P. 289 of Theorem List - Metamath Proof Explorer

Type

Label

Description

Statement

Theorem

axsegconlem2 28801*

Lemma for axsegcon 28810. Show that the square of the distance between two points is a real number. (Contributed by Scott Fenton, 17-Sep-2013.)

⊢ 𝑆 = Σ𝑝 ∈ (1...𝑁)(((𝐴‘𝑝) − (𝐵‘𝑝))↑2)    ⇒   ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝑆 ∈ ℝ)

Theorem

axsegconlem3 28802*

Lemma for axsegcon 28810. Show that the square of the distance between two points is nonnegative. (Contributed by Scott Fenton, 17-Sep-2013.)

⊢ 𝑆 = Σ𝑝 ∈ (1...𝑁)(((𝐴‘𝑝) − (𝐵‘𝑝))↑2)    ⇒   ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 0 ≤ 𝑆)

Theorem

axsegconlem4 28803*

Lemma for axsegcon 28810. Show that the distance between two points is a real number. (Contributed by Scott Fenton, 17-Sep-2013.)

⊢ 𝑆 = Σ𝑝 ∈ (1...𝑁)(((𝐴‘𝑝) − (𝐵‘𝑝))↑2)    ⇒   ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (√‘𝑆) ∈ ℝ)

Theorem

axsegconlem5 28804*

Lemma for axsegcon 28810. Show that the distance between two points is nonnegative. (Contributed by Scott Fenton, 17-Sep-2013.)

⊢ 𝑆 = Σ𝑝 ∈ (1...𝑁)(((𝐴‘𝑝) − (𝐵‘𝑝))↑2)    ⇒   ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 0 ≤ (√‘𝑆))

Theorem

axsegconlem6 28805*

Lemma for axsegcon 28810. Show that the distance between two distinct points is positive. (Contributed by Scott Fenton, 17-Sep-2013.)

⊢ 𝑆 = Σ𝑝 ∈ (1...𝑁)(((𝐴‘𝑝) − (𝐵‘𝑝))↑2)    ⇒   ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵) → 0 < (√‘𝑆))

Theorem

axsegconlem7 28806*

Lemma for axsegcon 28810. Show that a particular ratio of distances is in the closed unit interval. (Contributed by Scott Fenton, 18-Sep-2013.)

⊢ 𝑆 = Σ𝑝 ∈ (1...𝑁)(((𝐴‘𝑝) − (𝐵‘𝑝))↑2)    &   ⊢ 𝑇 = Σ𝑝 ∈ (1...𝑁)(((𝐶‘𝑝) − (𝐷‘𝑝))↑2)    ⇒   ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((√‘𝑆) / ((√‘𝑆) + (√‘𝑇))) ∈ (0[,]1))

Theorem

axsegconlem8 28807*

Lemma for axsegcon 28810. Show that a particular mapping generates a point. (Contributed by Scott Fenton, 18-Sep-2013.)

⊢ 𝑆 = Σ𝑝 ∈ (1...𝑁)(((𝐴‘𝑝) − (𝐵‘𝑝))↑2)    &   ⊢ 𝑇 = Σ𝑝 ∈ (1...𝑁)(((𝐶‘𝑝) − (𝐷‘𝑝))↑2)    &   ⊢ 𝐹 = (𝑘 ∈ (1...𝑁) ↦ (((((√‘𝑆) + (√‘𝑇)) · (𝐵‘𝑘)) − ((√‘𝑇) · (𝐴‘𝑘))) / (√‘𝑆)))    ⇒   ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → 𝐹 ∈ (𝔼‘𝑁))

Theorem

axsegconlem9 28808*

Lemma for axsegcon 28810. Show that 𝐵𝐹 is congruent to 𝐶𝐷. (Contributed by Scott Fenton, 19-Sep-2013.)

⊢ 𝑆 = Σ𝑝 ∈ (1...𝑁)(((𝐴‘𝑝) − (𝐵‘𝑝))↑2)    &   ⊢ 𝑇 = Σ𝑝 ∈ (1...𝑁)(((𝐶‘𝑝) − (𝐷‘𝑝))↑2)    &   ⊢ 𝐹 = (𝑘 ∈ (1...𝑁) ↦ (((((√‘𝑆) + (√‘𝑇)) · (𝐵‘𝑘)) − ((√‘𝑇) · (𝐴‘𝑘))) / (√‘𝑆)))    ⇒   ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → Σ𝑖 ∈ (1...𝑁)(((𝐵‘𝑖) − (𝐹‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2))

Theorem

axsegconlem10 28809*

Lemma for axsegcon 28810. Show that the scaling constant from axsegconlem7 28806 produces the betweenness condition for 𝐴, 𝐵 and 𝐹. (Contributed by Scott Fenton, 21-Sep-2013.)

⊢ 𝑆 = Σ𝑝 ∈ (1...𝑁)(((𝐴‘𝑝) − (𝐵‘𝑝))↑2)    &   ⊢ 𝑇 = Σ𝑝 ∈ (1...𝑁)(((𝐶‘𝑝) − (𝐷‘𝑝))↑2)    &   ⊢ 𝐹 = (𝑘 ∈ (1...𝑁) ↦ (((((√‘𝑆) + (√‘𝑇)) · (𝐵‘𝑘)) − ((√‘𝑇) · (𝐴‘𝑘))) / (√‘𝑆)))    ⇒   ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − ((√‘𝑆) / ((√‘𝑆) + (√‘𝑇)))) · (𝐴‘𝑖)) + (((√‘𝑆) / ((√‘𝑆) + (√‘𝑇))) · (𝐹‘𝑖))))

Theorem

axsegcon 28810*

Any segment 𝐴𝐵 can be extended to a point 𝑥 such that 𝐵𝑥 is congruent to 𝐶𝐷. Axiom A4 of [Schwabhauser] p. 11. (Contributed by Scott Fenton, 4-Jun-2013.)

⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ∃𝑥 ∈ (𝔼‘𝑁)(𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐵, 𝑥⟩Cgr⟨𝐶, 𝐷⟩))

Theorem

ax5seglem1 28811*

Lemma for ax5seg 28821. Rexpress a one congruence sum given betweenness. (Contributed by Scott Fenton, 11-Jun-2013.)

⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝑇 ∈ (0[,]1) ∧ ∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))))) → Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐵‘𝑗))↑2) = ((𝑇↑2) · Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)))

Theorem

ax5seglem2 28812*

Lemma for ax5seg 28821. Rexpress another congruence sum given betweenness. (Contributed by Scott Fenton, 11-Jun-2013.)

⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝑇 ∈ (0[,]1) ∧ ∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))))) → Σ𝑗 ∈ (1...𝑁)(((𝐵‘𝑗) − (𝐶‘𝑗))↑2) = (((1 − 𝑇)↑2) · Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)))

Theorem

ax5seglem3a 28813

Lemma for ax5seg 28821. (Contributed by Scott Fenton, 7-May-2015.)

⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑗 ∈ (1...𝑁)) → (((𝐴‘𝑗) − (𝐶‘𝑗)) ∈ ℝ ∧ ((𝐷‘𝑗) − (𝐹‘𝑗)) ∈ ℝ))

Theorem

ax5seglem3 28814*

Lemma for ax5seg 28821. Combine congruences for points on a line. (Contributed by Scott Fenton, 11-Jun-2013.)

⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2) = Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))

Theorem

ax5seglem4 28815*

Lemma for ax5seg 28821. Given two distinct points, the scaling constant in a betweenness statement is nonzero. (Contributed by Scott Fenton, 11-Jun-2013.)

⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ 𝐴 ≠ 𝐵) → 𝑇 ≠ 0)

Theorem

ax5seglem5 28816*

Lemma for ax5seg 28821. If 𝐵 is between 𝐴 and 𝐶, and 𝐴 is distinct from 𝐵, then 𝐴 is distinct from 𝐶. (Contributed by Scott Fenton, 11-Jun-2013.)

⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ (𝐴 ≠ 𝐵 ∧ 𝑇 ∈ (0[,]1) ∧ ∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))))) → Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2) ≠ 0)

Theorem

ax5seglem6 28817*

Lemma for ax5seg 28821. Given two line segments that are divided into pieces, if the pieces are congruent, then the scaling constant is the same. (Contributed by Scott Fenton, 12-Jun-2013.)

⊢ (((𝑁 ∈ ℕ ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁)))) ∧ (𝐴 ≠ 𝐵 ∧ (𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → 𝑇 = 𝑆)

Theorem

ax5seglem7 28818

Lemma for ax5seg 28821. An algebraic calculation needed further down the line. (Contributed by Scott Fenton, 12-Jun-2013.)

⊢ 𝐴 ∈ ℂ    &   ⊢ 𝑇 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    &   ⊢ 𝐷 ∈ ℂ    ⇒   ⊢ (𝑇 · ((𝐶 − 𝐷)↑2)) = ((((((1 − 𝑇) · 𝐴) + (𝑇 · 𝐶)) − 𝐷)↑2) + ((1 − 𝑇) · ((𝑇 · ((𝐴 − 𝐶)↑2)) − ((𝐴 − 𝐷)↑2))))

Theorem

ax5seglem8 28819

Lemma for ax5seg 28821. Use the weak deduction theorem to eliminate the hypotheses from ax5seglem7 28818. (Contributed by Scott Fenton, 11-Jun-2013.)

⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (𝑇 · ((𝐶 − 𝐷)↑2)) = ((((((1 − 𝑇) · 𝐴) + (𝑇 · 𝐶)) − 𝐷)↑2) + ((1 − 𝑇) · ((𝑇 · ((𝐴 − 𝐶)↑2)) − ((𝐴 − 𝐷)↑2)))))

Theorem

ax5seglem9 28820*

Lemma for ax5seg 28821. Take the calculation in ax5seglem8 28819 and turn it into a series of measurements. (Contributed by Scott Fenton, 12-Jun-2013.) (Revised by Mario Carneiro, 22-May-2014.)

⊢ (((𝑁 ∈ ℕ ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)))) ∧ (𝑇 ∈ (0[,]1) ∧ ∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))))) → (𝑇 · Σ𝑗 ∈ (1...𝑁)(((𝐶‘𝑗) − (𝐷‘𝑗))↑2)) = (Σ𝑗 ∈ (1...𝑁)(((𝐵‘𝑗) − (𝐷‘𝑗))↑2) + ((1 − 𝑇) · ((𝑇 · Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)) − Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐷‘𝑗))↑2)))))

Theorem

ax5seg 28821

The five segment axiom. Take two triangles 𝐴𝐷𝐶 and 𝐸𝐻𝐺, a point 𝐵 on 𝐴𝐶, and a point 𝐹 on 𝐸𝐺. If all corresponding line segments except for 𝐶𝐷 and 𝐺𝐻 are congruent, then so are 𝐶𝐷 and 𝐺𝐻. Axiom A5 of [Schwabhauser] p. 11. (Contributed by Scott Fenton, 12-Jun-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 ∧ 𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝐺⟩) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝐺⟩) ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩)) → ⟨𝐶, 𝐷⟩Cgr⟨𝐺, 𝐻⟩))

Theorem

axbtwnid 28822

Points are indivisible. That is, if 𝐴 lies between 𝐵 and 𝐵, then 𝐴 = 𝐵. Axiom A6 of [Schwabhauser] p. 11. (Contributed by Scott Fenton, 3-Jun-2013.)

⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (𝐴 Btwn ⟨𝐵, 𝐵⟩ → 𝐴 = 𝐵))

Theorem

axpaschlem 28823*

Lemma for axpasch 28824. Set up coefficients used in the proof. (Contributed by Scott Fenton, 5-Jun-2013.)

⊢ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) → ∃𝑟 ∈ (0[,]1)∃𝑝 ∈ (0[,]1)(𝑝 = ((1 − 𝑟) · (1 − 𝑇)) ∧ 𝑟 = ((1 − 𝑝) · (1 − 𝑆)) ∧ ((1 − 𝑟) · 𝑇) = ((1 − 𝑝) · 𝑆)))

Theorem

axpasch 28824*

The inner Pasch axiom. Take a triangle 𝐴𝐶𝐸, a point 𝐷 on 𝐴𝐶, and a point 𝐵 extending 𝐶𝐸. Then 𝐴𝐸 and 𝐷𝐵 intersect at some point 𝑥. Axiom A7 of [Schwabhauser] p. 12. (Contributed by Scott Fenton, 3-Jun-2013.)

⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) → ((𝐷 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐸 Btwn ⟨𝐵, 𝐶⟩) → ∃𝑥 ∈ (𝔼‘𝑁)(𝑥 Btwn ⟨𝐷, 𝐵⟩ ∧ 𝑥 Btwn ⟨𝐸, 𝐴⟩)))

Theorem

axlowdimlem1 28825

Lemma for axlowdim 28844. Establish a particular constant function as a function. (Contributed by Scott Fenton, 29-Jun-2013.)

⊢ ((3...𝑁) × {0}):(3...𝑁)⟶ℝ

Theorem

axlowdimlem2 28826

Lemma for axlowdim 28844. Show that two sets are disjoint. (Contributed by Scott Fenton, 29-Jun-2013.)

⊢ ((1...2) ∩ (3...𝑁)) = ∅

Theorem

axlowdimlem3 28827

Lemma for axlowdim 28844. Set up a union property for an interval of integers. (Contributed by Scott Fenton, 29-Jun-2013.)

⊢ (𝑁 ∈ (ℤ_≥‘2) → (1...𝑁) = ((1...2) ∪ (3...𝑁)))

Theorem

axlowdimlem4 28828

Lemma for axlowdim 28844. Set up a particular constant function. (Contributed by Scott Fenton, 17-Apr-2013.)

⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ {⟨1, 𝐴⟩, ⟨2, 𝐵⟩}:(1...2)⟶ℝ

Theorem

axlowdimlem5 28829

Lemma for axlowdim 28844. Show that a particular union is a point in Euclidean space. (Contributed by Scott Fenton, 29-Jun-2013.)

⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝑁 ∈ (ℤ_≥‘2) → ({⟨1, 𝐴⟩, ⟨2, 𝐵⟩} ∪ ((3...𝑁) × {0})) ∈ (𝔼‘𝑁))

Theorem

axlowdimlem6 28830

Lemma for axlowdim 28844. Show that three points are non-colinear. (Contributed by Scott Fenton, 29-Jun-2013.)

⊢ 𝐴 = ({⟨1, 0⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))    &   ⊢ 𝐵 = ({⟨1, 1⟩, ⟨2, 0⟩} ∪ ((3...𝑁) × {0}))    &   ⊢ 𝐶 = ({⟨1, 0⟩, ⟨2, 1⟩} ∪ ((3...𝑁) × {0}))    ⇒   ⊢ (𝑁 ∈ (ℤ_≥‘2) → ¬ (𝐴 Btwn ⟨𝐵, 𝐶⟩ ∨ 𝐵 Btwn ⟨𝐶, 𝐴⟩ ∨ 𝐶 Btwn ⟨𝐴, 𝐵⟩))

Theorem

axlowdimlem7 28831

Lemma for axlowdim 28844. Set up a point in Euclidean space. (Contributed by Scott Fenton, 29-Jun-2013.)

⊢ 𝑃 = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))    ⇒   ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑃 ∈ (𝔼‘𝑁))

Theorem

axlowdimlem8 28832

Lemma for axlowdim 28844. Calculate the value of 𝑃 at three. (Contributed by Scott Fenton, 21-Apr-2013.)

⊢ 𝑃 = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))    ⇒   ⊢ (𝑃‘3) = -1

Theorem

axlowdimlem9 28833

Lemma for axlowdim 28844. Calculate the value of 𝑃 away from three. (Contributed by Scott Fenton, 21-Apr-2013.)

⊢ 𝑃 = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))    ⇒   ⊢ ((𝐾 ∈ (1...𝑁) ∧ 𝐾 ≠ 3) → (𝑃‘𝐾) = 0)

Theorem

axlowdimlem10 28834

Lemma for axlowdim 28844. Set up a family of points in Euclidean space. (Contributed by Scott Fenton, 21-Apr-2013.)

