P. 354 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 35301-35400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	trisegint 35301*	A line segment between two sides of a triange intersects a segment crossing from the remaining side to the opposite vertex. Theorem 3.17 of [Schwabhauser] p. 33. (Contributed by Scott Fenton, 24-Sep-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐶⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩) → ∃𝑞 ∈ (𝔼‘𝑁)(𝑞 Btwn ⟨𝑃, 𝐶⟩ ∧ 𝑞 Btwn ⟨𝐵, 𝐸⟩)))
21.10.18.3  Segment Transportation
Syntax	ctransport 35302	Declare the syntax for the segment transport function.
class TransportTo
Definition	df-transport 35303*	Define the segment transport function. See fvtransport 35305 for an explanation of the function. (Contributed by Scott Fenton, 18-Oct-2013.)
⊢ TransportTo = {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)))}
Theorem	funtransport 35304	The TransportTo relationship is a function. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ Fun TransportTo
Theorem	fvtransport 35305*	Calculate the value of the TransportTo function. This function takes four points, 𝐴 through 𝐷, where 𝐶 and 𝐷 are distinct. It then returns the point that extends 𝐶𝐷 by the length of 𝐴𝐵. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷)) → (⟨𝐴, 𝐵⟩TransportTo⟨𝐶, 𝐷⟩) = (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)))
Theorem	transportcl 35306	Closure law for segment transport. (Contributed by Scott Fenton, 19-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷)) → (⟨𝐴, 𝐵⟩TransportTo⟨𝐶, 𝐷⟩) ∈ (𝔼‘𝑁))
Theorem	transportprops 35307	Calculate the defining properties of the transport function. (Contributed by Scott Fenton, 19-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷)) → (𝐷 Btwn ⟨𝐶, (⟨𝐴, 𝐵⟩TransportTo⟨𝐶, 𝐷⟩)⟩ ∧ ⟨𝐷, (⟨𝐴, 𝐵⟩TransportTo⟨𝐶, 𝐷⟩)⟩Cgr⟨𝐴, 𝐵⟩))
21.10.18.4  Properties relating betweenness and congruence
Syntax	cifs 35308	Declare the syntax for the inner five segment predicate.
class InnerFiveSeg
Syntax	ccgr3 35309	Declare the syntax for the three place congruence predicate.
class Cgr3
Syntax	ccolin 35310	Declare the syntax for the colinearity predicate.
class Colinear
Syntax	cfs 35311	Declare the syntax for the five segment predicate.
class FiveSeg
Definition	df-colinear 35312*	The colinearity predicate states that the three points in its arguments sit on one line. Definition 4.10 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 25-Oct-2013.)
⊢ Colinear = ◡{⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩))}
Definition	df-ifs 35313*	The inner five segment configuration is an abbreviation for another congruence condition. See brifs 35316 and ifscgr 35317 for how it is used. Definition 4.1 of [Schwabhauser] p. 34. (Contributed by Scott Fenton, 26-Sep-2013.)
⊢ InnerFiveSeg = {⟨𝑝, 𝑞⟩ ∣ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)∃𝑥 ∈ (𝔼‘𝑛)∃𝑦 ∈ (𝔼‘𝑛)∃𝑧 ∈ (𝔼‘𝑛)∃𝑤 ∈ (𝔼‘𝑛)(𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑥, 𝑦⟩, ⟨𝑧, 𝑤⟩⟩ ∧ ((𝑏 Btwn ⟨𝑎, 𝑐⟩ ∧ 𝑦 Btwn ⟨𝑥, 𝑧⟩) ∧ (⟨𝑎, 𝑐⟩Cgr⟨𝑥, 𝑧⟩ ∧ ⟨𝑏, 𝑐⟩Cgr⟨𝑦, 𝑧⟩) ∧ (⟨𝑎, 𝑑⟩Cgr⟨𝑥, 𝑤⟩ ∧ ⟨𝑐, 𝑑⟩Cgr⟨𝑧, 𝑤⟩)))}
Definition	df-cgr3 35314*	The three place congruence predicate. This is an abbreviation for saying that all three pair in a triple are congruent with each other. Three place form of Definition 4.4 of [Schwabhauser] p. 35. (Contributed by Scott Fenton, 4-Oct-2013.)
