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Theorem brsegle2 34148
Description: Alternate characterization of segment comparison. Theorem 5.5 of [Schwabhauser] p. 41-42. (Contributed by Scott Fenton, 11-Oct-2013.)
Assertion
Ref Expression
brsegle2 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (⟨𝐴, 𝐵⟩ Seg𝐶, 𝐷⟩ ↔ ∃𝑥 ∈ (𝔼‘𝑁)(𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)))
Distinct variable groups:   𝑥,𝑁   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐷

Proof of Theorem brsegle2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 brsegle 34147 . 2 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (⟨𝐴, 𝐵⟩ Seg𝐶, 𝐷⟩ ↔ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
2 simprl 771 . . . . . . 7 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)) → 𝑦 Btwn ⟨𝐶, 𝐷⟩)
3 simpl1 1193 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
4 simpl3l 1230 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝐶 ∈ (𝔼‘𝑁))
5 simpl3r 1231 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝐷 ∈ (𝔼‘𝑁))
6 simpr 488 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝑦 ∈ (𝔼‘𝑁))
7 btwncolinear2 34109 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁))) → (𝑦 Btwn ⟨𝐶, 𝐷⟩ → 𝐶 Colinear ⟨𝑦, 𝐷⟩))
83, 4, 5, 6, 7syl13anc 1374 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → (𝑦 Btwn ⟨𝐶, 𝐷⟩ → 𝐶 Colinear ⟨𝑦, 𝐷⟩))
98adantr 484 . . . . . . 7 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)) → (𝑦 Btwn ⟨𝐶, 𝐷⟩ → 𝐶 Colinear ⟨𝑦, 𝐷⟩))
102, 9mpd 15 . . . . . 6 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)) → 𝐶 Colinear ⟨𝑦, 𝐷⟩)
11 simpl2l 1228 . . . . . . 7 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝐴 ∈ (𝔼‘𝑁))
12 simpl2r 1229 . . . . . . 7 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝐵 ∈ (𝔼‘𝑁))
13 simprr 773 . . . . . . 7 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)) → ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)
143, 11, 12, 4, 6, 13cgrcomand 34030 . . . . . 6 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)) → ⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩)
15 simpl2 1194 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)))
16 lineext 34115 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → ((𝐶 Colinear ⟨𝑦, 𝐷⟩ ∧ ⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩) → ∃𝑥 ∈ (𝔼‘𝑁)⟨𝐶, ⟨𝑦, 𝐷⟩⟩Cgr3⟨𝐴, ⟨𝐵, 𝑥⟩⟩))
173, 4, 6, 5, 15, 16syl131anc 1385 . . . . . . 7 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → ((𝐶 Colinear ⟨𝑦, 𝐷⟩ ∧ ⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩) → ∃𝑥 ∈ (𝔼‘𝑁)⟨𝐶, ⟨𝑦, 𝐷⟩⟩Cgr3⟨𝐴, ⟨𝐵, 𝑥⟩⟩))
1817adantr 484 . . . . . 6 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)) → ((𝐶 Colinear ⟨𝑦, 𝐷⟩ ∧ ⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩) → ∃𝑥 ∈ (𝔼‘𝑁)⟨𝐶, ⟨𝑦, 𝐷⟩⟩Cgr3⟨𝐴, ⟨𝐵, 𝑥⟩⟩))
1910, 14, 18mp2and 699 . . . . 5 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)) → ∃𝑥 ∈ (𝔼‘𝑁)⟨𝐶, ⟨𝑦, 𝐷⟩⟩Cgr3⟨𝐴, ⟨𝐵, 𝑥⟩⟩)
