axlowdimlem6 - Metamath Proof Explorer

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Theorem axlowdimlem6 29507

Description: Lemma for axlowdim 29521. 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 12708	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 1 ∈ ℤ)
2		eluzelz 12956	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℤ)
3		2nn 12397	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ
4		uznnssnn 13003	. . . . . . . . . . 11 ⊢ (2 ∈ ℕ → (ℤ_≥‘2) ⊆ ℕ)
5	3, 4	ax-mp 5	. . . . . . . . . 10 ⊢ (ℤ_≥‘2) ⊆ ℕ
6		nnuz 12985	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
7	5, 6	sseqtri 3979	. . . . . . . . 9 ⊢ (ℤ_≥‘2) ⊆ (ℤ_≥‘1)
8	7	sseli 3927	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ (ℤ_≥‘1))
9		eluzle 12959	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 1 ≤ 𝑁)
10	8, 9	syl 18	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 1 ≤ 𝑁)
11		1re 11289	. . . . . . . 8 ⊢ 1 ∈ ℝ
12	11	leidi 11831	. . . . . . 7 ⊢ 1 ≤ 1
13	10, 12	jctil 529	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (1 ≤ 1 ∧ 1 ≤ 𝑁))
14		elfz4 13630	. . . . . 6 ⊢ (((1 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 1 ∈ ℤ) ∧ (1 ≤ 1 ∧ 1 ≤ 𝑁)) → 1 ∈ (1...𝑁))
15	1, 2, 1, 13, 14	syl31anc 1400	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 1 ∈ (1...𝑁))
16		eluzel2 12951	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ∈ ℤ)
17		eluzle 12959	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ≤ 𝑁)
18		1le2 12535	. . . . . . 7 ⊢ 1 ≤ 2
19	17, 18	jctil 529	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (1 ≤ 2 ∧ 2 ≤ 𝑁))
20		elfz4 13630	. . . . . 6 ⊢ (((1 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 2 ∈ ℤ) ∧ (1 ≤ 2 ∧ 2 ≤ 𝑁)) → 2 ∈ (1...𝑁))
21	1, 2, 16, 19, 20	syl31anc 1400	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ∈ (1...𝑁))
22		ax-1ne0 11250	. . . . . . 7 ⊢ 1 ≠ 0
23		1t1e1 12485	. . . . . . . 8 ⊢ (1 · 1) = 1
24		0cn 11279	. . . . . . . . 9 ⊢ 0 ∈ ℂ
25	24	mul01i 11481	. . . . . . . 8 ⊢ (0 · 0) = 0
26	23, 25	neeq12i 3022	. . . . . . 7 ⊢ ((1 · 1) ≠ (0 · 0) ↔ 1 ≠ 0)
27	22, 26	mpbir 234	. . . . . 6 ⊢ (1 · 1) ≠ (0 · 0)
28		fveq2 6877	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) = (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘1))
29		0re 11291	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
30	11, 29	axlowdimlem4 29505	. . . . . . . . . . . . . . 15 ⊢ {⟨1, 1⟩, ⟨2, 0⟩}:(1...2)⟶ℝ
31		ffn 6701	. . . . . . . . . . . . . . 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 29502	. . . . . . . . . . . . . . 15 ⊢ ((3...𝑁) × {0}):(3...𝑁)⟶ℝ
34		ffn 6701	. . . . . . . . . . . . . . 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 29503	. . . . . . . . . . . . . . 15 ⊢ ((1...2) ∩ (3...𝑁)) = ∅
37		1z 12707	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℤ
38		2z 12709	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℤ
39	37, 38, 37	3pm3.2i 1358	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ ℤ ∧ 2 ∈ ℤ ∧ 1 ∈ ℤ)
40	12, 18	pm3.2i 476	. . . . . . . . . . . . . . . 16 ⊢ (1 ≤ 1 ∧ 1 ≤ 2)
41		elfz4 13630	. . . . . . . . . . . . . . . 16 ⊢ (((1 ∈ ℤ ∧ 2 ∈ ℤ ∧ 1 ∈ ℤ) ∧ (1 ≤ 1 ∧ 1 ≤ 2)) → 1 ∈ (1...2))
42	39, 40, 41	mp2an 705	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ (1...2)
43	36, 42	pm3.2i 476	. . . . . . . . . . . . . 14 ⊢ (((1...2) ∩ (3...𝑁)) = ∅ ∧ 1 ∈ (1...2))
44		fvun1 6968	. . . . . . . . . . . . . 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 1490	. . . . . . . . . . . . 13 ⊢ (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘1) = ({⟨1, 1⟩, ⟨2, 0⟩}‘1)
46		1ne2 12534	. . . . . . . . . . . . . 14 ⊢ 1 ≠ 2
47		1ex 11284	. . . . . . . . . . . . . . 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 2784	. . . . . . . . . . . 12 ⊢ (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘1) = 1
51	28, 50	eqtrdi 2812	. . . . . . . . . . 11 ⊢ (𝑖 = 1 → (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) = 1)
52		fveq2 6877	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) = (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘1))
53	29, 29	axlowdimlem4 29505	. . . . . . . . . . . . . . 15 ⊢ {⟨1, 0⟩, ⟨2, 0⟩}:(1...2)⟶ℝ
