Theorem btwnxfr 35333

Description: A condition for extending betweenness to a new set of points based on congruence with another set of points. Theorem 4.6 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 4-Oct-2013.)

Assertion

Ref	Expression
btwnxfr	⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩) → 𝐸 Btwn ⟨𝐷, 𝐹⟩))

Proof of Theorem btwnxfr

Dummy variable 𝑒 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brcgr3 35323	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩ ↔ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝐹⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)))
2		simp2 1136	. . . . . 6 ⊢ ((⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝐹⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩) → ⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝐹⟩)
3	1, 2	syl6bi 253	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩ → ⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝐹⟩))
4		simp1 1135	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → 𝑁 ∈ ℕ)
5		simp21 1205	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → 𝐴 ∈ (𝔼‘𝑁))
6		simp22 1206	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → 𝐵 ∈ (𝔼‘𝑁))
7		simp23 1207	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → 𝐶 ∈ (𝔼‘𝑁))
8		simp31 1208	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → 𝐷 ∈ (𝔼‘𝑁))
9		simp33 1210	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → 𝐹 ∈ (𝔼‘𝑁))
10		cgrxfr 35332	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝐹⟩) → ∃𝑒 ∈ (𝔼‘𝑁)(𝑒 Btwn ⟨𝐷, 𝐹⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩)))
11	4, 5, 6, 7, 8, 9, 10	syl132anc 1387	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝐹⟩) → ∃𝑒 ∈ (𝔼‘𝑁)(𝑒 Btwn ⟨𝐷, 𝐹⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩)))
12	3, 11	sylan2d 604	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩) → ∃𝑒 ∈ (𝔼‘𝑁)(𝑒 Btwn ⟨𝐷, 𝐹⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩)))
13	12	imp 406	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩)) → ∃𝑒 ∈ (𝔼‘𝑁)(𝑒 Btwn ⟨𝐷, 𝐹⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩))
14		simprrl 778	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) ∧ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩) ∧ (𝑒 Btwn ⟨𝐷, 𝐹⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩))) → 𝑒 Btwn ⟨𝐷, 𝐹⟩)
15	14, 14	jca 511	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) ∧ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩) ∧ (𝑒 Btwn ⟨𝐷, 𝐹⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩))) → (𝑒 Btwn ⟨𝐷, 𝐹⟩ ∧ 𝑒 Btwn ⟨𝐷, 𝐹⟩))
16		simpl1 1190	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
17		simpl31 1253	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) → 𝐷 ∈ (𝔼‘𝑁))
18		simpl33 1255	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) → 𝐹 ∈ (𝔼‘𝑁))
19	16, 17, 18	cgrrflxd 35265	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) → ⟨𝐷, 𝐹⟩Cgr⟨𝐷, 𝐹⟩)
20		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) → 𝑒 ∈ (𝔼‘𝑁))
21	16, 20, 18	cgrrflxd 35265	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) → ⟨𝑒, 𝐹⟩Cgr⟨𝑒, 𝐹⟩)
22	19, 21	jca 511	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) → (⟨𝐷, 𝐹⟩Cgr⟨𝐷, 𝐹⟩ ∧ ⟨𝑒, 𝐹⟩Cgr⟨𝑒, 𝐹⟩))
23	22	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) ∧ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩) ∧ (𝑒 Btwn ⟨𝐷, 𝐹⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩))) → (⟨𝐷, 𝐹⟩Cgr⟨𝐷, 𝐹⟩ ∧ ⟨𝑒, 𝐹⟩Cgr⟨𝑒, 𝐹⟩))