⊢ Cgr3 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)∃𝑒 ∈ (𝔼‘𝑛)∃𝑓 ∈ (𝔼‘𝑛)(𝑝 = ⟨𝑎, ⟨𝑏, 𝑐⟩⟩ ∧ 𝑞 = ⟨𝑑, ⟨𝑒, 𝑓⟩⟩ ∧ (⟨𝑎, 𝑏⟩Cgr⟨𝑑, 𝑒⟩ ∧ ⟨𝑎, 𝑐⟩Cgr⟨𝑑, 𝑓⟩ ∧ ⟨𝑏, 𝑐⟩Cgr⟨𝑒, 𝑓⟩))}
Definition	df-fs 35315*	The general five segment configuration is a generalization of the outer and inner five segment configurations. See brfs 35352 and fscgr 35353 for its use. Definition 4.15 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 5-Oct-2013.)
⊢ FiveSeg = {⟨𝑝, 𝑞⟩ ∣ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)∃𝑥 ∈ (𝔼‘𝑛)∃𝑦 ∈ (𝔼‘𝑛)∃𝑧 ∈ (𝔼‘𝑛)∃𝑤 ∈ (𝔼‘𝑛)(𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑥, 𝑦⟩, ⟨𝑧, 𝑤⟩⟩ ∧ (𝑎 Colinear ⟨𝑏, 𝑐⟩ ∧ ⟨𝑎, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑥, ⟨𝑦, 𝑧⟩⟩ ∧ (⟨𝑎, 𝑑⟩Cgr⟨𝑥, 𝑤⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑦, 𝑤⟩)))}
Theorem	brifs 35316	Binary relation form of the inner five segment predicate. (Contributed by Scott Fenton, 26-Sep-2013.)
⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁)) ∧ (𝐹 ∈ (𝔼‘𝑁) ∧ 𝐺 ∈ (𝔼‘𝑁) ∧ 𝐻 ∈ (𝔼‘𝑁))) → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ InnerFiveSeg ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝐺⟩) ∧ (⟨𝐴, 𝐶⟩Cgr⟨𝐸, 𝐺⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝐺⟩) ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐺, 𝐻⟩))))
Theorem	ifscgr 35317	Inner five segment congruence. Take two triangles, 𝐴𝐷𝐶 and 𝐸𝐻𝐺, with 𝐵 between 𝐴 and 𝐶 and 𝐹 between 𝐸 and 𝐺. If the other components of the triangles are congruent, then so are 𝐵𝐷 and 𝐹𝐻. Theorem 4.2 of [Schwabhauser] p. 34. (Contributed by Scott Fenton, 27-Sep-2013.)
⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁)) ∧ (𝐹 ∈ (𝔼‘𝑁) ∧ 𝐺 ∈ (𝔼‘𝑁) ∧ 𝐻 ∈ (𝔼‘𝑁))) → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ InnerFiveSeg ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ → ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩))
Theorem	cgrsub 35318	Removing identical parts from the end of a line segment preserves congruence. Theorem 4.3 of [Schwabhauser] p. 35. (Contributed by Scott Fenton, 4-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐹⟩) ∧ (⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝐹⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → ⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩))
Theorem	brcgr3 35319	Binary relation form of the three-place congruence predicate. (Contributed by Scott Fenton, 4-Oct-2013.)