54		ffn 6701	. . . . . . . . . . . . . . 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 6968	. . . . . . . . . . . . . 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 1490	. . . . . . . . . . . . 13 ⊢ (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘1) = ({⟨1, 0⟩, ⟨2, 0⟩}‘1)
58	29	elexi 3473	. . . . . . . . . . . . . . 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 2784	. . . . . . . . . . . 12 ⊢ (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘1) = 0
62	52, 61	eqtrdi 2812	. . . . . . . . . . 11 ⊢ (𝑖 = 1 → (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) = 0)
63	51, 62	oveq12d 7430	. . . . . . . . . 10 ⊢ (𝑖 = 1 → ((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) = (1 − 0))
64		1m0e1 12443	. . . . . . . . . 10 ⊢ (1 − 0) = 1
65	63, 64	eqtrdi 2812	. . . . . . . . 9 ⊢ (𝑖 = 1 → ((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) = 1)
66	65	oveq1d 7427	. . . . . . . 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 6877	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) = (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘1))
68	29, 11	axlowdimlem4 29505	. . . . . . . . . . . . . . 15 ⊢ {⟨1, 0⟩, ⟨2, 1⟩}:(1...2)⟶ℝ
69		ffn 6701	. . . . . . . . . . . . . . 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 6968	. . . . . . . . . . . . . 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 1490	. . . . . . . . . . . . 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 2784	. . . . . . . . . . . 12 ⊢ (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘1) = 0
76	67, 75	eqtrdi 2812	. . . . . . . . . . 11 ⊢ (𝑖 = 1 → (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) = 0)
77	76, 62	oveq12d 7430	. . . . . . . . . 10 ⊢ (𝑖 = 1 → ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) = (0 − 0))
78		0m0e0 12442	. . . . . . . . . 10 ⊢ (0 − 0) = 0
79	77, 78	eqtrdi 2812	. . . . . . . . 9 ⊢ (𝑖 = 1 → ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑖) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑖)) = 0)
80	79	oveq2d 7428	. . . . . . . 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 3017	. . . . . . 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 6877	. . . . . . . . . . . 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 1358	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ ℤ ∧ 2 ∈ ℤ ∧ 2 ∈ ℤ)
84		2re 12398	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℝ
85	84	leidi 11831	. . . . . . . . . . . . . . . . 17 ⊢ 2 ≤ 2
86	18, 85	pm3.2i 476	. . . . . . . . . . . . . . . 16 ⊢ (1 ≤ 2 ∧ 2 ≤ 2)
87		elfz4 13630	. . . . . . . . . . . . . . . 16 ⊢ (((1 ∈ ℤ ∧ 2 ∈ ℤ ∧ 2 ∈ ℤ) ∧ (1 ≤ 2 ∧ 2 ≤ 2)) → 2 ∈ (1...2))
88	83, 86, 87	mp2an 705	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ (1...2)
89	36, 88	pm3.2i 476	. . . . . . . . . . . . . 14 ⊢ (((1...2) ∩ (3...𝑁)) = ∅ ∧ 2 ∈ (1...2))
90		fvun1 6968	. . . . . . . . . . . . . 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 1490	. . . . . . . . . . . . 13 ⊢ (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘2) = ({⟨1, 0⟩, ⟨2, 1⟩}‘2)
92	38	elexi 3473	. . . . . . . . . . . . . . 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 2784	. . . . . . . . . . . 12 ⊢ (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘2) = 1
96	82, 95	eqtrdi 2812	. . . . . . . . . . 11 ⊢ (𝑗 = 2 → (({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) = 1)
97		fveq2 6877	. . . . . . . . . . . 12 ⊢ (𝑗 = 2 → (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) = (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘2))
98		fvun1 6968	. . . . . . . . . . . . . 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 1490	. . . . . . . . . . . . 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 2784	. . . . . . . . . . . 12 ⊢ (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘2) = 0
103	97, 102	eqtrdi 2812	. . . . . . . . . . 11 ⊢ (𝑗 = 2 → (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) = 0)
104	96, 103	oveq12d 7430	. . . . . . . . . 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 2812	. . . . . . . . 9 ⊢ (𝑗 = 2 → ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) = 1)
