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Theorem eengtrkge 25580
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.
StepHypRef Expression
1 fvex 6094 . . 3 (EEG‘𝑁) ∈ V
21a1i 11 . 2 (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ V)
3 simpl 471 . . . . . . 7 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑁 ∈ ℕ)
43adantr 479 . . . . . 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)
5 simprl 789 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (Base‘(EEG‘𝑁)))
6 eengbas 25575 . . . . . . . . 9 (𝑁 ∈ ℕ → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
76adantr 479 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
85, 7eleqtrrd 2686 . . . . . . 7 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (𝔼‘𝑁))
98adantr 479 . . . . . 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))
10 simprr 791 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (Base‘(EEG‘𝑁)))
1110, 7eleqtrrd 2686 . . . . . . 7 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (𝔼‘𝑁))
1211adantr 479 . . . . . 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))
135adantr 479 . . . . . . 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)))
1410adantr 479 . . . . . . 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)))
15 simpr1 1059 . . . . . . 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)))
16 simpr3 1061 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → 𝑧 ∈ (Base‘(EEG‘𝑁)))
176adantr 479 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
1816, 17eleqtrrd 2686 . . . . . . 7 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → 𝑧 ∈ (𝔼‘𝑁))
194, 13, 14, 15, 18syl13anc 1319 . . . . . 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))
20 simpr2 1060 . . . . . . 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)))
216ad2antrr 757 . . . . . . 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)))
2220, 21eleqtrrd 2686 . . . . . 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))
23 simpr3 1061 . . . . . . 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)))
2423, 21eleqtrrd 2686 . . . . . 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))
25 axeuclid 25557 . . . . . 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⟩ ∧ 𝑥𝑢) → ∃𝑎 ∈ (𝔼‘𝑁)∃𝑏 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑥, 𝑎⟩ ∧ 𝑧 Btwn ⟨𝑥, 𝑏⟩ ∧ 𝑣 Btwn ⟨𝑎, 𝑏⟩)))
264, 9, 12, 19, 22, 24, 25syl132anc 1335 . . . . 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⟩ ∧ 𝑥𝑢) → ∃𝑎 ∈ (𝔼‘𝑁)∃𝑏 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑥, 𝑎⟩ ∧ 𝑧 Btwn ⟨𝑥, 𝑏⟩ ∧ 𝑣 Btwn ⟨𝑎, 𝑏⟩)))
27 eqid 2605 . . . . . . 7 (Base‘(EEG‘𝑁)) = (Base‘(EEG‘𝑁))
28 eqid 2605 . . . . . . 7 (Itv‘(EEG‘𝑁)) = (Itv‘(EEG‘𝑁))
294, 27, 28, 13, 23, 20ebtwntg 25576 . . . . . 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)))
304, 27, 28, 14, 15, 20ebtwntg 25576 . . . . . 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)))
3129, 303anbi12d 1391 . . . . 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⟩ ∧ 𝑥𝑢) ↔ (𝑢 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑣) ∧ 𝑢 ∈ (𝑦(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑥𝑢)))
3221adantr 479 . . . . . . 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)))
334ad2antrr 757 . . . . . . . . 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)
3413ad2antrr 757 . . . . . . . . 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)))
35 simpr 475 . . . . . . . . . . 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))
3635, 32eleqtrd 2685 . . . . . . . . . 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)))
3736adantr 479 . . . . . . . . 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)))
3814ad2antrr 757 . . . . . . . . 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)))
3933, 27, 28, 34, 37, 38ebtwntg 25576 . . . . . . . 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)))
40 simpr 475 . . . . . . . . . 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))
4121ad2antrr 757 . . . . . . . . . 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)))
4240, 41eleqtrd 2685 . . . . . . . . 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)))
4315ad2antrr 757 . . . . . . . . 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)))
4433, 27, 28, 34, 42, 43ebtwntg 25576 . . . . . . . 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)))
4523ad2antrr 757 . . . . . . . . 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)))
4633, 27, 28, 37, 42, 45ebtwntg 25576 . . . . . . . 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)))
4739, 44, 463anbi123d 1390 . . . . . . 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))))
4832, 47rexeqbidva 3127 . . . . . 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))))
4921, 48rexeqbidva 3127 . . . . 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))))
5026, 31, 493imtr3d 280 . . . 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) ∧ 𝑥𝑢) → ∃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))))
5150ralrimivvva 2950 . . 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) ∧ 𝑥𝑢) → ∃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))))
5251ralrimivva 2949 . 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) ∧ 𝑥𝑢) → ∃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 eqid 2605 . . 3 (dist‘(EEG‘𝑁)) = (dist‘(EEG‘𝑁))
5427, 53, 28istrkge 25069 . 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) ∧ 𝑥𝑢) → ∃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)))))
552, 52, 54sylanbrc 694 1 (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiGE)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1975  wne 2775  wral 2891  wrex 2892  Vcvv 3168  cop 4126   class class class wbr 4573  cfv 5786  (class class class)co 6523  cn 10863  Basecbs 15637  distcds 15719  TarskiGEcstrkge 25047  Itvcitv 25048  𝔼cee 25482   Btwn cbtwn 25483  EEGceeng 25571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-oadd 7424  df-er 7602  df-map 7719  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-div 10530  df-nn 10864  df-2 10922  df-3 10923  df-4 10924  df-5 10925  df-6 10926  df-7 10927  df-8 10928  df-9 10929  df-n0 11136  df-z 11207  df-dec 11322  df-uz 11516  df-icc 12005  df-fz 12149  df-seq 12615  df-sum 14207  df-struct 15639  df-ndx 15640  df-slot 15641  df-base 15642  df-ds 15733  df-itv 25050  df-lng 25051  df-trkge 25063  df-ee 25485  df-btwn 25486  df-eeng 25572
This theorem is referenced by: (None)
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