⊢ ((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⟩)))
Theorem	cgr3permute3 35320	Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
⊢ ((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⟩⟩ ↔ ⟨𝐵, ⟨𝐶, 𝐴⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝐷⟩⟩))
Theorem	cgr3permute1 35321	Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
⊢ ((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, 𝐵⟩⟩Cgr3⟨𝐷, ⟨𝐹, 𝐸⟩⟩))
Theorem	cgr3permute2 35322	Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
⊢ ((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⟩⟩ ↔ ⟨𝐵, ⟨𝐴, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐷, 𝐹⟩⟩))
Theorem	cgr3permute4 35323	Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
⊢ ((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⟩⟩ ↔ ⟨𝐶, ⟨𝐴, 𝐵⟩⟩Cgr3⟨𝐹, ⟨𝐷, 𝐸⟩⟩))
Theorem	cgr3permute5 35324	Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
⊢ ((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⟩⟩ ↔ ⟨𝐶, ⟨𝐵, 𝐴⟩⟩Cgr3⟨𝐹, ⟨𝐸, 𝐷⟩⟩))
Theorem	cgr3tr4 35325	Transitivity law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁)) ∧ (𝐺 ∈ (𝔼‘𝑁) ∧ 𝐻 ∈ (𝔼‘𝑁) ∧ 𝐼 ∈ (𝔼‘𝑁)))) → ((⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐺, ⟨𝐻, 𝐼⟩⟩) → ⟨𝐷, ⟨𝐸, 𝐹⟩⟩Cgr3⟨𝐺, ⟨𝐻, 𝐼⟩⟩))
Theorem	cgr3com 35326	Commutativity law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
⊢ ((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⟩⟩ ↔ ⟨𝐷, ⟨𝐸, 𝐹⟩⟩Cgr3⟨𝐴, ⟨𝐵, 𝐶⟩⟩))
Theorem	cgr3rflx 35327	Identity law for three-place congruence. (Contributed by Scott Fenton, 6-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐴, ⟨𝐵, 𝐶⟩⟩)
Theorem	cgrxfr 35328*	A line segment can be divided at the same place as a congruent line segment is divided. Theorem 4.5 of [Schwabhauser] p. 35. (Contributed by Scott Fenton, 4-Oct-2013.)
⊢ ((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⟩⟩)))
Theorem	btwnxfr 35329	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.)
⊢ ((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⟩))
Theorem	colinrel 35330	Colinearity is a relationship. (Contributed by Scott Fenton, 7-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ Rel Colinear
Theorem	brcolinear2 35331*	Alternate colinearity binary relation. (Contributed by Scott Fenton, 7-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑃 Colinear ⟨𝑄, 𝑅⟩ ↔ ∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑅 ∈ (𝔼‘𝑛)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∨ 𝑄 Btwn ⟨𝑅, 𝑃⟩ ∨ 𝑅 Btwn ⟨𝑃, 𝑄⟩))))
Theorem	brcolinear 35332	The binary relation form of the colinearity predicate. (Contributed by Scott Fenton, 5-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Colinear ⟨𝐵, 𝐶⟩ ↔ (𝐴 Btwn ⟨𝐵, 𝐶⟩ ∨ 𝐵 Btwn ⟨𝐶, 𝐴⟩ ∨ 𝐶 Btwn ⟨𝐴, 𝐵⟩)))
Theorem	colinearex 35333	The colinear predicate exists. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ Colinear ∈ V
Theorem	colineardim1 35334	If 𝐴 is colinear with 𝐵 and 𝐶, then 𝐴 is in the same space as 𝐵. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ 𝑊)) → (𝐴 Colinear ⟨𝐵, 𝐶⟩ → 𝐴 ∈ (𝔼‘𝑁)))
Theorem	colinearperm1 35335	Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Colinear ⟨𝐵, 𝐶⟩ ↔ 𝐴 Colinear ⟨𝐶, 𝐵⟩))
Theorem	colinearperm3 35336	Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Colinear ⟨𝐵, 𝐶⟩ ↔ 𝐵 Colinear ⟨𝐶, 𝐴⟩))
Theorem	colinearperm2 35337	Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Colinear ⟨𝐵, 𝐶⟩ ↔ 𝐵 Colinear ⟨𝐴, 𝐶⟩))