24		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩) → ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩)
25		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑒 Btwn ⟨𝐷, 𝐹⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩) → ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩)
26		simpl2 1191	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) → (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)))
27		simpl3 1192	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) → (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁)))
28	17, 20, 18	3jca 1127	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) → (𝐷 ∈ (𝔼‘𝑁) ∧ 𝑒 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁)))
29		cgr3tr4 35329	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝑒 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁)))) → ((⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩) → ⟨𝐷, ⟨𝐸, 𝐹⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩))
30	16, 26, 27, 28, 29	syl13anc 1371	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) → ((⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩) → ⟨𝐷, ⟨𝐸, 𝐹⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩))
31		cgr3com 35330	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝑒 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (⟨𝐷, ⟨𝐸, 𝐹⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩ ↔ ⟨𝐷, ⟨𝑒, 𝐹⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩))
32	16, 27, 17, 20, 18, 31	syl113anc 1381	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) → (⟨𝐷, ⟨𝐸, 𝐹⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩ ↔ ⟨𝐷, ⟨𝑒, 𝐹⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩))
33		simpl32 1254	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) → 𝐸 ∈ (𝔼‘𝑁))
34		brcgr3 35323	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝑒 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (⟨𝐷, ⟨𝑒, 𝐹⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩ ↔ (⟨𝐷, 𝑒⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐷, 𝐹⟩Cgr⟨𝐷, 𝐹⟩ ∧ ⟨𝑒, 𝐹⟩Cgr⟨𝐸, 𝐹⟩)))
35	16, 17, 20, 18, 17, 33, 18, 34	syl133anc 1392	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) → (⟨𝐷, ⟨𝑒, 𝐹⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩ ↔ (⟨𝐷, 𝑒⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐷, 𝐹⟩Cgr⟨𝐷, 𝐹⟩ ∧ ⟨𝑒, 𝐹⟩Cgr⟨𝐸, 𝐹⟩)))
36		simpr1 1193	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) ∧ (⟨𝐷, 𝑒⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐷, 𝐹⟩Cgr⟨𝐷, 𝐹⟩ ∧ ⟨𝑒, 𝐹⟩Cgr⟨𝐸, 𝐹⟩)) → ⟨𝐷, 𝑒⟩Cgr⟨𝐷, 𝐸⟩)
37		simpr3 1195	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) ∧ (⟨𝐷, 𝑒⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐷, 𝐹⟩Cgr⟨𝐷, 𝐹⟩ ∧ ⟨𝑒, 𝐹⟩Cgr⟨𝐸, 𝐹⟩)) → ⟨𝑒, 𝐹⟩Cgr⟨𝐸, 𝐹⟩)
38	16, 20, 18, 33, 18, 37	cgrcomlrand 35278	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) ∧ (⟨𝐷, 𝑒⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐷, 𝐹⟩Cgr⟨𝐷, 𝐹⟩ ∧ ⟨𝑒, 𝐹⟩Cgr⟨𝐸, 𝐹⟩)) → ⟨𝐹, 𝑒⟩Cgr⟨𝐹, 𝐸⟩)
39	36, 38	jca 511	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) ∧ (⟨𝐷, 𝑒⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐷, 𝐹⟩Cgr⟨𝐷, 𝐹⟩ ∧ ⟨𝑒, 𝐹⟩Cgr⟨𝐸, 𝐹⟩)) → (⟨𝐷, 𝑒⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐹, 𝑒⟩Cgr⟨𝐹, 𝐸⟩))