⊢ 𝑄 = ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}))    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → 𝑄 ∈ (𝔼‘𝑁))

Theorem

axlowdimlem11 28835

Lemma for axlowdim 28844. Calculate the value of 𝑄 at its distinguished point. (Contributed by Scott Fenton, 21-Apr-2013.)

⊢ 𝑄 = ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}))    ⇒   ⊢ (𝑄‘(𝐼 + 1)) = 1

Theorem

axlowdimlem12 28836

Lemma for axlowdim 28844. Calculate the value of 𝑄 away from its distinguished point. (Contributed by Scott Fenton, 21-Apr-2013.)

⊢ 𝑄 = ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}))    ⇒   ⊢ ((𝐾 ∈ (1...𝑁) ∧ 𝐾 ≠ (𝐼 + 1)) → (𝑄‘𝐾) = 0)

Theorem

axlowdimlem13 28837

Lemma for axlowdim 28844. Establish that 𝑃 and 𝑄 are different points. (Contributed by Scott Fenton, 21-Apr-2013.)

⊢ 𝑃 = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))    &   ⊢ 𝑄 = ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}))    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → 𝑃 ≠ 𝑄)

Theorem

axlowdimlem14 28838

Lemma for axlowdim 28844. Take two possible 𝑄 from axlowdimlem10 28834. They are the same iff their distinguished values are the same. (Contributed by Scott Fenton, 21-Apr-2013.)

⊢ 𝑄 = ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}))    &   ⊢ 𝑅 = ({⟨(𝐽 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐽 + 1)}) × {0}))    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1)) ∧ 𝐽 ∈ (1...(𝑁 − 1))) → (𝑄 = 𝑅 → 𝐼 = 𝐽))

Theorem

axlowdimlem15 28839*

Lemma for axlowdim 28844. Set up a one-to-one function of points. (Contributed by Scott Fenton, 21-Apr-2013.)

⊢ 𝐹 = (𝑖 ∈ (1...(𝑁 − 1)) ↦ if(𝑖 = 1, ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})), ({⟨(𝑖 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝑖 + 1)}) × {0}))))    ⇒   ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝐹:(1...(𝑁 − 1))–1-1→(𝔼‘𝑁))

Theorem

axlowdimlem16 28840*

Lemma for axlowdim 28844. Set up a summation that will help establish distance. (Contributed by Scott Fenton, 21-Apr-2013.)

⊢ 𝑃 = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))    &   ⊢ 𝑄 = ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}))    ⇒   ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → Σ𝑖 ∈ (3...𝑁)((𝑃‘𝑖)↑2) = Σ𝑖 ∈ (3...𝑁)((𝑄‘𝑖)↑2))

Theorem

axlowdimlem17 28841

Lemma for axlowdim 28844. Establish a congruence result. (Contributed by Scott Fenton, 22-Apr-2013.) (Proof shortened by Mario Carneiro, 22-May-2014.)

⊢ 𝑃 = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))    &   ⊢ 𝑄 = ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}))    &   ⊢ 𝐴 = ({⟨1, 𝑋⟩, ⟨2, 𝑌⟩} ∪ ((3...𝑁) × {0}))    &   ⊢ 𝑋 ∈ ℝ    &   ⊢ 𝑌 ∈ ℝ    ⇒   ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝐼 ∈ (2...(𝑁 − 1))) → ⟨𝑃, 𝐴⟩Cgr⟨𝑄, 𝐴⟩)

Theorem

axlowdim1 28842*

The lower dimension axiom for one dimension. In any dimension, there are at least two distinct points. Theorem 3.13 of [Schwabhauser] p. 32, where it is derived from axlowdim2 28843. (Contributed by Scott Fenton, 22-Apr-2013.)

⊢ (𝑁 ∈ ℕ → ∃𝑥 ∈ (𝔼‘𝑁)∃𝑦 ∈ (𝔼‘𝑁)𝑥 ≠ 𝑦)

Theorem

axlowdim2 28843*

The lower two-dimensional axiom. In any space where the dimension is greater than one, there are three non-colinear points. Axiom A8 of [Schwabhauser] p. 12. (Contributed by Scott Fenton, 15-Apr-2013.)

⊢ (𝑁 ∈ (ℤ_≥‘2) → ∃𝑥 ∈ (𝔼‘𝑁)∃𝑦 ∈ (𝔼‘𝑁)∃𝑧 ∈ (𝔼‘𝑁) ¬ (𝑥 Btwn ⟨𝑦, 𝑧⟩ ∨ 𝑦 Btwn ⟨𝑧, 𝑥⟩ ∨ 𝑧 Btwn ⟨𝑥, 𝑦⟩))

Theorem

axlowdim 28844*

The general lower dimension axiom. Take a dimension 𝑁 greater than or equal to three. Then, there are three non-colinear points in 𝑁 dimensional space that are equidistant from 𝑁 − 1 distinct points. Derived from remarks in Tarski's System of Geometry, Alfred Tarski and Steven Givant, Bulletin of Symbolic Logic, Volume 5, Number 2 (1999), 175-214. (Contributed by Scott Fenton, 22-Apr-2013.)

⊢ (𝑁 ∈ (ℤ_≥‘3) → ∃𝑝∃𝑥 ∈ (𝔼‘𝑁)∃𝑦 ∈ (𝔼‘𝑁)∃𝑧 ∈ (𝔼‘𝑁)(𝑝:(1...(𝑁 − 1))–1-1→(𝔼‘𝑁) ∧ ∀𝑖 ∈ (2...(𝑁 − 1))(⟨(𝑝‘1), 𝑥⟩Cgr⟨(𝑝‘𝑖), 𝑥⟩ ∧ ⟨(𝑝‘1), 𝑦⟩Cgr⟨(𝑝‘𝑖), 𝑦⟩ ∧ ⟨(𝑝‘1), 𝑧⟩Cgr⟨(𝑝‘𝑖), 𝑧⟩) ∧ ¬ (𝑥 Btwn ⟨𝑦, 𝑧⟩ ∨ 𝑦 Btwn ⟨𝑧, 𝑥⟩ ∨ 𝑧 Btwn ⟨𝑥, 𝑦⟩)))

Theorem

axeuclidlem 28845*

Lemma for axeuclid 28846. Handle the algebraic aspects of the theorem. (Contributed by Scott Fenton, 9-Sep-2013.)

⊢ ((((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑇 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (0[,]1) ∧ 𝑄 ∈ (0[,]1) ∧ 𝑃 ≠ 0) ∧ ∀𝑖 ∈ (1...𝑁)(((1 − 𝑃) · (𝐴‘𝑖)) + (𝑃 · (𝑇‘𝑖))) = (((1 − 𝑄) · (𝐵‘𝑖)) + (𝑄 · (𝐶‘𝑖)))) → ∃𝑥 ∈ (𝔼‘𝑁)∃𝑦 ∈ (𝔼‘𝑁)∃𝑟 ∈ (0[,]1)∃𝑠 ∈ (0[,]1)∃𝑢 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)((𝐵‘𝑖) = (((1 − 𝑟) · (𝐴‘𝑖)) + (𝑟 · (𝑥‘𝑖))) ∧ (𝐶‘𝑖) = (((1 − 𝑠) · (𝐴‘𝑖)) + (𝑠 · (𝑦‘𝑖))) ∧ (𝑇‘𝑖) = (((1 − 𝑢) · (𝑥‘𝑖)) + (𝑢 · (𝑦‘𝑖)))))

Theorem

axeuclid 28846*

Euclid's axiom. Take an angle 𝐵𝐴𝐶 and a point 𝐷 between 𝐵 and 𝐶. Now, if you extend the segment 𝐴𝐷 to a point 𝑇, then 𝑇 lies between two points 𝑥 and 𝑦 that lie on the angle. Axiom A10 of [Schwabhauser] p. 13. (Contributed by Scott Fenton, 9-Sep-2013.)

⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝑇 ∈ (𝔼‘𝑁))) → ((𝐷 Btwn ⟨𝐴, 𝑇⟩ ∧ 𝐷 Btwn ⟨𝐵, 𝐶⟩ ∧ 𝐴 ≠ 𝐷) → ∃𝑥 ∈ (𝔼‘𝑁)∃𝑦 ∈ (𝔼‘𝑁)(𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ 𝐶 Btwn ⟨𝐴, 𝑦⟩ ∧ 𝑇 Btwn ⟨𝑥, 𝑦⟩)))

Theorem

axcontlem1 28847*

Lemma for axcont 28859. Change bound variables for later use. (Contributed by Scott Fenton, 20-Jun-2013.)

⊢ 𝐹 = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑍‘𝑖)) + (𝑡 · (𝑈‘𝑖)))))}    ⇒   ⊢ 𝐹 = {⟨𝑦, 𝑠⟩ ∣ (𝑦 ∈ 𝐷 ∧ (𝑠 ∈ (0[,)+∞) ∧ ∀𝑗 ∈ (1...𝑁)(𝑦‘𝑗) = (((1 − 𝑠) · (𝑍‘𝑗)) + (𝑠 · (𝑈‘𝑗)))))}

Theorem

axcontlem2 28848*

Lemma for axcont 28859. The idea here is to set up a mapping 𝐹 that will allow to transfer dedekind 11409 to two sets of points. Here, we set up 𝐹 and show its domain and codomain. (Contributed by Scott Fenton, 17-Jun-2013.)

⊢ 𝐷 = {𝑝 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn ⟨𝑍, 𝑝⟩ ∨ 𝑝 Btwn ⟨𝑍, 𝑈⟩)}    &   ⊢ 𝐹 = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑍‘𝑖)) + (𝑡 · (𝑈‘𝑖)))))}    ⇒   ⊢ (((𝑁 ∈ ℕ ∧ 𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ (𝔼‘𝑁)) ∧ 𝑍 ≠ 𝑈) → 𝐹:𝐷–1-1-onto→(0[,)+∞))

Theorem

axcontlem3 28849*

Lemma for axcont 28859. Given the separation assumption, 𝐵 is a subset of 𝐷. (Contributed by Scott Fenton, 18-Jun-2013.)

⊢ 𝐷 = {𝑝 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn ⟨𝑍, 𝑝⟩ ∨ 𝑝 Btwn ⟨𝑍, 𝑈⟩)}    ⇒   ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ (𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝑍 ≠ 𝑈)) → 𝐵 ⊆ 𝐷)

Theorem

axcontlem4 28850*

Lemma for axcont 28859. Given the separation assumption, 𝐴 is a subset of 𝐷. (Contributed by Scott Fenton, 18-Jun-2013.)

⊢ 𝐷 = {𝑝 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn ⟨𝑍, 𝑝⟩ ∨ 𝑝 Btwn ⟨𝑍, 𝑈⟩)}    ⇒   ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → 𝐴 ⊆ 𝐷)

Theorem

axcontlem5 28851*

Lemma for axcont 28859. Compute the value of 𝐹. (Contributed by Scott Fenton, 18-Jun-2013.)

⊢ 𝐷 = {𝑝 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn ⟨𝑍, 𝑝⟩ ∨ 𝑝 Btwn ⟨𝑍, 𝑈⟩)}    &   ⊢ 𝐹 = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑍‘𝑖)) + (𝑡 · (𝑈‘𝑖)))))}    ⇒   ⊢ ((((𝑁 ∈ ℕ ∧ 𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ (𝔼‘𝑁)) ∧ 𝑍 ≠ 𝑈) ∧ 𝑃 ∈ 𝐷) → ((𝐹‘𝑃) = 𝑇 ↔ (𝑇 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑃‘𝑖) = (((1 − 𝑇) · (𝑍‘𝑖)) + (𝑇 · (𝑈‘𝑖))))))

Theorem

axcontlem6 28852*

Lemma for axcont 28859. State the defining properties of the value of 𝐹. (Contributed by Scott Fenton, 19-Jun-2013.)

⊢ 𝐷 = {𝑝 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn ⟨𝑍, 𝑝⟩ ∨ 𝑝 Btwn ⟨𝑍, 𝑈⟩)}    &   ⊢ 𝐹 = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑍‘𝑖)) + (𝑡 · (𝑈‘𝑖)))))}    ⇒   ⊢ ((((𝑁 ∈ ℕ ∧ 𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ (𝔼‘𝑁)) ∧ 𝑍 ≠ 𝑈) ∧ 𝑃 ∈ 𝐷) → ((𝐹‘𝑃) ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑃‘𝑖) = (((1 − (𝐹‘𝑃)) · (𝑍‘𝑖)) + ((𝐹‘𝑃) · (𝑈‘𝑖)))))

Theorem

axcontlem7 28853*

Lemma for axcont 28859. Given two points in 𝐷, one preceeds the other iff its scaling constant is less than the other point's. (Contributed by Scott Fenton, 18-Jun-2013.)

⊢ 𝐷 = {𝑝 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn ⟨𝑍, 𝑝⟩ ∨ 𝑝 Btwn ⟨𝑍, 𝑈⟩)}    &   ⊢ 𝐹 = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑍‘𝑖)) + (𝑡 · (𝑈‘𝑖)))))}    ⇒   ⊢ ((((𝑁 ∈ ℕ ∧ 𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ (𝔼‘𝑁)) ∧ 𝑍 ≠ 𝑈) ∧ (𝑃 ∈ 𝐷 ∧ 𝑄 ∈ 𝐷)) → (𝑃 Btwn ⟨𝑍, 𝑄⟩ ↔ (𝐹‘𝑃) ≤ (𝐹‘𝑄)))

Theorem

axcontlem8 28854*

Lemma for axcont 28859. A point in 𝐷 is between two others if its function value falls in the middle. (Contributed by Scott Fenton, 18-Jun-2013.)

⊢ 𝐷 = {𝑝 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn ⟨𝑍, 𝑝⟩ ∨ 𝑝 Btwn ⟨𝑍, 𝑈⟩)}    &   ⊢ 𝐹 = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑍‘𝑖)) + (𝑡 · (𝑈‘𝑖)))))}    ⇒   ⊢ ((((𝑁 ∈ ℕ ∧ 𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ (𝔼‘𝑁)) ∧ 𝑍 ≠ 𝑈) ∧ (𝑃 ∈ 𝐷 ∧ 𝑄 ∈ 𝐷 ∧ 𝑅 ∈ 𝐷)) → (((𝐹‘𝑃) ≤ (𝐹‘𝑄) ∧ (𝐹‘𝑄) ≤ (𝐹‘𝑅)) → 𝑄 Btwn ⟨𝑃, 𝑅⟩))

Theorem

axcontlem9 28855*

Lemma for axcont 28859. Given the separation assumption, all values of 𝐹 over 𝐴 are less than or equal to all values of 𝐹 over 𝐵. (Contributed by Scott Fenton, 20-Jun-2013.)

⊢ 𝐷 = {𝑝 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn ⟨𝑍, 𝑝⟩ ∨ 𝑝 Btwn ⟨𝑍, 𝑈⟩)}    &   ⊢ 𝐹 = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑍‘𝑖)) + (𝑡 · (𝑈‘𝑖)))))}    ⇒   ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → ∀𝑛 ∈ (𝐹 “ 𝐴)∀𝑚 ∈ (𝐹 “ 𝐵)𝑛 ≤ 𝑚)

Theorem

axcontlem10 28856*

Lemma for axcont 28859. Given a handful of assumptions, derive the conclusion of the final theorem. (Contributed by Scott Fenton, 20-Jun-2013.)

⊢ 𝐷 = {𝑝 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn ⟨𝑍, 𝑝⟩ ∨ 𝑝 Btwn ⟨𝑍, 𝑈⟩)}    &   ⊢ 𝐹 = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑍‘𝑖)) + (𝑡 · (𝑈‘𝑖)))))}    ⇒   ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑏 Btwn ⟨𝑥, 𝑦⟩)

Theorem

axcontlem11 28857*

Lemma for axcont 28859. Eliminate the hypotheses from axcontlem10 28856. (Contributed by Scott Fenton, 20-Jun-2013.)

⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑏 Btwn ⟨𝑥, 𝑦⟩)

Theorem

axcontlem12 28858*

Lemma for axcont 28859. Eliminate the trivial cases from the previous lemmas. (Contributed by Scott Fenton, 20-Jun-2013.)

⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ 𝑍 ∈ (𝔼‘𝑁)) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑏 Btwn ⟨𝑥, 𝑦⟩)

Theorem

axcont 28859*

The axiom of continuity. Take two sets of points 𝐴 and 𝐵. If all the points in 𝐴 come before the points of 𝐵 on a line, then there is a point separating the two. Axiom A11 of [Schwabhauser] p. 13. (Contributed by Scott Fenton, 20-Jun-2013.)

⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∃𝑎 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑎, 𝑦⟩)) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑏 Btwn ⟨𝑥, 𝑦⟩)

16.4.2.3  EE^n fulfills Tarski's Axioms

Syntax

ceeng 28860

Extends class notation with the Tarski geometry structure for 𝔼↑𝑁.

class EEG

Definition

df-eeng 28861*

Define the geometry structure for 𝔼↑𝑁. (Contributed by Thierry Arnoux, 24-Aug-2017.)

⊢ EEG = (𝑛 ∈ ℕ ↦ ({⟨(Base‘ndx), (𝔼‘𝑛)⟩, ⟨(dist‘ndx), (𝑥 ∈ (𝔼‘𝑛), 𝑦 ∈ (𝔼‘𝑛) ↦ Σ𝑖 ∈ (1...𝑛)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))⟩} ∪ {⟨(Itv‘ndx), (𝑥 ∈ (𝔼‘𝑛), 𝑦 ∈ (𝔼‘𝑛) ↦ {𝑧 ∈ (𝔼‘𝑛) ∣ 𝑧 Btwn ⟨𝑥, 𝑦⟩})⟩, ⟨(LineG‘ndx), (𝑥 ∈ (𝔼‘𝑛), 𝑦 ∈ ((𝔼‘𝑛) ∖ {𝑥}) ↦ {𝑧 ∈ (𝔼‘𝑛) ∣ (𝑧 Btwn ⟨𝑥, 𝑦⟩ ∨ 𝑥 Btwn ⟨𝑧, 𝑦⟩ ∨ 𝑦 Btwn ⟨𝑥, 𝑧⟩)})⟩}))

Theorem

eengv 28862*

The value of the Euclidean geometry for dimension 𝑁. (Contributed by Thierry Arnoux, 15-Mar-2019.)

⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) = ({⟨(Base‘ndx), (𝔼‘𝑁)⟩, ⟨(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))⟩} ∪ {⟨(Itv‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ {𝑧 ∈ (𝔼‘𝑁) ∣ 𝑧 Btwn ⟨𝑥, 𝑦⟩})⟩, ⟨(LineG‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ ((𝔼‘𝑁) ∖ {𝑥}) ↦ {𝑧 ∈ (𝔼‘𝑁) ∣ (𝑧 Btwn ⟨𝑥, 𝑦⟩ ∨ 𝑥 Btwn ⟨𝑧, 𝑦⟩ ∨ 𝑦 Btwn ⟨𝑥, 𝑧⟩)})⟩}))

Theorem

eengstr 28863

The Euclidean geometry as a structure. (Contributed by Thierry Arnoux, 15-Mar-2019.)

⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) Struct ⟨1, ;17⟩)

Theorem

eengbas 28864

The Base of the Euclidean geometry. (Contributed by Thierry Arnoux, 15-Mar-2019.)

⊢ (𝑁 ∈ ℕ → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))

Theorem

ebtwntg 28865

The betweenness relation used in the Tarski structure for the Euclidean geometry is the same as Btwn. (Contributed by Thierry Arnoux, 15-Mar-2019.)

⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑃 = (Base‘(EEG‘𝑁))    &   ⊢ 𝐼 = (Itv‘(EEG‘𝑁))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝑍 Btwn ⟨𝑋, 𝑌⟩ ↔ 𝑍 ∈ (𝑋𝐼𝑌)))

Theorem

ecgrtg 28866

The congruence relation used in the Tarski structure for the Euclidean geometry is the same as Cgr. (Contributed by Thierry Arnoux, 15-Mar-2019.)

⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑃 = (Base‘(EEG‘𝑁))    &   ⊢ − = (dist‘(EEG‘𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩ ↔ (𝐴 − 𝐵) = (𝐶 − 𝐷)))

Theorem

elntg 28867*

The line definition in the Tarski structure for the Euclidean geometry. (Contributed by Thierry Arnoux, 7-Apr-2019.)

⊢ 𝑃 = (Base‘(EEG‘𝑁))    &   ⊢ 𝐼 = (Itv‘(EEG‘𝑁))    ⇒   ⊢ (𝑁 ∈ ℕ → (LineG‘(EEG‘𝑁)) = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))

Theorem

elntg2 28868*

The line definition in the Tarski structure for the Euclidean geometry. In contrast to elntg 28867, the betweenness can be strengthened by excluding 1 resp. 0 from the related intervals (because of 𝑥 ≠ 𝑦). (Contributed by AV, 14-Feb-2023.)

⊢ 𝑃 = (Base‘(EEG‘𝑁))    &   ⊢ 𝐼 = (1...𝑁)    ⇒   ⊢ (𝑁 ∈ ℕ → (LineG‘(EEG‘𝑁)) = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝 ∈ 𝑃 ∣ (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ 𝐼 (𝑝‘𝑖) = (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ 𝐼 (𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ 𝐼 (𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))))}))

Theorem

eengtrkg 28869

The geometry structure for 𝔼↑𝑁 is a Tarski geometry. (Contributed by Thierry Arnoux, 15-Mar-2019.)

⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiG)

Theorem

eengtrkge 28870

The geometry structure for 𝔼↑𝑁 is a Euclidean geometry. (Contributed by Thierry Arnoux, 15-Mar-2019.)

⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiG_E)

PART 17  GRAPH THEORY

To give an overview of the definitions and terms used in the context of graph theory, a glossary is provided in the following, mainly according to definitions in [Bollobas] p. 1-8 or in [Diestel] p. 2-28. Although this glossary concentrates on undirected graphs, many of the concepts are also useful for directed graphs.

Basic concepts:

Term	Reference	Definition	Remarks
Vertex	df-vtx 28883	A vertex of a graph 𝐺 is an element of the set of vertices (Vtx‘𝐺) of the graph 𝐺.	The set of vertices (Vtx‘𝐺) (corresponding to V(G) in [Bollobas] p. 1) is usually the first component 𝑉 of the graph 𝐺 if it is represented by an ordered pair ⟨𝑉, 𝐸⟩ (see opvtxfv 28889), or the base set (Base‘𝐺) of the graph 𝐺 if it is represented as extensible structure (see basvtxval 28901).
Edge	df-edg 28933	An edge of a graph 𝐺 is a nonempty set of vertices of the graph. It is said that these vertices are "joined" or "connected" by the edge, see [Bollobas] p. 1.	The set of edges (Edg‘𝐺) (corresponding to E(G) in [Bollobas] p. 1) is usually the range ran 𝐸 of the second component 𝐸 of the graph 𝐺 if it is represented by an ordered pair ⟨𝑉, 𝐸⟩, or the range of the component (.ef‘𝐺) of the graph 𝐺 if it is represented as extensible structure.
Loop		A loop in a graph 𝐺 is an edge which connects a single vertex with itself (or, according to [Bollobas] p. 7 "joins a vertex to itself").	In other words, a loop is an edge 𝑒 ∈ (Edg‘𝐺) which is a singleton consisting of a vertex 𝑣 ∈ (Vtx‘𝐺): 𝑒 = {𝑣}
Edge function resp. indexed edges	df-iedg 28884	An edge function (or indexed set of edges) of a graph 𝐺 is a mapping of an arbitrary index set to nonempty sets of vertices of the graph.	The edge function (iEdg‘𝐺) is usually the second component 𝐸 of the graph 𝐺 if it is represented by an ordered pair ⟨𝑉, 𝐸 > (see opiedgfv 28892), or the component (.ef‘𝐺) of the graph 𝐺 if it is represented as extensible structure (see edgfiedgval 28902). The set of edges of a graph 𝐺 is the range of its edge function: (Edg‘𝐺) = ran (iEdg‘𝐺), see edgval 28934. Whereas the concept of plain edges is sufficient for simple hypergraphs, indexed edges are required for e.g., multigraphs in which the same vertices may be connected by more than one edge.