Theorem	colinearperm4 35338	Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Colinear ⟨𝐵, 𝐶⟩ ↔ 𝐶 Colinear ⟨𝐴, 𝐵⟩))
Theorem	colinearperm5 35339	Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Colinear ⟨𝐵, 𝐶⟩ ↔ 𝐶 Colinear ⟨𝐵, 𝐴⟩))
Theorem	colineartriv1 35340	Trivial case of colinearity. Theorem 4.12 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 5-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝐴 Colinear ⟨𝐴, 𝐵⟩)
Theorem	colineartriv2 35341	Trivial case of colinearity. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝐴 Colinear ⟨𝐵, 𝐵⟩)
Theorem	btwncolinear1 35342	Betweenness implies colinearity. (Contributed by Scott Fenton, 7-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐶 Btwn ⟨𝐴, 𝐵⟩ → 𝐴 Colinear ⟨𝐵, 𝐶⟩))
Theorem	btwncolinear2 35343	Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐶 Btwn ⟨𝐴, 𝐵⟩ → 𝐴 Colinear ⟨𝐶, 𝐵⟩))
Theorem	btwncolinear3 35344	Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐶 Btwn ⟨𝐴, 𝐵⟩ → 𝐵 Colinear ⟨𝐴, 𝐶⟩))
Theorem	btwncolinear4 35345	Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐶 Btwn ⟨𝐴, 𝐵⟩ → 𝐵 Colinear ⟨𝐶, 𝐴⟩))
Theorem	btwncolinear5 35346	Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐶 Btwn ⟨𝐴, 𝐵⟩ → 𝐶 Colinear ⟨𝐴, 𝐵⟩))
Theorem	btwncolinear6 35347	Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐶 Btwn ⟨𝐴, 𝐵⟩ → 𝐶 Colinear ⟨𝐵, 𝐴⟩))
Theorem	colinearxfr 35348	Transfer law for colinearity. Theorem 4.13 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 5-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → ((𝐵 Colinear ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝐹⟩⟩) → 𝐸 Colinear ⟨𝐷, 𝐹⟩))
Theorem	lineext 35349*	Extend a line with a missing point. Theorem 4.14 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 6-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) → ((𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩) → ∃𝑓 ∈ (𝔼‘𝑁)⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐷, ⟨𝐸, 𝑓⟩⟩))
Theorem	brofs2 35350	Change some conditions for outer five segment predicate. (Contributed by Scott Fenton, 6-Oct-2013.)
⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁)) ∧ (𝐹 ∈ (𝔼‘𝑁) ∧ 𝐺 ∈ (𝔼‘𝑁) ∧ 𝐻 ∈ (𝔼‘𝑁))) → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ OuterFiveSeg ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝐺⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩))))
Theorem	brifs2 35351	Change some conditions for inner five segment predicate. (Contributed by Scott Fenton, 6-Oct-2013.)
⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁)) ∧ (𝐹 ∈ (𝔼‘𝑁) ∧ 𝐺 ∈ (𝔼‘𝑁) ∧ 𝐻 ∈ (𝔼‘𝑁))) → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ InnerFiveSeg ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝐺⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐺, 𝐻⟩))))
Theorem	brfs 35352	Binary relation form of the general five segment predicate. (Contributed by Scott Fenton, 5-Oct-2013.)
⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁)) ∧ (𝐹 ∈ (𝔼‘𝑁) ∧ 𝐺 ∈ (𝔼‘𝑁) ∧ 𝐻 ∈ (𝔼‘𝑁))) → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ FiveSeg ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ (𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝐺⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩))))
Theorem	fscgr 35353	Congruence law for the general five segment configuration. Theorem 4.16 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 5-Oct-2013.)
⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁)) ∧ (𝐹 ∈ (𝔼‘𝑁) ∧ 𝐺 ∈ (𝔼‘𝑁) ∧ 𝐻 ∈ (𝔼‘𝑁))) → ((⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ FiveSeg ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ∧ 𝐴 ≠ 𝐵) → ⟨𝐶, 𝐷⟩Cgr⟨𝐺, 𝐻⟩))
Theorem	linecgr 35354	Congruence rule for lines. Theorem 4.17 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 6-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁))) → (((𝐴 ≠ 𝐵 ∧ 𝐴 Colinear ⟨𝐵, 𝐶⟩) ∧ (⟨𝐴, 𝑃⟩Cgr⟨𝐴, 𝑄⟩ ∧ ⟨𝐵, 𝑃⟩Cgr⟨𝐵, 𝑄⟩)) → ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑄⟩))
Theorem	linecgrand 35355	Deduction form of linecgr 35354. (Contributed by Scott Fenton, 14-Oct-2013.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁))    &   ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁))    &   ⊢ (𝜑 → 𝐶 ∈ (𝔼‘𝑁))    &   ⊢ (𝜑 → 𝑃 ∈ (𝔼‘𝑁))    &   ⊢ (𝜑 → 𝑄 ∈ (𝔼‘𝑁))    &   ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ≠ 𝐵)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝐴 Colinear ⟨𝐵, 𝐶⟩)    &   ⊢ ((𝜑 ∧ 𝜓) → ⟨𝐴, 𝑃⟩Cgr⟨𝐴, 𝑄⟩)    &   ⊢ ((𝜑 ∧ 𝜓) → ⟨𝐵, 𝑃⟩Cgr⟨𝐵, 𝑄⟩)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑄⟩)
Theorem	lineid 35356	Identity law for points on lines. Theorem 4.18 of [Schwabhauser] p. 38. (Contributed by Scott Fenton, 7-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (((𝐴 ≠ 𝐵 ∧ 𝐴 Colinear ⟨𝐵, 𝐶⟩) ∧ (⟨𝐴, 𝐶⟩Cgr⟨𝐴, 𝐷⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐵, 𝐷⟩)) → 𝐶 = 𝐷))
Theorem	idinside 35357	Law for finding a point inside a segment. Theorem 4.19 of [Schwabhauser] p. 38. (Contributed by Scott Fenton, 7-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((𝐶 Btwn ⟨𝐴, 𝐵⟩ ∧ ⟨𝐴, 𝐶⟩Cgr⟨𝐴, 𝐷⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐵, 𝐷⟩) → 𝐶 = 𝐷))
Theorem	endofsegid 35358	If 𝐴, 𝐵, and 𝐶 fall in order on a line, and 𝐴𝐵 and 𝐴𝐶 are congruent, then 𝐶 = 𝐵. (Contributed by Scott Fenton, 7-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ ⟨𝐴, 𝐶⟩Cgr⟨𝐴, 𝐵⟩) → 𝐶 = 𝐵))
Theorem	endofsegidand 35359	Deduction form of endofsegid 35358. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁))    &   ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁))    &   ⊢ (𝜑 → 𝐶 ∈ (𝔼‘𝑁))    &   ⊢ ((𝜑 ∧ 𝜓) → 𝐶 Btwn ⟨𝐴, 𝐵⟩)    &   ⊢ ((𝜑 ∧ 𝜓) → ⟨𝐴, 𝐵⟩Cgr⟨𝐴, 𝐶⟩)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝐵 = 𝐶)
21.10.18.5  Connectivity of betweenness
Theorem	btwnconn1lem1 35360	Lemma for btwnconn1 35374. The next several lemmas introduce various properties of hypothetical points that end up eliminating alternatives to connectivity. We begin by showing a congruence property of those hypothetical points. (Contributed by Scott Fenton, 8-Oct-2013.)
⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩)))) → ⟨𝐵, 𝑐⟩Cgr⟨𝑋, 𝐶⟩)
Theorem	btwnconn1lem2 35361	Lemma for btwnconn1 35374. Now, we show that two of the hypotheticals we introduced in the first lemma are identical. (Contributed by Scott Fenton, 8-Oct-2013.)
⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩)))) → 𝑋 = 𝑏)
Theorem	btwnconn1lem3 35362	Lemma for btwnconn1 35374. Establish the next congruence in the series. (Contributed by Scott Fenton, 8-Oct-2013.)
⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩)))) → ⟨𝐵, 𝑑⟩Cgr⟨𝑏, 𝐷⟩)
Theorem	btwnconn1lem4 35363	Lemma for btwnconn1 35374. Assuming 𝐶 ≠ 𝑐, we now attempt to force 𝐷 = 𝑑 from here out via a series of congruences. (Contributed by Scott Fenton, 8-Oct-2013.)
⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩)))) → ⟨𝑑, 𝑐⟩Cgr⟨𝐷, 𝐶⟩)
Theorem	btwnconn1lem5 35364	Lemma for btwnconn1 35374. Now, we introduce 𝐸, the intersection of 𝐶𝑐 and 𝐷𝑑. We begin by showing that it is the midpoint of 𝐶 and 𝑐. (Contributed by Scott Fenton, 8-Oct-2013.)
⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ (𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩))) → ⟨𝐸, 𝐶⟩Cgr⟨𝐸, 𝑐⟩)
Theorem	btwnconn1lem6 35365	Lemma for btwnconn1 35374. Next, we show that 𝐸 is the midpoint of 𝐷 and 𝑑. (Contributed by Scott Fenton, 8-Oct-2013.)
⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ (𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩))) → ⟨𝐸, 𝐷⟩Cgr⟨𝐸, 𝑑⟩)
Theorem	btwnconn1lem7 35366	Lemma for btwnconn1 35374. Under our assumptions, 𝐶 and 𝑑 are distinct. (Contributed by Scott Fenton, 8-Oct-2013.)
⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ (𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩))) → 𝐶 ≠ 𝑑)
Theorem	btwnconn1lem8 35367	Lemma for btwnconn1 35374. Now, we introduce the last three points used in the construction: 𝑃, 𝑄, and 𝑅 will turn out to be equal further down, and will provide us with the key to the final statement. We begin by establishing congruence of 𝑅𝑃 and 𝐸𝑑. (Contributed by Scott Fenton, 8-Oct-2013.)
⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝑅, 𝑃⟩Cgr⟨𝐸, 𝑑⟩)
Theorem	btwnconn1lem9 35368	Lemma for btwnconn1 35374. Now, a quick use of transitivity to establish congruence on 𝑅𝑄 and 𝐸𝐷. (Contributed by Scott Fenton, 8-Oct-2013.)
⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝑅, 𝑄⟩Cgr⟨𝐸, 𝐷⟩)
Theorem	btwnconn1lem10 35369	Lemma for btwnconn1 35374. Now we establish a congruence that will give us 𝐷 = 𝑑 when we compute 𝑃 = 𝑄 later on. (Contributed by Scott Fenton, 8-Oct-2013.)
⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝑑, 𝐷⟩Cgr⟨𝑃, 𝑄⟩)
Theorem	btwnconn1lem11 35370	Lemma for btwnconn1 35374. Now, we establish that 𝐷 and 𝑄 are equidistant from 𝐶. (Contributed by Scott Fenton, 8-Oct-2013.)
⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝐷, 𝐶⟩Cgr⟨𝑄, 𝐶⟩)
Theorem	btwnconn1lem12 35371	Lemma for btwnconn1 35374. Using a long string of invocations of linecgr 35354, we show that 𝐷 = 𝑑. (Contributed by Scott Fenton, 9-Oct-2013.)
⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝐷 = 𝑑)
Theorem	btwnconn1lem13 35372	Lemma for btwnconn1 35374. Begin back-filling and eliminating hypotheses. (Contributed by Scott Fenton, 9-Oct-2013.)
⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁)))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩)))) → (𝐶 = 𝑐 ∨ 𝐷 = 𝑑))
Theorem	btwnconn1lem14 35373	Lemma for btwnconn1 35374. Final statement of the theorem when 𝐵 ≠ 𝐶. (Contributed by Scott Fenton, 9-Oct-2013.)
⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩))) → (𝐶 Btwn ⟨𝐴, 𝐷⟩ ∨ 𝐷 Btwn ⟨𝐴, 𝐶⟩))
Theorem	btwnconn1 35374	Connectitivy law for betweenness. Theorem 5.1 of [Schwabhauser] p. 39-41. (Contributed by Scott Fenton, 9-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((𝐴 ≠ 𝐵 ∧ 𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩) → (𝐶 Btwn ⟨𝐴, 𝐷⟩ ∨ 𝐷 Btwn ⟨𝐴, 𝐶⟩)))
Theorem	btwnconn2 35375	Another connectivity law for betweenness. Theorem 5.2 of [Schwabhauser] p. 41. (Contributed by Scott Fenton, 9-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((𝐴 ≠ 𝐵 ∧ 𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩) → (𝐶 Btwn ⟨𝐵, 𝐷⟩ ∨ 𝐷 Btwn ⟨𝐵, 𝐶⟩)))
