P. 362 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36101-36200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	brcgr3 36101	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 36102	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 36103	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 36104	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 36105	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 36106	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 36107	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 36108	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 36109	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 36110*	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 36111	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 36112	Colinearity is a relationship. (Contributed by Scott Fenton, 7-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ Rel Colinear
Theorem	brcolinear2 36113*	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 36114	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 36115	The colinear predicate exists. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ Colinear ∈ V
Theorem	colineardim1 36116	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 36117	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 36118	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 36119	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 36120	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 36121	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 36122	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 36123	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 36124	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 36125	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 36126	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 36127	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 36128	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 36129	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 36130	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 36131*	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 36132	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 36133	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 36134	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 36135	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 36136	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 36137	Deduction form of linecgr 36136. (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 36138	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 36139	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 36140	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 36141	Deduction form of endofsegid 36140. (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)
21.11.17.5  Connectivity of betweenness
Theorem	btwnconn1lem1 36142	Lemma for btwnconn1 36156. 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 36143	Lemma for btwnconn1 36156. 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 36144	Lemma for btwnconn1 36156. 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 36145	Lemma for btwnconn1 36156. 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 36146	Lemma for btwnconn1 36156. 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 36147	Lemma for btwnconn1 36156. 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 36148	Lemma for btwnconn1 36156. 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 36149	Lemma for btwnconn1 36156. 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 36150	Lemma for btwnconn1 36156. 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 36151	Lemma for btwnconn1 36156. 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 36152	Lemma for btwnconn1 36156. 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 36153	Lemma for btwnconn1 36156. Using a long string of invocations of linecgr 36136, 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 36154	Lemma for btwnconn1 36156. 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 36155	Lemma for btwnconn1 36156. 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 36156	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 36157	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 36158	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 36159	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 36160*	Generalization of axsegcon 28916. This time, we generate an endpoint for a segment on the ray 𝑄𝐴 congruent to 𝐵𝐶 and starting at 𝑄, as opposed to axsegcon 28916, 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⟩))
21.11.17.6  Segment less than or equal to
Syntax	csegle 36161	Declare the constant for the segment less than or equal to relationship.
class Seg≤
Definition	df-segle 36162*	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 36163*	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 36164*	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 36165	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 36166	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 36167	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 36168	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 36169	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 36170	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 36171	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 36172	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 36173	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⟩))))
21.11.17.7  Outside-of relationship
Syntax	coutsideof 36174	Declare the syntax for the outside of constant.
class OutsideOf
Definition	df-outsideof 36175	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 36176	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 36177	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 36178	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 36179	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 36180	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 36181*	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 36182	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⟩))
Theorem	outsideofcom 36183	Commutativity law for outsideness. Theorem 6.6 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) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf⟨𝐴, 𝐵⟩ ↔ 𝑃OutsideOf⟨𝐵, 𝐴⟩))
Theorem	outsideoftr 36184	Transitivity law for outsideness. Theorem 6.7 of [Schwabhauser] p. 44. (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))) → ((𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ 𝑃OutsideOf⟨𝐵, 𝐶⟩) → 𝑃OutsideOf⟨𝐴, 𝐶⟩))
Theorem	outsideofeq 36185	Uniqueness law for OutsideOf. Analogue of segconeq 36065. (Contributed by Scott Fenton, 24-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) → (((𝐴OutsideOf⟨𝑋, 𝑅⟩ ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐵, 𝐶⟩) ∧ (𝐴OutsideOf⟨𝑌, 𝑅⟩ ∧ ⟨𝐴, 𝑌⟩Cgr⟨𝐵, 𝐶⟩)) → 𝑋 = 𝑌))
Theorem	outsideofeu 36186*	Given a nondegenerate ray, there is a unique point congruent to the segment 𝐵𝐶 lying on the ray 𝐴𝑅. Theorem 6.11 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 23-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → ((𝑅 ≠ 𝐴 ∧ 𝐵 ≠ 𝐶) → ∃!𝑥 ∈ (𝔼‘𝑁)(𝐴OutsideOf⟨𝑥, 𝑅⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐵, 𝐶⟩)))
Theorem	outsidele 36187	Relate OutsideOf to Seg≤. Theorem 6.13 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 24-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf⟨𝐴, 𝐵⟩ → (⟨𝑃, 𝐴⟩ Seg≤ ⟨𝑃, 𝐵⟩ ↔ 𝐴 Btwn ⟨𝑃, 𝐵⟩)))
Theorem	outsideofcol 36188	Outside of implies colinearity. (Contributed by Scott Fenton, 26-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ (𝑃OutsideOf⟨𝑄, 𝑅⟩ → 𝑃 Colinear ⟨𝑄, 𝑅⟩)
21.11.17.8  Lines and Rays
Syntax	cline2 36189	Declare the constant for the line function.
class Line
Syntax	cray 36190	Declare the constant for the ray function.
class Ray
Syntax	clines2 36191	Declare the constant for the set of all lines.
class LinesEE
Definition	df-line2 36192*	Define the Line function. This function generates the line passing through the distinct points 𝑎 and 𝑏. Adapted from definition 6.14 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 25-Oct-2013.)
⊢ Line = {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )}
Definition	df-ray 36193*	Define the Ray function. This function generates the set of all points that lie on the ray starting at 𝑝 and passing through 𝑎. Definition 6.8 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 21-Oct-2013.)
⊢ Ray = {⟨⟨𝑝, 𝑎⟩, 𝑟⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})}
Definition	df-lines2 36194	Define the set of all lines. Definition 6.14, part 2 of [Schwabhauser] p. 45. See ellines 36207 for membership. (Contributed by Scott Fenton, 28-Oct-2013.)
⊢ LinesEE = ran Line
Theorem	funray 36195	Show that the Ray relationship is a function. (Contributed by Scott Fenton, 21-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ Fun Ray
Theorem	fvray 36196*	Calculate the value of the Ray function. (Contributed by Scott Fenton, 21-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝐴)) → (𝑃Ray𝐴) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})
Theorem	funline 36197	Show that the Line relationship is a function. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ Fun Line
Theorem	linedegen 36198	When Line is applied with the same argument, the result is the empty set. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ (𝐴Line𝐴) = ∅
Theorem	fvline 36199*	Calculate the value of the Line function. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → (𝐴Line𝐵) = {𝑥 ∣ 𝑥 Colinear ⟨𝐴, 𝐵⟩})
Theorem	liness 36200	A line is a subset of the space its two points lie in. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → (𝐴Line𝐵) ⊆ (𝔼‘𝑁))