20 an32 646 . . . . . . . 8 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ↔ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)))
21 simpll1 1214 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
22 simpl3l 1230 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝐶 ∈ (𝔼‘𝑁))
2322adantr 484 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝐶 ∈ (𝔼‘𝑁))
24 simpr 488 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝑦 ∈ (𝔼‘𝑁))
25 simpl3r 1231 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝐷 ∈ (𝔼‘𝑁))
2625adantr 484 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝐷 ∈ (𝔼‘𝑁))
27 simpl2l 1228 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝐴 ∈ (𝔼‘𝑁))
2827adantr 484 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝐴 ∈ (𝔼‘𝑁))
29 simpl2r 1229 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝐵 ∈ (𝔼‘𝑁))
3029adantr 484 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝐵 ∈ (𝔼‘𝑁))
31 simplr 769 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝑥 ∈ (𝔼‘𝑁))
32 brcgr3 34085 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁))) → (⟨𝐶, ⟨𝑦, 𝐷⟩⟩Cgr3⟨𝐴, ⟨𝐵, 𝑥⟩⟩ ↔ (⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐴, 𝑥⟩ ∧ ⟨𝑦, 𝐷⟩Cgr⟨𝐵, 𝑥⟩)))
3321, 23, 24, 26, 28, 30, 31, 32syl133anc 1395 . . . . . . . . . 10 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) → (⟨𝐶, ⟨𝑦, 𝐷⟩⟩Cgr3⟨𝐴, ⟨𝐵, 𝑥⟩⟩ ↔ (⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐴, 𝑥⟩ ∧ ⟨𝑦, 𝐷⟩Cgr⟨𝐵, 𝑥⟩)))
3433adantr 484 . . . . . . . . 9 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)) → (⟨𝐶, ⟨𝑦, 𝐷⟩⟩Cgr3⟨𝐴, ⟨𝐵, 𝑥⟩⟩ ↔ (⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐴, 𝑥⟩ ∧ ⟨𝑦, 𝐷⟩Cgr⟨𝐵, 𝑥⟩)))
35 simp2l 1201 . . . . . . . . . . . 12 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐴, 𝑥⟩ ∧ ⟨𝑦, 𝐷⟩Cgr⟨𝐵, 𝑥⟩)) → 𝑦 Btwn ⟨𝐶, 𝐷⟩)
36 simp3 1140 . . . . . . . . . . . . 13 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐴, 𝑥⟩ ∧ ⟨𝑦, 𝐷⟩Cgr⟨𝐵, 𝑥⟩)) → (⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐴, 𝑥⟩ ∧ ⟨𝑦, 𝐷⟩Cgr⟨𝐵, 𝑥⟩))
37333ad2ant1 1135 . . . . . . . . . . . . 13 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐴, 𝑥⟩ ∧ ⟨𝑦, 𝐷⟩Cgr⟨𝐵, 𝑥⟩)) → (⟨𝐶, ⟨𝑦, 𝐷⟩⟩Cgr3⟨𝐴, ⟨𝐵, 𝑥⟩⟩ ↔ (⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐴, 𝑥⟩ ∧ ⟨𝑦, 𝐷⟩Cgr⟨𝐵, 𝑥⟩)))
3836, 37mpbird 260 . . . . . . . . . . . 12 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐴, 𝑥⟩ ∧ ⟨𝑦, 𝐷⟩Cgr⟨𝐵, 𝑥⟩)) → ⟨𝐶, ⟨𝑦, 𝐷⟩⟩Cgr3⟨𝐴, ⟨𝐵, 𝑥⟩⟩)
39 btwnxfr 34095 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁))) → ((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐶, ⟨𝑦, 𝐷⟩⟩Cgr3⟨𝐴, ⟨𝐵, 𝑥⟩⟩) → 𝐵 Btwn ⟨𝐴, 𝑥⟩))
4021, 23, 24, 26, 28, 30, 31, 39syl133anc 1395 . . . . . . . . . . . . 13 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) → ((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐶, ⟨𝑦, 𝐷⟩⟩Cgr3⟨𝐴, ⟨𝐵, 𝑥⟩⟩) → 𝐵 Btwn ⟨𝐴, 𝑥⟩))
41403ad2ant1 1135 . . . . . . . . . . . 12 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐴, 𝑥⟩ ∧ ⟨𝑦, 𝐷⟩Cgr⟨𝐵, 𝑥⟩)) → ((𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐶, ⟨𝑦, 𝐷⟩⟩Cgr3⟨𝐴, ⟨𝐵, 𝑥⟩⟩) → 𝐵 Btwn ⟨𝐴, 𝑥⟩))