Theorem	btwnconn3 35376	Inner connectivity law for betweenness. Theorem 5.3 of [Schwabhauser] p. 41. (Contributed by Scott Fenton, 9-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn ⟨𝐴, 𝐷⟩ ∧ 𝐶 Btwn ⟨𝐴, 𝐷⟩) → (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∨ 𝐶 Btwn ⟨𝐴, 𝐵⟩)))
Theorem	midofsegid 35377	If two points fall in the same place in the middle of a segment, then they are identical. (Contributed by Scott Fenton, 16-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) → ((𝐷 Btwn ⟨𝐴, 𝐵⟩ ∧ 𝐸 Btwn ⟨𝐴, 𝐵⟩ ∧ ⟨𝐴, 𝐷⟩Cgr⟨𝐴, 𝐸⟩) → 𝐷 = 𝐸))
Theorem	segcon2 35378*	Generalization of axsegcon 28449. This time, we generate an endpoint for a segment on the ray 𝑄𝐴 congruent to 𝐵𝐶 and starting at 𝑄, as opposed to axsegcon 28449, where the segment starts at 𝐴 (Contributed by Scott Fenton, 14-Oct-2013.) Remove unneeded inequality. (Revised by Scott Fenton, 15-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝑄 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) ∧ (𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → ∃𝑥 ∈ (𝔼‘𝑁)((𝐴 Btwn ⟨𝑄, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑄, 𝐴⟩) ∧ ⟨𝑄, 𝑥⟩Cgr⟨𝐵, 𝐶⟩))
21.10.18.6  Segment less than or equal to
Syntax	csegle 35379	Declare the constant for the segment less than or equal to relationship.
class Seg≤
Definition	df-segle 35380*	Define the segment length comparison relationship. This relationship expresses that the segment 𝐴𝐵 is no longer than 𝐶𝐷. In this section, we establish various properties of this relationship showing that it is a transitive, reflexive relationship on pairs of points that is substitutive under congruence. Definition 5.4 of [Schwabhauser] p. 41. (Contributed by Scott Fenton, 11-Oct-2013.)
⊢ Seg≤ = {⟨𝑝, 𝑞⟩ ∣ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))}
Theorem	brsegle 35381*	Binary relation form of the segment comparison relationship. (Contributed by Scott Fenton, 11-Oct-2013.)
⊢ ((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⟩)))
Theorem	brsegle2 35382*	Alternate characterization of segment comparison. Theorem 5.5 of [Schwabhauser] p. 41-42. (Contributed by Scott Fenton, 11-Oct-2013.)
⊢ ((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⟩)))
Theorem	seglecgr12im 35383	Substitution law for segment comparison under congruence. Theorem 5.6 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 11-Oct-2013.)
⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁)) ∧ (𝐹 ∈ (𝔼‘𝑁) ∧ 𝐺 ∈ (𝔼‘𝑁) ∧ 𝐻 ∈ (𝔼‘𝑁))) → ((⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐺, 𝐻⟩ ∧ ⟨𝐴, 𝐵⟩ Seg≤ ⟨𝐶, 𝐷⟩) → ⟨𝐸, 𝐹⟩ Seg≤ ⟨𝐺, 𝐻⟩))
Theorem	seglecgr12 35384	Substitution law for segment comparison under congruence. Biconditional version. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁)) ∧ (𝐹 ∈ (𝔼‘𝑁) ∧ 𝐺 ∈ (𝔼‘𝑁) ∧ 𝐻 ∈ (𝔼‘𝑁))) → ((⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐺, 𝐻⟩) → (⟨𝐴, 𝐵⟩ Seg≤ ⟨𝐶, 𝐷⟩ ↔ ⟨𝐸, 𝐹⟩ Seg≤ ⟨𝐺, 𝐻⟩)))
Theorem	seglerflx 35385	Segment comparison is reflexive. Theorem 5.7 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 11-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ⟨𝐴, 𝐵⟩ Seg≤ ⟨𝐴, 𝐵⟩)
Theorem	seglemin 35386	Any segment is at least as long as a degenerate segment. Theorem 5.11 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 11-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → ⟨𝐴, 𝐴⟩ Seg≤ ⟨𝐵, 𝐶⟩)