106	105	oveq2d 7428	. . . . . . . 8 ⊢ (𝑗 = 2 → (1 · ((({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗))) = (1 · 1))
107		fveq2 6877	. . . . . . . . . . . 12 ⊢ (𝑗 = 2 → (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) = (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘2))
108		fvun1 6968	. . . . . . . . . . . . . 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 1490	. . . . . . . . . . . . 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 2784	. . . . . . . . . . . 12 ⊢ (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘2) = 0
113	107, 112	eqtrdi 2812	. . . . . . . . . . 11 ⊢ (𝑗 = 2 → (({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) = 0)
114	113, 103	oveq12d 7430	. . . . . . . . . 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 2812	. . . . . . . . 9 ⊢ (𝑗 = 2 → ((({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗) − (({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))‘𝑗)) = 0)
116	115	oveq1d 7427	. . . . . . . 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 3017	. . . . . . 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 1479	. . . . 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 596	. . . 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 2957	. . . . . . . 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 3110	. . . . . . 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 3115	. . . . . . 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 278	. . . . . 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 3110	. . . . 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 3115	. . . . 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 278	. . . 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 221	. . 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 29506	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})) ∈ (𝔼‘𝑁))
130	11, 29	axlowdimlem5 29506	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})) ∈ (𝔼‘𝑁))
131	29, 11	axlowdimlem5 29506	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0})) ∈ (𝔼‘𝑁))
132		colinearalg 29470	. . . 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 1398	. . 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 328	. 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 4838	. . . 4 ⊢ ⟨𝐵, 𝐶⟩ = ⟨({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})), ({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))⟩
139	135, 138	breq12i 5112	. . 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 4838	. . . 4 ⊢ ⟨𝐶, 𝐴⟩ = ⟨({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0})), ({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))⟩
141	136, 140	breq12i 5112	. . 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 4838	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ = ⟨({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0})), ({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))⟩
143	137, 142	breq12i 5112	. . 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 1174	. 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 332	1 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ¬ (𝐴 Btwn ⟨𝐵, 𝐶⟩ ∨ 𝐵 Btwn ⟨𝐶, 𝐴⟩ ∨ 𝐶 Btwn ⟨𝐴, 𝐵⟩))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ w3o 1102 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∀wral 3077 ∃wrex 3087 ∪ cun 3897 ∩ cin 3898 ⊆ wss 3899 ∅c0 4279 {csn 4584 {cpr 4586 ⟨cop 4590 class class class wbr 5103 × cxp 5649 Fn wfn 6526 ⟶wf 6527 ‘cfv 6531 (class class class)co 7412 ℝcr 11180 0cc0 11181 1c1 11182 · cmul 11186 ≤ cle 11325 − cmin 11522 ℕcn 12316 2c2 12378 3c3 12379 ℤcz 12674 ℤ_≥cuz 12946 ...cfz 13620 𝔼cee 29447 Btwn cbtwn 29448

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-er 8701 df-map 8833 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-div 11955 df-nn 12317 df-2 12386 df-3 12387 df-n0 12588 df-z 12675 df-uz 12947 df-icc 13464 df-fz 13621 df-seq 14125 df-exp 14185 df-ee 29450 df-btwn 29451

This theorem is used by: axlowdim2 29520 axlowdim 29521

W3C validator