brsegle2 - Mathbox for Scott Fenton

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Theorem brsegle2 36622

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⟩ Seg_≤ ⟨𝐶, 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.

Step	Hyp	Ref	Expression
1		brsegle 36621	. 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (⟨𝐴, 𝐵⟩ Seg_≤ ⟨𝐶, 𝐷⟩ ↔ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
2		simprl 783	. . . . . . 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 1210	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
4		simpl3l 1247	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝐶 ∈ (𝔼‘𝑁))
5		simpl3r 1248	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝐷 ∈ (𝔼‘𝑁))
6		simpr 490	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝑦 ∈ (𝔼‘𝑁))
7		btwncolinear2 36583	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁))) → (𝑦 Btwn ⟨𝐶, 𝐷⟩ → 𝐶 Colinear ⟨𝑦, 𝐷⟩))
8	3, 4, 5, 6, 7	syl13anc 1399	. . . . . . . 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⟩))
9	8	adantr 486	. . . . . . 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⟩))
10	2, 9	mpd 16	. . . . . 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 1245	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝐴 ∈ (𝔼‘𝑁))
12		simpl2r 1246	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝐵 ∈ (𝔼‘𝑁))
13		simprr 785	. . . . . . 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⟩)
14	3, 11, 12, 4, 6, 13	cgrcomand 36504	. . . . . 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 1211	. . . . . . . 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 36589	. . . . . . . 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⟩⟩))
17	3, 4, 6, 5, 15, 16	syl131anc 1410	. . . . . . 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⟩⟩))
18	17	adantr 486	. . . . . 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⟩⟩))
19	10, 14, 18	mp2and 712	. . . . 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 659	. . . . . . . 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 1231	. . . . . . . . . . 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 1247	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝐶 ∈ (𝔼‘𝑁))
23	22	adantr 486	. . . . . . . . . . 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 490	. . . . . . . . . . 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 1248	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝐷 ∈ (𝔼‘𝑁))
26	25	adantr 486	. . . . . . . . . . 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 1245	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝐴 ∈ (𝔼‘𝑁))
28	27	adantr 486	. . . . . . . . . . 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 1246	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝐵 ∈ (𝔼‘𝑁))
30	29	adantr 486	. . . . . . . . . . 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 781	. . . . . . . . . . 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 36559	. . . . . . . . . . 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⟩)))
33	21, 23, 24, 26, 28, 30, 31, 32	syl133anc 1420	. . . . . . . . . 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⟩)))
34	33	adantr 486	. . . . . . . . 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 1218	. . . . . . . . . . . 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 1156	. . . . . . . . . . . . 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⟩))
37	33	3ad2ant1 1151	. . . . . . . . . . . . 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⟩)))
38	36, 37	mpbird 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 36569	. . . . . . . . . . . . . 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⟩))
40	21, 23, 24, 26, 28, 30, 31, 39	syl133anc 1420	. . . . . . . . . . . . 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⟩))
41	40	3ad2ant1 1151	. . . . . . . . . . . 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⟩))
42	35, 38, 41	mp2and 712	. . . . . . . . . . 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 1229	. . . . . . . . . . . 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 36503	. . . . . . . . . . . . . 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⟩))
45	21, 23, 26, 28, 31, 44	syl122anc 1406	. . . . . . . . . . . . 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⟩))
46	45	3ad2ant1 1151	. . . . . . . . . . . 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⟩))
47	43, 46	mpbid 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⟩)
48	42, 47	jca 521	. . . . . . . . . 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⟩))
49	48	3expia 1139	. . . . . . . . 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⟩)))
50	34, 49	sylbid 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⟩)))
51	20, 50	sylanb 593	. . . . . . 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⟩)))
52	51	an32s 665	. . . . . 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⟩)))
53	52	reximdva 3181	. . . . 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⟩)))
54	19, 53	mpd 16	. . . 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⟩))
