (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) = (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘2))
108		fvun1 6975	. . . . . . . . . . . . . 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 7190	. . . . . . . . . . . . . 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 7428	. . . . . . . . . 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 7425	. . . . . . . 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 2994	. . . . . . 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 3619	. . . . . 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 2934	. . . . . . . 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 3084	. . . . . . 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 3090	. . . . . . 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 3084	. . . . 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 3090	. . . . 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 28930	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})) ∈ (𝔼‘𝑁))
130	11, 29	axlowdimlem5 28930	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})) ∈ (𝔼‘𝑁))
131	29, 11	axlowdimlem5 28930	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0})) ∈ (𝔼‘𝑁))
132		colinearalg 28894	. . . 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 4859	. . . 4 ⊢ ⟨𝐵, 𝐶⟩ = ⟨({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})), ({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))⟩
139	135, 138	breq12i 5133	. . 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 4859	. . . 4 ⊢ ⟨𝐶, 𝐴⟩ = ⟨({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0})), ({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))⟩
141	136, 140	breq12i 5133	. . 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 4859	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ = ⟨({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})), ({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))⟩
143	137, 142	breq12i 5133	. . 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 1540 ∈ wcel 2109 ≠ wne 2933 ∀wral 3052 ∃wrex 3061 ∪ cun 3929 ∩ cin 3930 ⊆ wss 3931 ∅c0 4313 {csn 4606 {cpr 4608 ⟨cop 4612 class class class wbr 5124 × cxp 5657 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ℝcr 11133 0cc0 11134 1c1 11135 · cmul 11139 ≤ cle 11275 − cmin 11471 ℕcn 12245 2c2 12300 3c3 12301 ℤcz 12593 ℤ_≥cuz 12857 ...cfz 13529 𝔼cee 28872 Btwn cbtwn 28873

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8724 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-n0 12507 df-z 12594 df-uz 12858 df-icc 13374 df-fz 13530 df-seq 14025 df-exp 14085 df-ee 28875 df-btwn 28876

This theorem is referenced by: axlowdim2 28944 axlowdim 28945

W3C validator