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Theorem eengtrkge 29134

Description: The geometry structure for 𝔼↑𝑁 is a Euclidean geometry. (Contributed by Thierry Arnoux, 15-Mar-2019.)

**Assertion**
Ref	Expression
eengtrkge	⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiG_E)

Proof of Theorem eengtrkge Dummy variables 𝑎 𝑏 𝑢 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvexd 6878	. 2 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ V)
2		simpll 776	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑁 ∈ ℕ)
3		simprl 780	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (Base‘(EEG‘𝑁)))
4		eengbas 29128	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
5	4	adantr 484	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
6	3, 5	eleqtrrd 2864	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (𝔼‘𝑁))
7	6	adantr 484	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (𝔼‘𝑁))
8		simprr 782	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (Base‘(EEG‘𝑁)))
9	8, 5	eleqtrrd 2864	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (𝔼‘𝑁))
10	9	adantr 484	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (𝔼‘𝑁))
11	3	adantr 484	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (Base‘(EEG‘𝑁)))
12	8	adantr 484	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (Base‘(EEG‘𝑁)))
13		simpr1 1207	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑧 ∈ (Base‘(EEG‘𝑁)))
14		simpr3 1209	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → 𝑧 ∈ (Base‘(EEG‘𝑁)))
15	4	adantr 484	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
16	14, 15	eleqtrrd 2864	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → 𝑧 ∈ (𝔼‘𝑁))
17	2, 11, 12, 13, 16	syl13anc 1390	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑧 ∈ (𝔼‘𝑁))
18		simpr2 1208	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑢 ∈ (Base‘(EEG‘𝑁)))
19	4	ad2antrr 736	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
20	18, 19	eleqtrrd 2864	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑢 ∈ (𝔼‘𝑁))
21		simpr3 1209	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑣 ∈ (Base‘(EEG‘𝑁)))
22	21, 19	eleqtrrd 2864	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑣 ∈ (𝔼‘𝑁))
23		axeuclid 29110	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁) ∧ 𝑧 ∈ (𝔼‘𝑁)) ∧ (𝑢 ∈ (𝔼‘𝑁) ∧ 𝑣 ∈ (𝔼‘𝑁))) → ((𝑢 Btwn ⟨𝑥, 𝑣⟩ ∧ 𝑢 Btwn ⟨𝑦, 𝑧⟩ ∧ 𝑥 ≠ 𝑢) → ∃𝑎 ∈ (𝔼‘𝑁)∃𝑏 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑥, 𝑎⟩ ∧ 𝑧 Btwn ⟨𝑥, 𝑏⟩ ∧ 𝑣 Btwn ⟨𝑎, 𝑏⟩)))
24	2, 7, 10, 17, 20, 22, 23	syl132anc 1406	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → ((𝑢 Btwn ⟨𝑥, 𝑣⟩ ∧ 𝑢 Btwn ⟨𝑦, 𝑧⟩ ∧ 𝑥 ≠ 𝑢) → ∃𝑎 ∈ (𝔼‘𝑁)∃𝑏 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑥, 𝑎⟩ ∧ 𝑧 Btwn ⟨𝑥, 𝑏⟩ ∧ 𝑣 Btwn ⟨𝑎, 𝑏⟩)))
25		eqid 2761	. . . . . . 7 ⊢ (Base‘(EEG‘𝑁)) = (Base‘(EEG‘𝑁))
26		eqid 2761	. . . . . . 7 ⊢ (Itv‘(EEG‘𝑁)) = (Itv‘(EEG‘𝑁))
27	2, 25, 26, 11, 21, 18	ebtwntg 29129	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (𝑢 Btwn ⟨𝑥, 𝑣⟩ ↔ 𝑢 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑣)))
28	2, 25, 26, 12, 13, 18	ebtwntg 29129	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (𝑢 Btwn ⟨𝑦, 𝑧⟩ ↔ 𝑢 ∈ (𝑦(Itv‘(EEG‘𝑁))𝑧)))