55	54	rexlimdva2 3171	. . 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 783	. . . . . . 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 1231	. . . . . . . 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)
58	27	adantr 486	. . . . . . . 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 781	. . . . . . . 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))
60	29	adantr 486	. . . . . . . 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 36582	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐵 Btwn ⟨𝐴, 𝑥⟩ → 𝐴 Colinear ⟨𝑥, 𝐵⟩))
62	57, 58, 59, 60, 61	syl13anc 1399	. . . . . . 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⟩))
63	56, 62	mpd 16	. . . . . 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 785	. . . . . 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 1210	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
66		simpr 490	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑥 ∈ (𝔼‘𝑁))
67		simpl3 1212	. . . . . . . 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 36589	. . . . . . . 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⟩⟩))
69	65, 27, 66, 29, 67, 68	syl131anc 1410	. . . . . . 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⟩⟩))
70	69	adantr 486	. . . . . 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⟩⟩))
71	63, 64, 70	mp2and 712	. . . . 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⟩⟩)
72	27, 66, 29	3jca 1146	. . . . . . . . . . 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)))
73	72	adantr 486	. . . . . . . . . 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 36559	. . . . . . . . . 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⟩)))
75	21, 73, 23, 26, 24, 74	syl113anc 1409	. . . . . . . . 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⟩)))
76	75	adantr 486	. . . . . . . 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 1218	. . . . . . . . . . 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 1229	. . . . . . . . . . . . 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 1219	. . . . . . . . . . . . 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 1230	. . . . . . . . . . . . . 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 36511	. . . . . . . . . . . . . . . 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⟩))
82	21, 31, 30, 26, 24, 81	syl122anc 1406	. . . . . . . . . . . . . . 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⟩))
83	82	3ad2ant1 1151	. . . . . . . . . . . . . 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⟩))
84	80, 83	mpbid 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⟩)
85	78, 79, 84	3jca 1146	. . . . . . . . . . . 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 36559	. . . . . . . . . . . . . 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⟩)))
87	21, 28, 30, 31, 23, 24, 26, 86	syl133anc 1420	. . . . . . . . . . . . 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⟩)))
88	87	3ad2ant1 1151	. . . . . . . . . . . 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⟩)))
89	85, 88	mpbird 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 36569	. . . . . . . . . . . . 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⟩))
91	21, 28, 30, 31, 23, 24, 26, 90	syl133anc 1420	. . . . . . . . . . . 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⟩))
92	91	3ad2ant1 1151	. . . . . . . . . . 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⟩))
93	77, 89, 92	mp2and 712	. . . . . . . . . 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⟩)
94	93, 78	jca 521	. . . . . . . . 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⟩))
95	94	3expia 1139	. . . . . . . 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⟩)))
96	76, 95	sylbid 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⟩)))
97	96	an32s 665	. . . . . 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⟩)))
98	97	reximdva 3181	. . . . 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⟩)))
99	71, 98	mpd 16	. . . 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⟩))
100	99	rexlimdva2 3171	. . 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⟩)))
101	55, 100	impbid 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⟩)))
102	1, 101	bitrd 282	1 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (⟨𝐴, 𝐵⟩ Seg_≤ ⟨𝐶, 𝐷⟩ ↔ ∃𝑥 ∈ (𝔼‘𝑁)(𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 ∈ wcel 2146 ∃wrex 3092 ⟨cop 4600 class class class wbr 5114 ‘cfv 6543 ℕcn 12251 𝔼cee 29274 Btwn cbtwn 29275 Cgrccgr 29276 Cgr3ccgr3 36549 Colinear ccolin 36550 Seg_≤ csegle 36619

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-inf2 9620 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8703 df-map 8835 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-oi 9482 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-n0 12523 df-z 12610 df-uz 12881 df-rp 13035 df-ico 13396 df-icc 13397 df-fz 13554 df-fzo 13702 df-seq 14058 df-exp 14118 df-hash 14387 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-clim 15565 df-sum 15764 df-ee 29277 df-btwn 29278 df-cgr 29279 df-ofs 36496 df-colinear 36552 df-ifs 36553 df-cgr3 36554 df-segle 36620

This theorem is used by: segleantisym 36628 seglelin 36629 outsidele 36645

W3C validator