4235, 38, 41mp2and 699 . . . . . . . . . . 11 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐴, 𝑥⟩ ∧ ⟨𝑦, 𝐷⟩Cgr⟨𝐵, 𝑥⟩)) → 𝐵 Btwn ⟨𝐴, 𝑥⟩)
43 simp32 1212 . . . . . . . . . . . 12 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐴, 𝑥⟩ ∧ ⟨𝑦, 𝐷⟩Cgr⟨𝐵, 𝑥⟩)) → ⟨𝐶, 𝐷⟩Cgr⟨𝐴, 𝑥⟩)
44 cgrcom 34029 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁))) → (⟨𝐶, 𝐷⟩Cgr⟨𝐴, 𝑥⟩ ↔ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩))
4521, 23, 26, 28, 31, 44syl122anc 1381 . . . . . . . . . . . . 13 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) → (⟨𝐶, 𝐷⟩Cgr⟨𝐴, 𝑥⟩ ↔ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩))
46453ad2ant1 1135 . . . . . . . . . . . 12 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐴, 𝑥⟩ ∧ ⟨𝑦, 𝐷⟩Cgr⟨𝐵, 𝑥⟩)) → (⟨𝐶, 𝐷⟩Cgr⟨𝐴, 𝑥⟩ ↔ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩))
4743, 46mpbid 235 . . . . . . . . . . 11 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐴, 𝑥⟩ ∧ ⟨𝑦, 𝐷⟩Cgr⟨𝐵, 𝑥⟩)) → ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)
4842, 47jca 515 . . . . . . . . . 10 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ∧ (⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐴, 𝑥⟩ ∧ ⟨𝑦, 𝐷⟩Cgr⟨𝐵, 𝑥⟩)) → (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩))
49483expia 1123 . . . . . . . . 9 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)) → ((⟨𝐶, 𝑦⟩Cgr⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐴, 𝑥⟩ ∧ ⟨𝑦, 𝐷⟩Cgr⟨𝐵, 𝑥⟩) → (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)))
5034, 49sylbid 243 . . . . . . . 8 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)) → (⟨𝐶, ⟨𝑦, 𝐷⟩⟩Cgr3⟨𝐴, ⟨𝐵, 𝑥⟩⟩ → (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)))
5120, 50sylanb 584 . . . . . . 7 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)) → (⟨𝐶, ⟨𝑦, 𝐷⟩⟩Cgr3⟨𝐴, ⟨𝐵, 𝑥⟩⟩ → (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)))
5251an32s 652 . . . . . 6 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (⟨𝐶, ⟨𝑦, 𝐷⟩⟩Cgr3⟨𝐴, ⟨𝐵, 𝑥⟩⟩ → (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)))
5352reximdva 3193 . . . . 5 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)) → (∃𝑥 ∈ (𝔼‘𝑁)⟨𝐶, ⟨𝑦, 𝐷⟩⟩Cgr3⟨𝐴, ⟨𝐵, 𝑥⟩⟩ → ∃𝑥 ∈ (𝔼‘𝑁)(𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)))
5419, 53mpd 15 . . . 4 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)) → ∃𝑥 ∈ (𝔼‘𝑁)(𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩))
5554rexlimdva2 3206 . . 3 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) → ∃𝑥 ∈ (𝔼‘𝑁)(𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)))
56 simprl 771 . . . . . . 7 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)) → 𝐵 Btwn ⟨𝐴, 𝑥⟩)
57 simpll1 1214 . . . . . . . 8 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)) → 𝑁 ∈ ℕ)
5827adantr 484 . . . . . . . 8 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)) → 𝐴 ∈ (𝔼‘𝑁))