Basic kinds of graphs:

Term	Reference	Definition	Remarks
Undirected hypergraph	df-uhgr 28943	a class 𝐺 with an edge function 𝐸 = (iEdg‘𝐺) which is a function into the power set of the vertices 𝑉 = (Vtx‘𝐺): ran 𝐸 ⊆ (𝒫 𝑉 ∖ {∅}).	In this most general definition of a graph, an "edge" may connect three or more vertices with each other, see [Berge] p. 1. In Wikipedia "Hypergraph", see https://en.wikipedia.org/wiki/Hypergraph 28943 (18-Jan-2020) such a hypergraph is called a "non-simple hypergraph", "multiple hypergraph" or "multi-hypergraphs". According to Wikipedia "Incidence structure", see https://en.wikipedia.org/wiki/Incidence_structure 28943 (18-Jan-2020) "Each hypergraph [...] can be regarded as an incidence structure in which the [vertices] play the role of "points", the corresponding family of [edges] plays the role of "lines" and the incidence relation is set membership". Notice that by using (Edg‘𝐺) the (possibly more than one) edges connecting the same vertices could not be distinguished anymore. Therefore, this representation will only be used for undirected simple hypergraphs.
Undirected simple hypergraph	df-ushgr 28944	a class 𝐺 with an edge function 𝐸 = (iEdg‘𝐺) which is a one-to-one function into the power set of the vertices 𝑉 = (Vtx‘𝐺): ran 𝐸 ⊆ (𝒫 𝑉 ∖ {∅}).	See also Wikipedia "Hypergraph", https://en.wikipedia.org/wiki/Hypergraph 28944 (18-Jan-2020). This is how a "hypergraph" is defined in Section I.1 in [Bollobas] p. 7 or the definition in section 1.10 in [Diestel] p. 27. A simple hypergraph has at most one edge between the same vertices, hence a pseudograph needs not be a simple hypergraph. According to [Berge] p. 1, "A simple hypergraph (or "Sperner family") is a hypergraph H = { E_1, E_2, ..., E_m } such that E_i C_ E_j => i = j". By this definition, a simple hypergraph cannot contain the edges E_1 = { v_1 , v_2 } and E_2 = { v_1, v_2, v_3 }, because E_1 C_ E_2, but 1 =/= 2.
Undirected loop-free hypergraph	---	an undirected hypergraph without a loop, i.e. all edges connect at least two vertices.
Undirected pseudograph	df-upgr 28967	a class 𝐺 with an edge function 𝐸 = (iEdg‘𝐺) which is a function into the set of (proper or not proper) unordered pairs of vertices 𝑉 = (Vtx‘𝐺).	A proper unordered pair contains two different elements, a not proper unordered pair contains two times the same element, so it is a singleton (see preqsn 4864). This means a pseudograph may contain loops. This definition corresponds to the definition of a "multigraph" in Section I.1 in [Bollobas] p. 7, "In a multigraph both multiple edges [joining two vertices] and multiple loops [joining a vertex to itself] are allowed", or in [Diestel] p. 28, "A multigraph is a pair (V,E) of disjoint sets (of vertices and edges) together with a map E -> V u. [V]^2 assigning to every edge either one or two vertices, its end(s).".
Undirected multigraph	df-umgr 28968	a class 𝐺 with an edge function 𝐸 = (iEdg‘𝐺) which is a function into the set of (proper!) unordered pairs of vertices 𝑉 = (Vtx‘𝐺).	This definition is according to Chartrand, Gary and Zhang, Ping (2012): "A First Course in Graph Theory.", Dover, ISBN 978-0-486-48368-9, section 1.4, p. 26: "A multigraph M consists of a finite nonempty set V of vertices and a set E of edges, where every two vertices of M are joined by a finite number of edges (possibly zero). If two or more edges join the same pair of (distinct) vertices, then these edges are called parallel edges." A proper unordered pair contains two different elements, therefore a multigraph does not have loops.
Undirected simple pseudograph	df-uspgr 29035	a class 𝐺 with an edge function 𝐸 = (iEdg‘𝐺) which is a one-to-one function into the set of (proper or not proper) unordered pairs of vertices 𝑉 = (Vtx‘𝐺).	This means that there is at most one edge between two vertices, and at most one loop from a vertex to itself.
Undirected simple graph	df-usgr 29036	a class 𝐺 with an edge function 𝐸 = (iEdg‘𝐺) which is a one-to-one function into the set of (proper!) unordered pairs of vertices 𝑉 = (Vtx‘𝐺).	An ordered pair ⟨𝑉, 𝐸⟩ of two distinct sets 𝑉 (the vertices) and 𝐸 (the edges), the "usual" definition of a "graph", see, for example, the definition in section I.1 of [Bollobas] p. 1 or in section 1.1 of [Diestel] p. 2, can be identified with an undirected simple graph without loops by "indexing" the edges with themselves, see usgrausgrb 29054.
Finite graph	df-fusgr 29202	a graph 𝐺 with a finite set of vertices 𝑉 = (Vtx‘𝐺).	See definitions in [Bollobas] p. 1 or [Diestel] p. 2. In simple graphs, the set of (indexed) edges (iEdg‘𝐺) (and therefore also the set of edges (Edg‘𝐺)) is finite if 𝑉 = (Vtx‘𝐺) is finite, see fusgrfis 29215. The number of edges is limited by (𝑛C2) (or "𝑛 choose 2") with 𝑛 = (♯‘𝑉), see fusgrmaxsize 29350. Analogously, the number of edges 𝐸 = (iEdg‘𝐺) of an undirected simple pseudograph (which may have loops) is limited by ((𝑛 + 1)C2). In pseudographs or multigraphs, however, 𝐸 can be infinite although 𝑉 is finite.
Graph of finite size	---	a graph 𝐺 with a finite set 𝐸 = (iEdg‘𝐺), i.e. with a finite number of edges.	A graph can be of finite size although its set of vertices is infinite (most of the vertices would not be connected by an edge).

Terms and properties of graphs:

