brsegle2 - Mathbox for Scott Fenton

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Theorem brsegle2 34694

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 34693	. 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 769	. . . . . . 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 1191	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
4		simpl3l 1228	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝐶 ∈ (𝔼‘𝑁))
5		simpl3r 1229	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝐷 ∈ (𝔼‘𝑁))
6		simpr 485	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝑦 ∈ (𝔼‘𝑁))
7		btwncolinear2 34655	. . . . . . . . 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 1372	. . . . . . . 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 481	. . . . . . 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 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 1226	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝐴 ∈ (𝔼‘𝑁))
12		simpl2r 1227	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝐵 ∈ (𝔼‘𝑁))
13		simprr 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⟩)) → ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)
14	3, 11, 12, 4, 6, 13	cgrcomand 34576	. . . . . 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 1192	. . . . . . . 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 34661	. . . . . . . 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 1383	. . . . . . 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 481	. . . . . 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 697	. . . . 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 644	. . . . . . . 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 1212	. . . . . . . . . . 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 1228	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝐶 ∈ (𝔼‘𝑁))
23	22	adantr 481	. . . . . . . . . . 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 485	. . . . . . . . . . 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 1229	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝐷 ∈ (𝔼‘𝑁))
26	25	adantr 481	. . . . . . . . . . 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 1226	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝐴 ∈ (𝔼‘𝑁))
28	27	adantr 481	. . . . . . . . . . 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 1227	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝐵 ∈ (𝔼‘𝑁))
30	29	adantr 481	. . . . . . . . . . 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 767	. . . . . . . . . . 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 34631	. . . . . . . . . . 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 1393	. . . . . . . . . 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 481	. . . . . . . . 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 1199	. . . . . . . . . . . 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 1138	. . . . . . . . . . . . 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 1133	. . . . . . . . . . . . 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 256	. . . . . . . . . . . 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 34641	. . . . . . . . . . . . . 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 1393	. . . . . . . . . . . . 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 1133	. . . . . . . . . . . 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 697	. . . . . . . . . . 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 1210	. . . . . . . . . . . 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 34575	. . . . . . . . . . . . . 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 1379	. . . . . . . . . . . . 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 1133	. . . . . . . . . . . 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 231	. . . . . . . . . . 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 512	. . . . . . . . . 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 1121	. . . . . . . . 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 239	. . . . . . . 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 581	. . . . . . 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 650	. . . . . 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 3165	. . . . 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 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⟩))
55	54	rexlimdva2 3154	. . 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 769	. . . . . . 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 1212	. . . . . . . 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 481	. . . . . . . 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 767	. . . . . . . 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 481	. . . . . . . 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 34654	. . . . . . . 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 1372	. . . . . . 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 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 771	. . . . . 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 1191	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
66		simpr 485	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑥 ∈ (𝔼‘𝑁))
67		simpl3 1193	. . . . . . . 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 34661	. . . . . . . 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 1383	. . . . . . 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 481	. . . . . 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 697	. . . . 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 1128	. . . . . . . . . . 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 481	. . . . . . . . . 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 34631	. . . . . . . . . 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 1382	. . . . . . . . 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 481	. . . . . . . 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 1199	. . . . . . . . . . 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 1210	. . . . . . . . . . . . 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 1200	. . . . . . . . . . . . 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 1211	. . . . . . . . . . . . . 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 34583	. . . . . . . . . . . . . . . 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 1379	. . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . 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 231	. . . . . . . . . . . . 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 1128	. . . . . . . . . . . 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 34631	. . . . . . . . . . . . . 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 1393	. . . . . . . . . . . . 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 1133	. . . . . . . . . . . 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 256	. . . . . . . . . . 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 34641	. . . . . . . . . . . . 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 1393	. . . . . . . . . . . 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 1133	. . . . . . . . . . 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 697	. . . . . . . . . 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 512	. . . . . . . . 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 1121	. . . . . . . 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 239	. . . . . . 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 650	. . . . . 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 3165	. . . . 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 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⟩))
100	99	rexlimdva2 3154	. . 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 211	. 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 278	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 205 ∧ wa 396 ∧ w3a 1087 ∈ wcel 2106 ∃wrex 3073 ⟨cop 4592 class class class wbr 5105 ‘cfv 6496 ℕcn 12153 𝔼cee 27837 Btwn cbtwn 27838 Cgrccgr 27839 Cgr3ccgr3 34621 Colinear ccolin 34622 Seg_≤ csegle 34691

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-z 12500 df-uz 12764 df-rp 12916 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-sum 15571 df-ee 27840 df-btwn 27841 df-cgr 27842 df-ofs 34568 df-colinear 34624 df-ifs 34625 df-cgr3 34626 df-segle 34692

This theorem is referenced by: segleantisym 34700 seglelin 34701 outsidele 34717

W3C validator