59 simplr 769 . . . . . . . 8 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)) → 𝑥 ∈ (𝔼‘𝑁))
6029adantr 484 . . . . . . . 8 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)) → 𝐵 ∈ (𝔼‘𝑁))
61 btwncolinear1 34108 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐵 Btwn ⟨𝐴, 𝑥⟩ → 𝐴 Colinear ⟨𝑥, 𝐵⟩))
6257, 58, 59, 60, 61syl13anc 1374 . . . . . . 7 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)) → (𝐵 Btwn ⟨𝐴, 𝑥⟩ → 𝐴 Colinear ⟨𝑥, 𝐵⟩))
6356, 62mpd 15 . . . . . 6 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)) → 𝐴 Colinear ⟨𝑥, 𝐵⟩)
64 simprr 773 . . . . . 6 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)) → ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)
65 simpl1 1193 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
66 simpr 488 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑥 ∈ (𝔼‘𝑁))
67 simpl3 1195 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)))
68 lineext 34115 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((𝐴 Colinear ⟨𝑥, 𝐵⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩) → ∃𝑦 ∈ (𝔼‘𝑁)⟨𝐴, ⟨𝑥, 𝐵⟩⟩Cgr3⟨𝐶, ⟨𝐷, 𝑦⟩⟩))
6965, 27, 66, 29, 67, 68syl131anc 1385 . . . . . . 7 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) → ((𝐴 Colinear ⟨𝑥, 𝐵⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩) → ∃𝑦 ∈ (𝔼‘𝑁)⟨𝐴, ⟨𝑥, 𝐵⟩⟩Cgr3⟨𝐶, ⟨𝐷, 𝑦⟩⟩))
7069adantr 484 . . . . . 6 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)) → ((𝐴 Colinear ⟨𝑥, 𝐵⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩) → ∃𝑦 ∈ (𝔼‘𝑁)⟨𝐴, ⟨𝑥, 𝐵⟩⟩Cgr3⟨𝐶, ⟨𝐷, 𝑦⟩⟩))
7163, 64, 70mp2and 699 . . . . 5 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)) → ∃𝑦 ∈ (𝔼‘𝑁)⟨𝐴, ⟨𝑥, 𝐵⟩⟩Cgr3⟨𝐶, ⟨𝐷, 𝑦⟩⟩)
7227, 66, 293jca 1130 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)))
7372adantr 484 . . . . . . . . . 10 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) → (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)))
74 brcgr3 34085 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁))) → (⟨𝐴, ⟨𝑥, 𝐵⟩⟩Cgr3⟨𝐶, ⟨𝐷, 𝑦⟩⟩ ↔ (⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩ ∧ ⟨𝑥, 𝐵⟩Cgr⟨𝐷, 𝑦⟩)))
7521, 73, 23, 26, 24, 74syl113anc 1384 . . . . . . . . 9 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) → (⟨𝐴, ⟨𝑥, 𝐵⟩⟩Cgr3⟨𝐶, ⟨𝐷, 𝑦⟩⟩ ↔ (⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩ ∧ ⟨𝑥, 𝐵⟩Cgr⟨𝐷, 𝑦⟩)))
7675adantr 484 . . . . . . . 8 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)) → (⟨𝐴, ⟨𝑥, 𝐵⟩⟩Cgr3⟨𝐶, ⟨𝐷, 𝑦⟩⟩ ↔ (⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩ ∧ ⟨𝑥, 𝐵⟩Cgr⟨𝐷, 𝑦⟩)))
77 simp2l 1201 . . . . . . . . . . 11 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩) ∧ (⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩ ∧ ⟨𝑥, 𝐵⟩Cgr⟨𝐷, 𝑦⟩)) → 𝐵 Btwn ⟨𝐴, 𝑥⟩)
78 simp32 1212 . . . . . . . . . . . . 13 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩) ∧ (⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩ ∧ ⟨𝑥, 𝐵⟩Cgr⟨𝐷, 𝑦⟩)) → ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)
