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Theorem eengtrkge 27078

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 6732	. 2 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ V)
2		simpll 767	. . . . . 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 771	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (Base‘(EEG‘𝑁)))
4		eengbas 27072	. . . . . . . . 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 2841	. . . . . . 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 773	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (Base‘(EEG‘𝑁)))
9	8, 5	eleqtrrd 2841	. . . . . . 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 1196	. . . . . . 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 1198	. . . . . . . 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 2841	. . . . . . 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 1374	. . . . . 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 1197	. . . . . . 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 726	. . . . . . 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 2841	. . . . . 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 1198	. . . . . . 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 2841	. . . . . 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 27054	. . . . . 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 1390	. . . . 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 2737	. . . . . . 7 ⊢ (Base‘(EEG‘𝑁)) = (Base‘(EEG‘𝑁))
26		eqid 2737	. . . . . . 7 ⊢ (Itv‘(EEG‘𝑁)) = (Itv‘(EEG‘𝑁))
27	2, 25, 26, 11, 21, 18	ebtwntg 27073	. . . . . 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 27073	. . . . . 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 1439	. . . . 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 726	. . . . . . . . 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 726	. . . . . . . . 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 2840	. . . . . . . . . 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 726	. . . . . . . . 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 27073	. . . . . . . 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 726	. . . . . . . . . 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 2840	. . . . . . . . 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 726	. . . . . . . . 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 27073	. . . . . . . 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 726	. . . . . . . . 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 27073	. . . . . . . 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 1438	. . . . . . 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 3332	. . . . . 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 3332	. . . . 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 296	. . . 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 3113	. . 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 3112	. 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 2737	. . 3 ⊢ (dist‘(EEG‘𝑁)) = (dist‘(EEG‘𝑁))
52	25, 51, 26	istrkge 26548	. 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 586	1 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiG_E)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 ∀wral 3061 ∃wrex 3062 Vcvv 3408 ⟨cop 4547 class class class wbr 5053 ‘cfv 6380 (class class class)co 7213 ℕcn 11830 Basecbs 16760 distcds 16811 TarskiG_Ecstrkge 26526 Itvcitv 26527 𝔼cee 26979 Btwn cbtwn 26980 EEGceeng 27068

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-icc 12942 df-fz 13096 df-seq 13575 df-sum 15250 df-struct 16700 df-slot 16735 df-ndx 16745 df-base 16761 df-ds 16824 df-itv 26529 df-lng 26530 df-trkge 26542 df-ee 26982 df-btwn 26983 df-eeng 27069

This theorem is referenced by: (None)

W3C validator