Term	Reference	Definition	Remarks
Edge joining resp. connecting (two) vertices	---	An edge 𝑒 ∈ (Edg‘𝐺) joins resp. connects the vertices v_1, v_2, ... v_n (𝑛 ∈ ℕ) if 𝑒 = { v_1, v_2, ... v_n }.	If 𝑛 = 1, 𝑒 = { v_1 } is a loop, if 𝑛 = 2, 𝑒 = { v_1 , v_2 } is an edge as it is usually defined, see definition in Section I.1 in [Bollobas] p. 1.
(Two) Endvertices of an edge	see definition in Section I.1 in [Bollobas] p. 1.	If an edge 𝑒 ∈ (Edg‘𝐺) joins the vertices v_1, v_2, ... v_n (𝑛 ∈ ℕ), then the vertices v_1, v_2, ... v_n are called the endvertices of the edge 𝑒.
(Two) Adjacent vertices	see definition in Section I.1 in [Bollobas] p. 1/2.	The vertices v_1, v_2, ... v_n (𝑛 ∈ ℕ) are adjacent if there is an edge e = { v_1, v_2, ... v_n } joining these vertices. In this case, the vertices are incident with the edge e (see definition in Section I.1 in [Bollobas] p. 2) or connected by the edge e.
Edge ending at a vertex		An edge 𝑒 ∈ (Edg‘𝐺) is ending at a vertex 𝑣 if the vertex is an endvertex of the edge: 𝑣 ∈ 𝑒. In other words, the vertex 𝑣 is incident with the edge 𝑒.
(Two) Adjacent edges		The edges e_0, e_1, ... e_n (𝑛 ∈ ℕ) are adjacent if they have exactly one common endvertex.	Generalization of definition in Section I.1 in [Bollobas] p. 2.
Order of a graph	see definition in Section I.1 in [Bollobas] p. 3	The order of a graph 𝐺 is the number of vertices in the graph: (♯‘(Vtx‘𝐺)).
Size of a graph	see definition in Section I.1 in [Bollobas] p. 3	The size of a graph 𝐺 is the number of edges in the graph: (♯‘(iEdg‘𝐺)). Or, for a simple graph 𝐺: (♯‘(Edg‘𝐺))).
Neighborhood of a vertex	df-nbgr 29218 resp. definition in Section I.1 in [Bollobas] p. 3	A vertex connected with a vertex 𝑣 by an edge is called a neighbor of the vertex 𝑣. The set of neighbors of a vertex 𝑣 is called the neighborhood (or open neighborhood) of the vertex 𝑣. The closed neighborhood is the union of the (open) neighborhood of the vertex 𝑣 with {𝑣} (see df-clnbgr 47296).
Degree of a vertex	df-vtxdg 29352	The degree of a vertex is the number of the edges ending at this vertex.	In a simple graph, the degree of a vertex is the number of neighbors of this vertex, see definition in Section I.1 in [Bollobas] p. 3
Isolated vertex	usgrvd0nedg 29419	A vertex is called isolated if it is not an endvertex of any edge, thus having degree 0.
Universal vertex	df-uvtx 29271	A vertex is called universal if it is connected with every other vertex of the graph by an edge, thus having degree ((♯‘(Vtx‘𝐺)) − ).

Special kinds of graphs:

Term	Reference	Definition	Remarks
Complete graph	df-cplgr 29296	A graph is called complete if each pair of vertices is connected by an edge.	The size of a complete undirected simple graph of order 𝑛 is (𝑛C2) (or "𝑛 choose 2"), see cusgrsize 29340.
Empty graph	uhgr0e 28956	A graph is called empty if it has no edges.
Null graph	uhgr0 28958 and uhgr0vb 28957	A graph is called a null graph if it has no vertices (and therefore also no edges).
Trivial graph	usgr1v 29141	A graph is called the "trivial graph" if it has only one vertex and no edges.
Connected graph	df-conngr 30069 resp. definition in Section I.1 in [Bollobas] p. 6	A graph is called connected if for each pair of vertices there is a path between these vertices.

For the terms "Path", "Walk", "Trail", "Circuit", "Cycle" see the remarks below and the definitions in Section I.1 in [Bollobas] p. 4-5.

17.1  Vertices and edges

In the following, the vertices and (indexed) edges for an arbitrary class 𝐺 (called "graph" in the following) are defined and examined. The main result of this section is to show that the set of vertices (Vtx‘𝐺) of a graph 𝐺 is the first component 𝑉 of the graph 𝐺 if it is represented by an ordered pair ⟨𝑉, 𝐸⟩ (see opvtxfv 28889), or the base set (Base‘𝐺) of the graph 𝐺 if it is represented as extensible structure (see basvtxval 28901), and that the set of indexed edges resp. the edge function (iEdg‘𝐺) is the second component 𝐸 of the graph 𝐺 if it is represented by an ordered pair ⟨𝑉, 𝐸⟩ (see opiedgfv 28892), or the component (.ef‘𝐺) of the graph 𝐺 if it is represented as extensible structure (see edgfiedgval 28902). Finally, it is shown that the set of edges of a graph 𝐺 is the range of its edge function: (Edg‘𝐺) = ran (iEdg‘𝐺), see edgval 28934.

Usually, a graph 𝐺 is a set. If 𝐺 is a proper class, however, it represents the null graph (without vertices and edges), because (Vtx‘𝐺) = ∅ and (iEdg‘𝐺) = ∅ holds, see vtxvalprc 28930 and iedgvalprc 28931.

Up to the end of this section, the edges need not be related to the vertices. Once undirected hypergraphs are defined (see df-uhgr 28943), the edges become nonempty sets of vertices, and by this obtain their meaning as "connectors" of vertices.

17.1.1  The edge function extractor for extensible structures

Syntax

cedgf 28871

Extend class notation with an edge function.

class .ef

Definition

df-edgf 28872

Define the edge function (indexed edges) of a graph. (Contributed by AV, 18-Jan-2020.) Use its index-independent form edgfid 28873 instead. (New usage is discouraged.)

⊢ .ef = Slot ;18

Theorem

edgfid 28873

Utility theorem: index-independent form of df-edgf 28872. (Contributed by AV, 16-Nov-2021.)

⊢ .ef = Slot (.ef‘ndx)

Theorem

edgfndx 28874

Index value of the df-edgf 28872 slot. (Contributed by AV, 13-Oct-2024.) (New usage is discouraged.)

⊢ (.ef‘ndx) = ;18

Theorem

edgfndxnn 28875

The index value of the edge function extractor is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 13-Oct-2024.)

⊢ (.ef‘ndx) ∈ ℕ

Theorem

edgfndxid 28876

The value of the edge function extractor is the value of the corresponding slot of the structure. (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 28-Oct-2024.)

⊢ (𝐺 ∈ 𝑉 → (.ef‘𝐺) = (𝐺‘(.ef‘ndx)))

Theorem

edgfndxidOLD 28877

Obsolete version of edgfndxid 28876 as of 28-Oct-2024. The value of the edge function extractor is the value of the corresponding slot of the structure. (Contributed by AV, 21-Sep-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐺 ∈ 𝑉 → (.ef‘𝐺) = (𝐺‘(.ef‘ndx)))

Theorem

basendxltedgfndx 28878

The index value of the Base slot is less than the index value of the .ef slot. (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 30-Oct-2024.)

⊢ (Base‘ndx) < (.ef‘ndx)

Theorem

baseltedgfOLD 28879

Obsolete proof of basendxltedgfndx 28878 as of 30-Oct-2024. The index value of the Base slot is less than the index value of the .ef slot. (Contributed by AV, 21-Sep-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (Base‘ndx) < (.ef‘ndx)

Theorem

basendxnedgfndx 28880

The slots Base and .ef are different. (Contributed by AV, 21-Sep-2020.)

⊢ (Base‘ndx) ≠ (.ef‘ndx)

17.1.2  Vertices and indexed edges

The key concepts in graph theory are vertices and edges. In general, a graph "consists" (at least) of two sets: the set of vertices and the set of edges. The edges "connect" vertices. The meaning of "connect" is different for different kinds of graphs (directed/undirected graphs, hyper-/pseudo-/ multi-/simple graphs, etc.). The simplest way to represent a graph (of any kind) is to define a graph as "an ordered pair of disjoint sets (V, E)" (see section I.1 in [Bollobas] p. 1), or in the notation of Metamath: ⟨𝑉, 𝐸⟩.

Another way is to regard a graph as a mathematical structure, which consists at least of a set (of vertices) and a relation between the vertices (edge function), but which can be enhanced by additional features (see Wikipedia "Mathematical structure", 24-Sep-2020, https://en.wikipedia.org/wiki/Mathematical_structure): "In mathematics, a structure is a set endowed with some additional features on the set (e.g., operation, relation, metric, topology). Often, the additional features are attached or related to the set, so as to provide it with some additional meaning or significance.". Such structures are provided as "extensible structures" in Metamath, see df-struct 17119.

To allow for expressing and proving most of the theorems for graphs independently from their representation, the functions Vtx and iEdg are defined (see df-vtx 28883 and df-iedg 28884), which provide the vertices resp. (indexed) edges of an arbitrary class 𝐺 which represents a graph: (Vtx‘𝐺) resp. (iEdg‘𝐺). In literature, these functions are often denoted also by "V" and "E", see section I.1 in [Bollobas] p. 1 ("If G is a graph, then V = V(G) is the vertex set of G, and E = E(G) is the edge set.") or section 1.1 in [Diestel] p. 2 ("The vertex set of graph G is referred to as V(G), its edge set as E(G).").