79 simp2r 1202 . . . . . . . . . . . . 13 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩) ∧ (⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩ ∧ ⟨𝑥, 𝐵⟩Cgr⟨𝐷, 𝑦⟩)) → ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)
80 simp33 1213 . . . . . . . . . . . . . 14 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩) ∧ (⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩ ∧ ⟨𝑥, 𝐵⟩Cgr⟨𝐷, 𝑦⟩)) → ⟨𝑥, 𝐵⟩Cgr⟨𝐷, 𝑦⟩)
81 cgrcomlr 34037 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁))) → (⟨𝑥, 𝐵⟩Cgr⟨𝐷, 𝑦⟩ ↔ ⟨𝐵, 𝑥⟩Cgr⟨𝑦, 𝐷⟩))
8221, 31, 30, 26, 24, 81syl122anc 1381 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) → (⟨𝑥, 𝐵⟩Cgr⟨𝐷, 𝑦⟩ ↔ ⟨𝐵, 𝑥⟩Cgr⟨𝑦, 𝐷⟩))
83823ad2ant1 1135 . . . . . . . . . . . . . 14 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩) ∧ (⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩ ∧ ⟨𝑥, 𝐵⟩Cgr⟨𝐷, 𝑦⟩)) → (⟨𝑥, 𝐵⟩Cgr⟨𝐷, 𝑦⟩ ↔ ⟨𝐵, 𝑥⟩Cgr⟨𝑦, 𝐷⟩))
8480, 83mpbid 235 . . . . . . . . . . . . 13 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩) ∧ (⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩ ∧ ⟨𝑥, 𝐵⟩Cgr⟨𝐷, 𝑦⟩)) → ⟨𝐵, 𝑥⟩Cgr⟨𝑦, 𝐷⟩)
8578, 79, 843jca 1130 . . . . . . . . . . . 12 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩) ∧ (⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩ ∧ ⟨𝑥, 𝐵⟩Cgr⟨𝐷, 𝑦⟩)) → (⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩ ∧ ⟨𝐵, 𝑥⟩Cgr⟨𝑦, 𝐷⟩))
86 brcgr3 34085 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (⟨𝐴, ⟨𝐵, 𝑥⟩⟩Cgr3⟨𝐶, ⟨𝑦, 𝐷⟩⟩ ↔ (⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩ ∧ ⟨𝐵, 𝑥⟩Cgr⟨𝑦, 𝐷⟩)))
8721, 28, 30, 31, 23, 24, 26, 86syl133anc 1395 . . . . . . . . . . . . 13 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) → (⟨𝐴, ⟨𝐵, 𝑥⟩⟩Cgr3⟨𝐶, ⟨𝑦, 𝐷⟩⟩ ↔ (⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩ ∧ ⟨𝐵, 𝑥⟩Cgr⟨𝑦, 𝐷⟩)))
88873ad2ant1 1135 . . . . . . . . . . . 12 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩) ∧ (⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩ ∧ ⟨𝑥, 𝐵⟩Cgr⟨𝐷, 𝑦⟩)) → (⟨𝐴, ⟨𝐵, 𝑥⟩⟩Cgr3⟨𝐶, ⟨𝑦, 𝐷⟩⟩ ↔ (⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩ ∧ ⟨𝐵, 𝑥⟩Cgr⟨𝑦, 𝐷⟩)))
8985, 88mpbird 260 . . . . . . . . . . 11 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩) ∧ (⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩ ∧ ⟨𝑥, 𝐵⟩Cgr⟨𝐷, 𝑦⟩)) → ⟨𝐴, ⟨𝐵, 𝑥⟩⟩Cgr3⟨𝐶, ⟨𝑦, 𝐷⟩⟩)
90 btwnxfr 34095 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, ⟨𝐵, 𝑥⟩⟩Cgr3⟨𝐶, ⟨𝑦, 𝐷⟩⟩) → 𝑦 Btwn ⟨𝐶, 𝐷⟩))
9121, 28, 30, 31, 23, 24, 26, 90syl133anc 1395 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) → ((𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, ⟨𝐵, 𝑥⟩⟩Cgr3⟨𝐶, ⟨𝑦, 𝐷⟩⟩) → 𝑦 Btwn ⟨𝐶, 𝐷⟩))
92913ad2ant1 1135 . . . . . . . . . . 11 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩) ∧ (⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩ ∧ ⟨𝑥, 𝐵⟩Cgr⟨𝐷, 𝑦⟩)) → ((𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, ⟨𝐵, 𝑥⟩⟩Cgr3⟨𝐶, ⟨𝑦, 𝐷⟩⟩) → 𝑦 Btwn ⟨𝐶, 𝐷⟩))