40	39	ex 412	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) → ((⟨𝐷, 𝑒⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐷, 𝐹⟩Cgr⟨𝐷, 𝐹⟩ ∧ ⟨𝑒, 𝐹⟩Cgr⟨𝐸, 𝐹⟩) → (⟨𝐷, 𝑒⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐹, 𝑒⟩Cgr⟨𝐹, 𝐸⟩)))
41	35, 40	sylbid 239	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) → (⟨𝐷, ⟨𝑒, 𝐹⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩ → (⟨𝐷, 𝑒⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐹, 𝑒⟩Cgr⟨𝐹, 𝐸⟩)))
42	32, 41	sylbid 239	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) → (⟨𝐷, ⟨𝐸, 𝐹⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩ → (⟨𝐷, 𝑒⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐹, 𝑒⟩Cgr⟨𝐹, 𝐸⟩)))
43	30, 42	syld 47	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) → ((⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩) → (⟨𝐷, 𝑒⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐹, 𝑒⟩Cgr⟨𝐹, 𝐸⟩)))
44	24, 25, 43	syl2ani 606	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) → (((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩) ∧ (𝑒 Btwn ⟨𝐷, 𝐹⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩)) → (⟨𝐷, 𝑒⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐹, 𝑒⟩Cgr⟨𝐹, 𝐸⟩)))
45	44	imp 406	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) ∧ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩) ∧ (𝑒 Btwn ⟨𝐷, 𝐹⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩))) → (⟨𝐷, 𝑒⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐹, 𝑒⟩Cgr⟨𝐹, 𝐸⟩))
46	15, 23, 45	3jca 1127	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) ∧ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩) ∧ (𝑒 Btwn ⟨𝐷, 𝐹⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩))) → ((𝑒 Btwn ⟨𝐷, 𝐹⟩ ∧ 𝑒 Btwn ⟨𝐷, 𝐹⟩) ∧ (⟨𝐷, 𝐹⟩Cgr⟨𝐷, 𝐹⟩ ∧ ⟨𝑒, 𝐹⟩Cgr⟨𝑒, 𝐹⟩) ∧ (⟨𝐷, 𝑒⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐹, 𝑒⟩Cgr⟨𝐹, 𝐸⟩)))
47	46	ex 412	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) → (((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩) ∧ (𝑒 Btwn ⟨𝐷, 𝐹⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩)) → ((𝑒 Btwn ⟨𝐷, 𝐹⟩ ∧ 𝑒 Btwn ⟨𝐷, 𝐹⟩) ∧ (⟨𝐷, 𝐹⟩Cgr⟨𝐷, 𝐹⟩ ∧ ⟨𝑒, 𝐹⟩Cgr⟨𝑒, 𝐹⟩) ∧ (⟨𝐷, 𝑒⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐹, 𝑒⟩Cgr⟨𝐹, 𝐸⟩))))
48		brifs 35320	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑒 ∈ (𝔼‘𝑁)) ∧ (𝐹 ∈ (𝔼‘𝑁) ∧ 𝑒 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ (𝑒 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) → (⟨⟨𝐷, 𝑒⟩, ⟨𝐹, 𝑒⟩⟩ InnerFiveSeg ⟨⟨𝐷, 𝑒⟩, ⟨𝐹, 𝐸⟩⟩ ↔ ((𝑒 Btwn ⟨𝐷, 𝐹⟩ ∧ 𝑒 Btwn ⟨𝐷, 𝐹⟩) ∧ (⟨𝐷, 𝐹⟩Cgr⟨𝐷, 𝐹⟩ ∧ ⟨𝑒, 𝐹⟩Cgr⟨𝑒, 𝐹⟩) ∧ (⟨𝐷, 𝑒⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐹, 𝑒⟩Cgr⟨𝐹, 𝐸⟩))))
49		ifscgr 35321	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑒 ∈ (𝔼‘𝑁)) ∧ (𝐹 ∈ (𝔼‘𝑁) ∧ 𝑒 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ (𝑒 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) → (⟨⟨𝐷, 𝑒⟩, ⟨𝐹, 𝑒⟩⟩ InnerFiveSeg ⟨⟨𝐷, 𝑒⟩, ⟨𝐹, 𝐸⟩⟩ → ⟨𝑒, 𝑒⟩Cgr⟨𝑒, 𝐸⟩))
50	48, 49	sylbird 260	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑒 ∈ (𝔼‘𝑁)) ∧ (𝐹 ∈ (𝔼‘𝑁) ∧ 𝑒 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ (𝑒 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) → (((𝑒 Btwn ⟨𝐷, 𝐹⟩ ∧ 𝑒 Btwn ⟨𝐷, 𝐹⟩) ∧ (⟨𝐷, 𝐹⟩Cgr⟨𝐷, 𝐹⟩ ∧ ⟨𝑒, 𝐹⟩Cgr⟨𝑒, 𝐹⟩) ∧ (⟨𝐷, 𝑒⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐹, 𝑒⟩Cgr⟨𝐹, 𝐸⟩)) → ⟨𝑒, 𝑒⟩Cgr⟨𝑒, 𝐸⟩))
51	16, 17, 20, 18, 20, 17, 20, 18, 33, 50	syl333anc 1401	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) → (((𝑒 Btwn ⟨𝐷, 𝐹⟩ ∧ 𝑒 Btwn ⟨𝐷, 𝐹⟩) ∧ (⟨𝐷, 𝐹⟩Cgr⟨𝐷, 𝐹⟩ ∧ ⟨𝑒, 𝐹⟩Cgr⟨𝑒, 𝐹⟩) ∧ (⟨𝐷, 𝑒⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐹, 𝑒⟩Cgr⟨𝐹, 𝐸⟩)) → ⟨𝑒, 𝑒⟩Cgr⟨𝑒, 𝐸⟩))
52	47, 51	syld 47	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) → (((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩) ∧ (𝑒 Btwn ⟨𝐷, 𝐹⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩)) → ⟨𝑒, 𝑒⟩Cgr⟨𝑒, 𝐸⟩))
53		cgrid2 35280	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑒 ∈ (𝔼‘𝑁) ∧ 𝑒 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) → (⟨𝑒, 𝑒⟩Cgr⟨𝑒, 𝐸⟩ → 𝑒 = 𝐸))
54	16, 20, 20, 33, 53	syl13anc 1371	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) → (⟨𝑒, 𝑒⟩Cgr⟨𝑒, 𝐸⟩ → 𝑒 = 𝐸))
55	52, 54	syld 47	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) → (((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩) ∧ (𝑒 Btwn ⟨𝐷, 𝐹⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩)) → 𝑒 = 𝐸))
56	55	imp 406	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) ∧ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩) ∧ (𝑒 Btwn ⟨𝐷, 𝐹⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩))) → 𝑒 = 𝐸)
57	56, 14	eqbrtrrd 5172	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) ∧ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩) ∧ (𝑒 Btwn ⟨𝐷, 𝐹⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩))) → 𝐸 Btwn ⟨𝐷, 𝐹⟩)
58	57	expr 456	. . . . 5 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑒 ∈ (𝔼‘𝑁)) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩)) → ((𝑒 Btwn ⟨𝐷, 𝐹⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩) → 𝐸 Btwn ⟨𝐷, 𝐹⟩))
59	58	an32s 649	. . . 4 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩)) ∧ 𝑒 ∈ (𝔼‘𝑁)) → ((𝑒 Btwn ⟨𝐷, 𝐹⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩) → 𝐸 Btwn ⟨𝐷, 𝐹⟩))
60	59	rexlimdva 3154	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩)) → (∃𝑒 ∈ (𝔼‘𝑁)(𝑒 Btwn ⟨𝐷, 𝐹⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩) → 𝐸 Btwn ⟨𝐷, 𝐹⟩))
61	13, 60	mpd 15	. 2 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩)) → 𝐸 Btwn ⟨𝐷, 𝐹⟩)
62	61	ex 412	1 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩) → 𝐸 Btwn ⟨𝐷, 𝐹⟩))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∃wrex 3069  ⟨cop 4634   class class class wbr 5148  ‘cfv 6543  ℕcn 12217  𝔼cee 28414   Btwn cbtwn 28415  Cgrccgr 28416   InnerFiveSeg cifs 35312  Cgr3ccgr3 35313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728  ax-inf2 9639  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190  ax-pre-sup 11191

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  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 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-er 8706  df-map 8825  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-sup 9440  df-oi 9508  df-card 9937  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-div 11877  df-nn 12218  df-2 12280  df-3 12281  df-n0 12478  df-z 12564  df-uz 12828  df-rp 12980  df-ico 13335  df-icc 13336  df-fz 13490  df-fzo 13633  df-seq 13972  df-exp 14033  df-hash 14296  df-cj 15051  df-re 15052  df-im 15053  df-sqrt 15187  df-abs 15188  df-clim 15437  df-sum 15638  df-ee 28417  df-btwn 28418  df-cgr 28419  df-ofs 35260  df-ifs 35317  df-cgr3 35318

This theorem is referenced by:  colinearxfr  35352  brofs2  35354  brifs2  35355  endofsegid  35362  brsegle2  35386