29	27, 28	3anbi12d 1457	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → ((𝑢 Btwn ⟨𝑥, 𝑣⟩ ∧ 𝑢 Btwn ⟨𝑦, 𝑧⟩ ∧ 𝑥 ≠ 𝑢) ↔ (𝑢 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑣) ∧ 𝑢 ∈ (𝑦(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑥 ≠ 𝑢)))
30	19	adantr 484	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
31	2	ad2antrr 736	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ 𝑏 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
32	11	ad2antrr 736	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ 𝑏 ∈ (𝔼‘𝑁)) → 𝑥 ∈ (Base‘(EEG‘𝑁)))
33		simpr 488	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → 𝑎 ∈ (𝔼‘𝑁))
34	33, 30	eleqtrd 2863	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → 𝑎 ∈ (Base‘(EEG‘𝑁)))
35	34	adantr 484	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ 𝑏 ∈ (𝔼‘𝑁)) → 𝑎 ∈ (Base‘(EEG‘𝑁)))
36	12	ad2antrr 736	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ 𝑏 ∈ (𝔼‘𝑁)) → 𝑦 ∈ (Base‘(EEG‘𝑁)))
37	31, 25, 26, 32, 35, 36	ebtwntg 29129	. . . . . . . 8 ⊢ (((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ 𝑏 ∈ (𝔼‘𝑁)) → (𝑦 Btwn ⟨𝑥, 𝑎⟩ ↔ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑎)))
38		simpr 488	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ 𝑏 ∈ (𝔼‘𝑁)) → 𝑏 ∈ (𝔼‘𝑁))
39	19	ad2antrr 736	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ 𝑏 ∈ (𝔼‘𝑁)) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
40	38, 39	eleqtrd 2863	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ 𝑏 ∈ (𝔼‘𝑁)) → 𝑏 ∈ (Base‘(EEG‘𝑁)))
41	13	ad2antrr 736	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ 𝑏 ∈ (𝔼‘𝑁)) → 𝑧 ∈ (Base‘(EEG‘𝑁)))
42	31, 25, 26, 32, 40, 41	ebtwntg 29129	. . . . . . . 8 ⊢ (((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ 𝑏 ∈ (𝔼‘𝑁)) → (𝑧 Btwn ⟨𝑥, 𝑏⟩ ↔ 𝑧 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑏)))
43	21	ad2antrr 736	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ 𝑏 ∈ (𝔼‘𝑁)) → 𝑣 ∈ (Base‘(EEG‘𝑁)))
44	31, 25, 26, 35, 40, 43	ebtwntg 29129	. . . . . . . 8 ⊢ (((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ 𝑏 ∈ (𝔼‘𝑁)) → (𝑣 Btwn ⟨𝑎, 𝑏⟩ ↔ 𝑣 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑏)))
45	37, 42, 44	3anbi123d 1456	. . . . . . 7 ⊢ (((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ 𝑏 ∈ (𝔼‘𝑁)) → ((𝑦 Btwn ⟨𝑥, 𝑎⟩ ∧ 𝑧 Btwn ⟨𝑥, 𝑏⟩ ∧ 𝑣 Btwn ⟨𝑎, 𝑏⟩) ↔ (𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑎) ∧ 𝑧 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑏) ∧ 𝑣 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑏))))
46	30, 45	rexeqbidva 3326	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → (∃𝑏 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑥, 𝑎⟩ ∧ 𝑧 Btwn ⟨𝑥, 𝑏⟩ ∧ 𝑣 Btwn ⟨𝑎, 𝑏⟩) ↔ ∃𝑏 ∈ (Base‘(EEG‘𝑁))(𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑎) ∧ 𝑧 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑏) ∧ 𝑣 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑏))))
47	19, 46	rexeqbidva 3326	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (∃𝑎 ∈ (𝔼‘𝑁)∃𝑏 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑥, 𝑎⟩ ∧ 𝑧 Btwn ⟨𝑥, 𝑏⟩ ∧ 𝑣 Btwn ⟨𝑎, 𝑏⟩) ↔ ∃𝑎 ∈ (Base‘(EEG‘𝑁))∃𝑏 ∈ (Base‘(EEG‘𝑁))(𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑎) ∧ 𝑧 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑏) ∧ 𝑣 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑏))))