Theorem	segletr 35387	Segment less than is transitive. Theorem 5.8 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 11-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → ((⟨𝐴, 𝐵⟩ Seg≤ ⟨𝐶, 𝐷⟩ ∧ ⟨𝐶, 𝐷⟩ Seg≤ ⟨𝐸, 𝐹⟩) → ⟨𝐴, 𝐵⟩ Seg≤ ⟨𝐸, 𝐹⟩))
Theorem	segleantisym 35388	Antisymmetry law for segment comparison. Theorem 5.9 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 14-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((⟨𝐴, 𝐵⟩ Seg≤ ⟨𝐶, 𝐷⟩ ∧ ⟨𝐶, 𝐷⟩ Seg≤ ⟨𝐴, 𝐵⟩) → ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩))
Theorem	seglelin 35389	Linearity law for segment comparison. Theorem 5.10 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 14-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (⟨𝐴, 𝐵⟩ Seg≤ ⟨𝐶, 𝐷⟩ ∨ ⟨𝐶, 𝐷⟩ Seg≤ ⟨𝐴, 𝐵⟩))
Theorem	btwnsegle 35390	If 𝐵 falls between 𝐴 and 𝐶, then 𝐴𝐵 is no longer than 𝐴𝐶. (Contributed by Scott Fenton, 16-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐵 Btwn ⟨𝐴, 𝐶⟩ → ⟨𝐴, 𝐵⟩ Seg≤ ⟨𝐴, 𝐶⟩))
Theorem	colinbtwnle 35391	Given three colinear points 𝐴, 𝐵, and 𝐶, 𝐵 falls in the middle iff the two segments to 𝐵 are no longer than 𝐴𝐶. Theorem 5.12 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Colinear ⟨𝐵, 𝐶⟩ → (𝐵 Btwn ⟨𝐴, 𝐶⟩ ↔ (⟨𝐴, 𝐵⟩ Seg≤ ⟨𝐴, 𝐶⟩ ∧ ⟨𝐵, 𝐶⟩ Seg≤ ⟨𝐴, 𝐶⟩))))
21.10.18.7  Outside-of relationship
Syntax	coutsideof 35392	Declare the syntax for the outside of constant.
class OutsideOf
Definition	df-outsideof 35393	The outside of relationship. This relationship expresses that 𝑃, 𝐴, and 𝐵 fall on a line, but 𝑃 is not on the segment 𝐴𝐵. This definition is taken from theorem 6.4 of [Schwabhauser] p. 43, since it requires no dummy variables. (Contributed by Scott Fenton, 17-Oct-2013.)
⊢ OutsideOf = ( Colinear ∖ Btwn )
Theorem	broutsideof 35394	Binary relation form of OutsideOf. Theorem 6.4 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 17-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ↔ (𝑃 Colinear ⟨𝐴, 𝐵⟩ ∧ ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩))
Theorem	broutsideof2 35395	Alternate form of OutsideOf. Definition 6.1 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 17-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf⟨𝐴, 𝐵⟩ ↔ (𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃 ∧ (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩))))
Theorem	outsidene1 35396	Outsideness implies inequality. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf⟨𝐴, 𝐵⟩ → 𝐴 ≠ 𝑃))
Theorem	outsidene2 35397	Outsideness implies inequality. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf⟨𝐴, 𝐵⟩ → 𝐵 ≠ 𝑃))
Theorem	btwnoutside 35398	A principle linking outsideness to betweenness. Theorem 6.2 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁))) → (((𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃 ∧ 𝐶 ≠ 𝑃) ∧ 𝑃 Btwn ⟨𝐴, 𝐶⟩) → (𝑃 Btwn ⟨𝐵, 𝐶⟩ ↔ 𝑃OutsideOf⟨𝐴, 𝐵⟩)))
Theorem	broutsideof3 35399*	Characterization of outsideness in terms of relationship to a fourth point. Theorem 6.3 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf⟨𝐴, 𝐵⟩ ↔ (𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃 ∧ ∃𝑐 ∈ (𝔼‘𝑁)(𝑐 ≠ 𝑃 ∧ 𝑃 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑃 Btwn ⟨𝐵, 𝑐⟩))))
Theorem	outsideofrflx 35400	Reflexivity of outsideness. Theorem 6.5 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) → (𝐴 ≠ 𝑃 → 𝑃OutsideOf⟨𝐴, 𝐴⟩))