Instead of providing edges themselves, iEdg is intended to provide a function as mapping of "indices" (the domain of the function) to the edges (therefore called "set of indexed edges"), which allows for hyper-/pseudo-/multigraphs with more than one edge between two (or more) vertices. For example, e₁ = e(1) = { a, b } and e₂ = e(2) = { a, b } are two different edges connecting the same two vertices a and b (in a pseudograph). In section 1.10 of [Diestel] p. 28, the edge function is defined differently: as "map E -> V u. [V]^2 assigning to every edge either one or two vertices, its end.". Here, the domain is the set of abstract edges: for two different edges e₁ and e₂ connecting the same two vertices a and b, we would have e(e₁) = e(e₂) = { a, b }. Since the set of abstract edges can be chosen as index set, these definitions are equivalent.

The result of these functions are as expected: for a graph represented as ordered pair (𝐺 ∈ (V × V)), the set of vertices is (Vtx‘𝐺) = (1^st ‘𝐺) (see opvtxval 28888) and the set of (indexed) edges is (iEdg‘𝐺) = (2^nd ‘𝐺) (see opiedgval 28891), or if 𝐺 is given as ordered pair 𝐺 = ⟨𝑉, 𝐸⟩, the set of vertices is (Vtx‘𝐺) = 𝑉 (see opvtxfv 28889) and the set of (indexed) edges is (iEdg‘𝐺) = 𝐸 (see opiedgfv 28892).

And for a graph represented as extensible structure (𝐺 Struct ⟨(Base‘ndx), (.ef‘ndx)⟩), the set of vertices is (Vtx‘𝐺) = (Base‘𝐺) (see funvtxval 28903) and the set of (indexed) edges is (iEdg‘𝐺) = (.ef‘𝐺) (see funiedgval 28904), or if 𝐺 is given in its simplest form as extensible structure with two slots (𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}), the set of vertices is (Vtx‘𝐺) = 𝑉 (see struct2grvtx 28912) and the set of (indexed) edges is (iEdg‘𝐺) = 𝐸 (see struct2griedg 28913).

These two representations are convertible, see graop 28914 and grastruct 28915: If 𝐺 is a graph (for example 𝐺 = ⟨𝑉, 𝐸⟩), then 𝐻 = {⟨(Base‘ndx), (Vtx‘𝐺)⟩, ⟨(.ef‘ndx), (iEdg‘𝐺)⟩} represents essentially the same graph, and if 𝐺 is a graph (for example 𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}), then 𝐻 = ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ represents essentially the same graph. In both cases, (Vtx‘𝐺) = (Vtx‘𝐻) and (iEdg‘𝐺) = (iEdg‘𝐻) hold. Theorems gropd 28916 and gropeld 28918 show that if any representation of a graph with vertices 𝑉 and edges 𝐸 has a certain property, then the ordered pair ⟨𝑉, 𝐸⟩ of the set of vertices and the set of edges (which is such a representation of a graph with vertices 𝑉 and edges 𝐸) has this property. Analogously, theorems grstructd 28917 and grstructeld 28919 show that if any representation of a graph with vertices 𝑉 and edges 𝐸 has a certain property, then any extensible structure with base set 𝑉 and value 𝐸 in the slot for edge functions (which is also such a representation of a graph with vertices 𝑉 and edges 𝐸) has this property.

Besides the usual way to represent graphs without edges (consisting of unconnected vertices only), which would be 𝐺 = ⟨𝑉, ∅⟩ or 𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), ∅⟩}, a structure without a slot for edges can be used: 𝐺 = {⟨(Base‘ndx), 𝑉⟩}, see snstrvtxval 28922 and snstriedgval 28923. Analogously, the empty set ∅ can be used to represent the null graph, see vtxval0 28924 and iedgval0 28925, which can also be represented by 𝐺 = ⟨∅, ∅⟩ or 𝐺 = {⟨(Base‘ndx), ∅⟩, ⟨(.ef‘ndx), ∅⟩}. Even proper classes can be used to represent the null graph, see vtxvalprc 28930 and iedgvalprc 28931.

Other classes should not be used to represent graphs, because there could be a degenerate behavior of the vertex set and (indexed) edge functions, see vtxvalsnop 28926 resp. iedgvalsnop 28927, and vtxval3sn 28928 resp. iedgval3sn 28929. Avoid directly depending on this detail so that theorems will not depend on the Kuratowski construction of ordered pairs, see also the comment for df-op 4637.

17.1.2.1  Definitions and basic properties

Syntax

cvtx 28881

Extend class notation with the vertices of "graphs".

class Vtx

Syntax

ciedg 28882

Extend class notation with the indexed edges of "graphs".

class iEdg

Definition

df-vtx 28883

Define the function mapping a graph to the set of its vertices. This definition is very general: It defines the set of vertices for any ordered pair as its first component, and for any other class as its "base set". It is meaningful, however, only if the ordered pair represents a graph resp. the class is an extensible structure representing a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 20-Sep-2020.)

⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1^st ‘𝑔), (Base‘𝑔)))

Definition

df-iedg 28884

Define the function mapping a graph to its indexed edges. This definition is very general: It defines the indexed edges for any ordered pair as its second component, and for any other class as its "edge function". It is meaningful, however, only if the ordered pair represents a graph resp. the class is an extensible structure (containing a slot for "edge functions") representing a graph. (Contributed by AV, 20-Sep-2020.)

⊢ iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2^nd ‘𝑔), (.ef‘𝑔)))

Theorem

vtxval 28885

The set of vertices of a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.)

⊢ (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺))

Theorem

iedgval 28886

The set of indexed edges of a graph. (Contributed by AV, 21-Sep-2020.)

⊢ (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2^nd ‘𝐺), (.ef‘𝐺))

Theorem

1vgrex 28887

A graph with at least one vertex is a set. (Contributed by AV, 2-Mar-2021.)

⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → 𝐺 ∈ V)

17.1.2.2  The vertices and edges of a graph represented as ordered pair

Theorem

opvtxval 28888

The set of vertices of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.)

⊢ (𝐺 ∈ (V × V) → (Vtx‘𝐺) = (1^st ‘𝐺))

Theorem

opvtxfv 28889

The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 21-Sep-2020.)

⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)

Theorem

opvtxov 28890

The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as operation value. (Contributed by AV, 21-Sep-2020.)

⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉Vtx𝐸) = 𝑉)

Theorem

opiedgval 28891

The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 21-Sep-2020.)

⊢ (𝐺 ∈ (V × V) → (iEdg‘𝐺) = (2^nd ‘𝐺))

Theorem

opiedgfv 28892

The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 21-Sep-2020.)

⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)

Theorem

opiedgov 28893

The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges as operation value. (Contributed by AV, 21-Sep-2020.)

⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉iEdg𝐸) = 𝐸)

Theorem

opvtxfvi 28894

The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 4-Mar-2021.)

⊢ 𝑉 ∈ V    &   ⊢ 𝐸 ∈ V    ⇒   ⊢ (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉

Theorem

opiedgfvi 28895

The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 4-Mar-2021.)

⊢ 𝑉 ∈ V    &   ⊢ 𝐸 ∈ V    ⇒   ⊢ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸

17.1.2.3  The vertices and edges of a graph represented as extensible structure

Theorem

funvtxdmge2val 28896

The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)

⊢ ((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (♯‘dom 𝐺)) → (Vtx‘𝐺) = (Base‘𝐺))

Theorem

funiedgdmge2val 28897

The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)

⊢ ((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (♯‘dom 𝐺)) → (iEdg‘𝐺) = (.ef‘𝐺))

Theorem

funvtxdm2val 28898

The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)

⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺))

Theorem

funiedgdm2val 28899

The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)

⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → (iEdg‘𝐺) = (.ef‘𝐺))

Theorem

funvtxval0 28900

The set of vertices of an extensible structure with a base set and (at least) another slot. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)

⊢ 𝑆 ∈ V    ⇒   ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 𝑆 ≠ (Base‘ndx) ∧ {(Base‘ndx), 𝑆} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺))