Theorem btwnxfr 36419

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 36409	. . . . . 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 1153	. . . . . 6 ⊢ ((⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝐹⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩) → ⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝐹⟩)
3	1, 2	biimtrdi 256	. . . . 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 1152	. . . . . 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 1223	. . . . . 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 1224	. . . . . 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 1225	. . . . . 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 1226	. . . . . 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 1228	. . . . . 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 36418	. . . . . 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 1411	. . . . 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 616	. . . 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 411	. . 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 792	. . . . . . . . . . . . 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 520	. . . . . . . . . . . 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 1208	. . . . . . . . . . . . . . 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 1271	. . . . . . . . . . . . . . 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 1273	. . . . . . . . . . . . . . 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 36351	. . . . . . . . . . . . . 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 489	. . . . . . . . . . . . . . 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 36351	. . . . . . . . . . . . . 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 520	. . . . . . . . . . . . 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 485	. . . . . . . . . . . 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 489	. . . . . . . . . . . . . 14 ⊢ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩) → ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩)
25		simpr 489	. . . . . . . . . . . . . 14 ⊢ ((𝑒 Btwn ⟨𝐷, 𝐹⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩) → ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝑒, 𝐹⟩⟩)
26		simpl2 1209	. . . . . . . . . . . . . . . 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 1210	. . . . . . . . . . . . . . . 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 1144	. . . . . . . . . . . . . . . 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 36415	. . . . . . . . . . . . . . . 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 1395	. . . . . . . . . . . . . . 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 36416	. . . . . . . . . . . . . . . . 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 1405	. . . . . . . . . . . . . . . 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 1272	. . . . . . . . . . . . . . . . . 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 36409	. . . . . . . . . . . . . . . . . 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 1416	. . . . . . . . . . . . . . . . 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 1211	. . . . . . . . . . . . . . . . . . 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 1213	. . . . . . . . . . . . . . . . . . . 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 36364	. . . . . . . . . . . . . . . . . . 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 520	. . . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . . 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 243	. . . . . . . . . . . . . . . 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 243	. . . . . . . . . . . . . . 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 48	. . . . . . . . . . . . . 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 618	. . . . . . . . . . . . 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 411	. . . . . . . . . . . 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 1144	. . . . . . . . . . 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 417	. . . . . . . . . 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 36406	. . . . . . . . . . . 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 36407	. . . . . . . . . . . 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 263	. . . . . . . . . . 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 1425	. . . . . . . . . 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 48	. . . . . . . . 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 36366	. . . . . . . . . 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 1395	. . . . . . . . 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 48	. . . . . . . 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 411	. . . . . . 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 5129	. . . . . 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 461	. . . . 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 664	. . . 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 3166	. . 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 16	. 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 417	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 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∃wrex 3089  ⟨cop 4591   class class class wbr 5105  ‘cfv 6525  ℕcn 12224  𝔼cee 29146   Btwn cbtwn 29147  Cgrccgr 29148   InnerFiveSeg cifs 36398  Cgr3ccgr3 36399

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-ifs 36403  df-cgr3 36404

This theorem is referenced by:  colinearxfr  36438  brofs2  36440  brifs2  36441  endofsegid  36448  brsegle2  36472