brsegle2 - Mathbox for Scott Fenton

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Theorem brsegle2 36472

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 36471	. 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 782	. . . . . . 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 1208	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
4		simpl3l 1245	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝐶 ∈ (𝔼‘𝑁))
5		simpl3r 1246	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝐷 ∈ (𝔼‘𝑁))
6		simpr 489	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝑦 ∈ (𝔼‘𝑁))
7		btwncolinear2 36433	. . . . . . . . 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 1395	. . . . . . . 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 485	. . . . . . 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 1243	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝐴 ∈ (𝔼‘𝑁))
12		simpl2r 1244	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝐵 ∈ (𝔼‘𝑁))
13		simprr 784	. . . . . . 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 36354	. . . . . 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 1209	. . . . . . . 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 36439	. . . . . . . 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 1406	. . . . . . 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 485	. . . . . 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 711	. . . . 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 658	. . . . . . . 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 1229	. . . . . . . . . . 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 1245	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝐶 ∈ (𝔼‘𝑁))
23	22	adantr 485	. . . . . . . . . . 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 489	. . . . . . . . . . 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 1246	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝐷 ∈ (𝔼‘𝑁))
26	25	adantr 485	. . . . . . . . . . 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 1243	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝐴 ∈ (𝔼‘𝑁))
28	27	adantr 485	. . . . . . . . . . 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 1244	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝐵 ∈ (𝔼‘𝑁))
30	29	adantr 485	. . . . . . . . . . 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 780	. . . . . . . . . . 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 36409	. . . . . . . . . . 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 1416	. . . . . . . . . 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 485	. . . . . . . . 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 1216	. . . . . . . . . . . 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 1154	. . . . . . . . . . . . 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 1149	. . . . . . . . . . . . 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 36419	. . . . . . . . . . . . . 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 1416	. . . . . . . . . . . . 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 1149	. . . . . . . . . . . 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 711	. . . . . . . . . . 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 1227	. . . . . . . . . . . 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 36353	. . . . . . . . . . . . . 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 1402	. . . . . . . . . . . . 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 1149	. . . . . . . . . . . 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 520	. . . . . . . . . 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 1137	. . . . . . . . 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 592	. . . . . . 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 664	. . . . . 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 3178	. . . . 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 3168	. . 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 782	. . . . . . 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 1229	. . . . . . . 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 485	. . . . . . . 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 780	. . . . . . . 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 485	. . . . . . . 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 36432	. . . . . . . 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 1395	. . . . . . 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 784	. . . . . 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 1208	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
66		simpr 489	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑥 ∈ (𝔼‘𝑁))
67		simpl3 1210	. . . . . . . 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 36439	. . . . . . . 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 1406	. . . . . . 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 485	. . . . . 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 711	. . . . 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 1144	. . . . . . . . . . 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 485	. . . . . . . . . 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 36409	. . . . . . . . . 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 1405	. . . . . . . . 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 485	. . . . . . . 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 1216	. . . . . . . . . . 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 1227	. . . . . . . . . . . . 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 1217	. . . . . . . . . . . . 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 1228	. . . . . . . . . . . . . 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 36361	. . . . . . . . . . . . . . . 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 1402	. . . . . . . . . . . . . . 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 1149	. . . . . . . . . . . . . 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 1144	. . . . . . . . . . . 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 36409	. . . . . . . . . . . . . 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 1416	. . . . . . . . . . . . 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 1149	. . . . . . . . . . . 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 36419	. . . . . . . . . . . . 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 1416	. . . . . . . . . . . 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 1149	. . . . . . . . . . 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 711	. . . . . . . . . 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 520	. . . . . . . . 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 1137	. . . . . . . 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 664	. . . . . 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 3178	. . . . 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 3168	. . 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

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1101 ∈ wcel 2145 ∃wrex 3089 ⟨cop 4591 class class class wbr 5105 ‘cfv 6525 ℕcn 12224 𝔼cee 29146 Btwn cbtwn 29147 Cgrccgr 29148 Cgr3ccgr3 36399 Colinear ccolin 36400 Seg_≤ csegle 36469

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-inf2 9598 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-se 5606 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-isom 6534 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-er 8682 df-map 8814 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-sup 9390 df-oi 9460 df-card 9913 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12225 df-2 12294 df-3 12295 df-n0 12496 df-z 12583 df-uz 12854 df-rp 13008 df-ico 13369 df-icc 13370 df-fz 13527 df-fzo 13674 df-seq 14029 df-exp 14089 df-hash 14358 df-cj 15140 df-re 15141 df-im 15142 df-sqrt 15276 df-abs 15277 df-clim 15529 df-sum 15728 df-ee 29149 df-btwn 29150 df-cgr 29151 df-ofs 36346 df-colinear 36402 df-ifs 36403 df-cgr3 36404 df-segle 36470

This theorem is referenced by: segleantisym 36478 seglelin 36479 outsidele 36495

W3C validator