48	24, 29, 47	3imtr3d 295	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → ((𝑢 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑣) ∧ 𝑢 ∈ (𝑦(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑥 ≠ 𝑢) → ∃𝑎 ∈ (Base‘(EEG‘𝑁))∃𝑏 ∈ (Base‘(EEG‘𝑁))(𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑎) ∧ 𝑧 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑏) ∧ 𝑣 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑏))))
49	48	ralrimivvva 3207	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → ∀𝑧 ∈ (Base‘(EEG‘𝑁))∀𝑢 ∈ (Base‘(EEG‘𝑁))∀𝑣 ∈ (Base‘(EEG‘𝑁))((𝑢 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑣) ∧ 𝑢 ∈ (𝑦(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑥 ≠ 𝑢) → ∃𝑎 ∈ (Base‘(EEG‘𝑁))∃𝑏 ∈ (Base‘(EEG‘𝑁))(𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑎) ∧ 𝑧 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑏) ∧ 𝑣 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑏))))
50	49	ralrimivva 3204	. 2 ⊢ (𝑁 ∈ ℕ → ∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))∀𝑧 ∈ (Base‘(EEG‘𝑁))∀𝑢 ∈ (Base‘(EEG‘𝑁))∀𝑣 ∈ (Base‘(EEG‘𝑁))((𝑢 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑣) ∧ 𝑢 ∈ (𝑦(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑥 ≠ 𝑢) → ∃𝑎 ∈ (Base‘(EEG‘𝑁))∃𝑏 ∈ (Base‘(EEG‘𝑁))(𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑎) ∧ 𝑧 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑏) ∧ 𝑣 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑏))))
51		eqid 2761	. . 3 ⊢ (dist‘(EEG‘𝑁)) = (dist‘(EEG‘𝑁))
52	25, 51, 26	istrkge 28603	. 2 ⊢ ((EEG‘𝑁) ∈ TarskiG_E ↔ ((EEG‘𝑁) ∈ V ∧ ∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))∀𝑧 ∈ (Base‘(EEG‘𝑁))∀𝑢 ∈ (Base‘(EEG‘𝑁))∀𝑣 ∈ (Base‘(EEG‘𝑁))((𝑢 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑣) ∧ 𝑢 ∈ (𝑦(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑥 ≠ 𝑢) → ∃𝑎 ∈ (Base‘(EEG‘𝑁))∃𝑏 ∈ (Base‘(EEG‘𝑁))(𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑎) ∧ 𝑧 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑏) ∧ 𝑣 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑏)))))
53	1, 50, 52	sylanbrc 592	1 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiG_E)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∀wral 3075 ∃wrex 3085 Vcvv 3453 ⟨cop 4587 class class class wbr 5099 ‘cfv 6517 (class class class)co 7392 ℕcn 12207 Basecbs 17228 distcds 17278 TarskiG_Ecstrkge 28578 Itvcitv 28579 𝔼cee 29034 Btwn cbtwn 29035 EEGceeng 29124

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-om 7843 df-1st 7966 df-2nd 7967 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-er 8673 df-map 8805 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12208 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12479 df-z 12566 df-dec 12686 df-uz 12837 df-icc 13353 df-fz 13510 df-seq 14012 df-sum 15697 df-struct 17166 df-slot 17201 df-ndx 17213 df-base 17229 df-ds 17291 df-itv 28581 df-lng 28582 df-trkge 28597 df-ee 29037 df-btwn 29038 df-eeng 29125

This theorem is referenced by: (None)

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