9377, 89, 92mp2and 699 . . . . . . . . . 10 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩) ∧ (⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩ ∧ ⟨𝑥, 𝐵⟩Cgr⟨𝐷, 𝑦⟩)) → 𝑦 Btwn ⟨𝐶, 𝐷⟩)
9493, 78jca 515 . . . . . . . . 9 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩) ∧ (⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩ ∧ ⟨𝑥, 𝐵⟩Cgr⟨𝐷, 𝑦⟩)) → (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩))
95943expia 1123 . . . . . . . 8 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)) → ((⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩ ∧ ⟨𝑥, 𝐵⟩Cgr⟨𝐷, 𝑦⟩) → (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
9676, 95sylbid 243 . . . . . . 7 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)) → (⟨𝐴, ⟨𝑥, 𝐵⟩⟩Cgr3⟨𝐶, ⟨𝐷, 𝑦⟩⟩ → (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
9796an32s 652 . . . . . 6 (((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)) ∧ 𝑦 ∈ (𝔼‘𝑁)) → (⟨𝐴, ⟨𝑥, 𝐵⟩⟩Cgr3⟨𝐶, ⟨𝐷, 𝑦⟩⟩ → (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
9897reximdva 3193 . . . . 5 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)) → (∃𝑦 ∈ (𝔼‘𝑁)⟨𝐴, ⟨𝑥, 𝐵⟩⟩Cgr3⟨𝐶, ⟨𝐷, 𝑦⟩⟩ → ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
9971, 98mpd 15 . . . 4 ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)) → ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩))
10099rexlimdva2 3206 . . 3 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (∃𝑥 ∈ (𝔼‘𝑁)(𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩) → ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
10155, 100impbid 215 . 2 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ↔ ∃𝑥 ∈ (𝔼‘𝑁)(𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)))
1021, 101bitrd 282 1 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (⟨𝐴, 𝐵⟩ Seg𝐶, 𝐷⟩ ↔ ∃𝑥 ∈ (𝔼‘𝑁)(𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089  wcel 2110  wrex 3062  cop 4547   class class class wbr 5053  cfv 6380  cn 11830  𝔼cee 26979   Btwn cbtwn 26980  Cgrccgr 26981  Cgr3ccgr3 34075   Colinear ccolin 34076   Seg csegle 34145
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-inf2 9256  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  ax-pre-sup 10807
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-int 4860  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-se 5510  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-isom 6389  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-sup 9058  df-oi 9126  df-card 9555  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-n0 12091  df-z 12177  df-uz 12439  df-rp 12587  df-ico 12941  df-icc 12942  df-fz 13096  df-fzo 13239  df-seq 13575  df-exp 13636  df-hash 13897  df-cj 14662  df-re 14663  df-im 14664  df-sqrt 14798  df-abs 14799  df-clim 15049  df-sum 15250  df-ee 26982  df-btwn 26983  df-cgr 26984  df-ofs 34022  df-colinear 34078  df-ifs 34079  df-cgr3 34080  df-segle 34146
This theorem is referenced by:  segleantisym  34154  seglelin  34155  outsidele  34171
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