Theorem eengtrkge 29020

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) ∈ TarskiGE)

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 6935	. 2 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ V)
2		simpll 766	. . . . . 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 770	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (Base‘(EEG‘𝑁)))
4		eengbas 29014	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
5	4	adantr 480	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
6	3, 5	eleqtrrd 2847	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (𝔼‘𝑁))
7	6	adantr 480	. . . . . 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 772	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (Base‘(EEG‘𝑁)))
9	8, 5	eleqtrrd 2847	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (𝔼‘𝑁))
10	9	adantr 480	. . . . . 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 480	. . . . . . 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 480	. . . . . . 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 1194	. . . . . . 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 1196	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → 𝑧 ∈ (Base‘(EEG‘𝑁)))
15	4	adantr 480	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
16	14, 15	eleqtrrd 2847	. . . . . . 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 1372	. . . . . 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 1195	. . . . . . 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 725	. . . . . . 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 2847	. . . . . 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 1196	. . . . . . 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 2847	. . . . . 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 28996	. . . . . 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 1388	. . . . 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 2740	. . . . . . 7 ⊢ (Base‘(EEG‘𝑁)) = (Base‘(EEG‘𝑁))
26		eqid 2740	. . . . . . 7 ⊢ (Itv‘(EEG‘𝑁)) = (Itv‘(EEG‘𝑁))
27	2, 25, 26, 11, 21, 18	ebtwntg 29015	. . . . . 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 29015	. . . . . 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 1437	. . . . 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 480	. . . . . . 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 725	. . . . . . . . 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 725	. . . . . . . . 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 484	. . . . . . . . . . 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 2846	. . . . . . . . . 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 480	. . . . . . . . 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 725	. . . . . . . . 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 29015	. . . . . . . 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 484	. . . . . . . . . 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 725	. . . . . . . . . 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 2846	. . . . . . . . 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 725	. . . . . . . . 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 29015	. . . . . . . 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 725	. . . . . . . . 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 29015	. . . . . . . 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 1436	. . . . . . 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 3341	. . . . . 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 3341	. . . . 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 293	. . . 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 3211	. . 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 3208	. 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 2740	. . 3 ⊢ (dist‘(EEG‘𝑁)) = (dist‘(EEG‘𝑁))
52	25, 51, 26	istrkge 28483	. 2 ⊢ ((EEG‘𝑁) ∈ TarskiGE ↔ ((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 582	1 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiGE)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076  Vcvv 3488  ⟨cop 4654   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448  ℕcn 12293  Basecbs 17258  distcds 17320  TarskiG_Ecstrkge 28458  Itvcitv 28459  𝔼cee 28921   Btwn cbtwn 28922  EEGceeng 29010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-icc 13414  df-fz 13568  df-seq 14053  df-sum 15735  df-struct 17194  df-slot 17229  df-ndx 17241  df-base 17259  df-ds 17333  df-itv 28461  df-lng 28462  df-trkge 28477  df-ee 28924  df-btwn 28925  df-eeng 29011

This theorem is referenced by: (None)