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Theorem List for Metamath Proof Explorer - 25601-25700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremaxsegconlem5 25601* Lemma for axsegcon 25607. Show that the distance between two points is nonnegative. (Contributed by Scott Fenton, 17-Sep-2013.)
𝑆 = Σ𝑝 ∈ (1...𝑁)(((𝐴𝑝) − (𝐵𝑝))↑2)       ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 0 ≤ (√‘𝑆))

Theoremaxsegconlem6 25602* Lemma for axsegcon 25607. Show that the distance between two distinct points is positive. (Contributed by Scott Fenton, 17-Sep-2013.)
𝑆 = Σ𝑝 ∈ (1...𝑁)(((𝐴𝑝) − (𝐵𝑝))↑2)       ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵) → 0 < (√‘𝑆))

Theoremaxsegconlem7 25603* Lemma for axsegcon 25607. Show that a particular ratio of distances is in the closed unit interval. (Contributed by Scott Fenton, 18-Sep-2013.)
𝑆 = Σ𝑝 ∈ (1...𝑁)(((𝐴𝑝) − (𝐵𝑝))↑2)    &   𝑇 = Σ𝑝 ∈ (1...𝑁)(((𝐶𝑝) − (𝐷𝑝))↑2)       (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((√‘𝑆) / ((√‘𝑆) + (√‘𝑇))) ∈ (0[,]1))

Theoremaxsegconlem8 25604* Lemma for axsegcon 25607. Show that a particular mapping generates a point. (Contributed by Scott Fenton, 18-Sep-2013.)
𝑆 = Σ𝑝 ∈ (1...𝑁)(((𝐴𝑝) − (𝐵𝑝))↑2)    &   𝑇 = Σ𝑝 ∈ (1...𝑁)(((𝐶𝑝) − (𝐷𝑝))↑2)    &   𝐹 = (𝑘 ∈ (1...𝑁) ↦ (((((√‘𝑆) + (√‘𝑇)) · (𝐵𝑘)) − ((√‘𝑇) · (𝐴𝑘))) / (√‘𝑆)))       (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → 𝐹 ∈ (𝔼‘𝑁))

Theoremaxsegconlem9 25605* Lemma for axsegcon 25607. Show that 𝐵𝐹 is congruent to 𝐶𝐷. (Contributed by Scott Fenton, 19-Sep-2013.)
𝑆 = Σ𝑝 ∈ (1...𝑁)(((𝐴𝑝) − (𝐵𝑝))↑2)    &   𝑇 = Σ𝑝 ∈ (1...𝑁)(((𝐶𝑝) − (𝐷𝑝))↑2)    &   𝐹 = (𝑘 ∈ (1...𝑁) ↦ (((((√‘𝑆) + (√‘𝑇)) · (𝐵𝑘)) − ((√‘𝑇) · (𝐴𝑘))) / (√‘𝑆)))       (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → Σ𝑖 ∈ (1...𝑁)(((𝐵𝑖) − (𝐹𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶𝑖) − (𝐷𝑖))↑2))

Theoremaxsegconlem10 25606* Lemma for axsegcon 25607. Show that the scaling constant from axsegconlem7 25603 produces the betweenness condition for 𝐴, 𝐵 and 𝐹. (Contributed by Scott Fenton, 21-Sep-2013.)
𝑆 = Σ𝑝 ∈ (1...𝑁)(((𝐴𝑝) − (𝐵𝑝))↑2)    &   𝑇 = Σ𝑝 ∈ (1...𝑁)(((𝐶𝑝) − (𝐷𝑝))↑2)    &   𝐹 = (𝑘 ∈ (1...𝑁) ↦ (((((√‘𝑆) + (√‘𝑇)) · (𝐵𝑘)) − ((√‘𝑇) · (𝐴𝑘))) / (√‘𝑆)))       (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ∀𝑖 ∈ (1...𝑁)(𝐵𝑖) = (((1 − ((√‘𝑆) / ((√‘𝑆) + (√‘𝑇)))) · (𝐴𝑖)) + (((√‘𝑆) / ((√‘𝑆) + (√‘𝑇))) · (𝐹𝑖))))

Theoremaxsegcon 25607* 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⟩))

Theoremax5seglem1 25608* Lemma for ax5seg 25618. 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)(𝐵𝑖) = (((1 − 𝑇) · (𝐴𝑖)) + (𝑇 · (𝐶𝑖))))) → Σ𝑗 ∈ (1...𝑁)(((𝐴𝑗) − (𝐵𝑗))↑2) = ((𝑇↑2) · Σ𝑗 ∈ (1...𝑁)(((𝐴𝑗) − (𝐶𝑗))↑2)))

Theoremax5seglem2 25609* Lemma for ax5seg 25618. 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)(𝐵𝑖) = (((1 − 𝑇) · (𝐴𝑖)) + (𝑇 · (𝐶𝑖))))) → Σ𝑗 ∈ (1...𝑁)(((𝐵𝑗) − (𝐶𝑗))↑2) = (((1 − 𝑇)↑2) · Σ𝑗 ∈ (1...𝑁)(((𝐴𝑗) − (𝐶𝑗))↑2)))

Theoremax5seglem3a 25610 Lemma for ax5seg 25618. (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)) → (((𝐴𝑗) − (𝐶𝑗)) ∈ ℝ ∧ ((𝐷𝑗) − (𝐹𝑗)) ∈ ℝ))

Theoremax5seglem3 25611* Lemma for ax5seg 25618. 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)(𝐵𝑖) = (((1 − 𝑇) · (𝐴𝑖)) + (𝑇 · (𝐶𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸𝑖) = (((1 − 𝑆) · (𝐷𝑖)) + (𝑆 · (𝐹𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → Σ𝑗 ∈ (1...𝑁)(((𝐴𝑗) − (𝐶𝑗))↑2) = Σ𝑗 ∈ (1...𝑁)(((𝐷𝑗) − (𝐹𝑗))↑2))

Theoremax5seglem4 25612* Lemma for ax5seg 25618. 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)(𝐵𝑖) = (((1 − 𝑇) · (𝐴𝑖)) + (𝑇 · (𝐶𝑖))) ∧ 𝐴𝐵) → 𝑇 ≠ 0)

Theoremax5seglem5 25613* Lemma for ax5seg 25618. 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))) ∧ (𝐴𝐵𝑇 ∈ (0[,]1) ∧ ∀𝑖 ∈ (1...𝑁)(𝐵𝑖) = (((1 − 𝑇) · (𝐴𝑖)) + (𝑇 · (𝐶𝑖))))) → Σ𝑗 ∈ (1...𝑁)(((𝐴𝑗) − (𝐶𝑗))↑2) ≠ 0)

Theoremax5seglem6 25614* Lemma for ax5seg 25618. 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)))) ∧ (𝐴𝐵 ∧ (𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵𝑖) = (((1 − 𝑇) · (𝐴𝑖)) + (𝑇 · (𝐶𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸𝑖) = (((1 − 𝑆) · (𝐷𝑖)) + (𝑆 · (𝐹𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → 𝑇 = 𝑆)

Theoremax5seglem7 25615 Lemma for ax5seg 25618. An algebraic calculation needed further down the line. (Contributed by Scott Fenton, 12-Jun-2013.)
𝐴 ∈ ℂ    &   𝑇 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ       (𝑇 · ((𝐶𝐷)↑2)) = ((((((1 − 𝑇) · 𝐴) + (𝑇 · 𝐶)) − 𝐷)↑2) + ((1 − 𝑇) · ((𝑇 · ((𝐴𝐶)↑2)) − ((𝐴𝐷)↑2))))

Theoremax5seglem8 25616 Lemma for ax5seg 25618. Use the weak deduction theorem to eliminate the hypotheses from ax5seglem7 25615. (Contributed by Scott Fenton, 11-Jun-2013.)
(((𝐴 ∈ ℂ ∧ 𝑇 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (𝑇 · ((𝐶𝐷)↑2)) = ((((((1 − 𝑇) · 𝐴) + (𝑇 · 𝐶)) − 𝐷)↑2) + ((1 − 𝑇) · ((𝑇 · ((𝐴𝐶)↑2)) − ((𝐴𝐷)↑2)))))

Theoremax5seglem9 25617* Lemma for ax5seg 25618. Take the calculation in ax5seglem8 25616 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)(𝐵𝑖) = (((1 − 𝑇) · (𝐴𝑖)) + (𝑇 · (𝐶𝑖))))) → (𝑇 · Σ𝑗 ∈ (1...𝑁)(((𝐶𝑗) − (𝐷𝑗))↑2)) = (Σ𝑗 ∈ (1...𝑁)(((𝐵𝑗) − (𝐷𝑗))↑2) + ((1 − 𝑇) · ((𝑇 · Σ𝑗 ∈ (1...𝑁)(((𝐴𝑗) − (𝐶𝑗))↑2)) − Σ𝑗 ∈ (1...𝑁)(((𝐴𝑗) − (𝐷𝑗))↑2)))))

Theoremax5seg 25618 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))) → (((𝐴𝐵𝐵 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⟩))

Theoremaxbtwnid 25619 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))

Theoremaxpaschlem 25620* Lemma for axpasch 25621. Set up coefficents 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)))

Theoremaxpasch 25621* 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⟩)))

Theoremaxlowdimlem1 25622 Lemma for axlowdim 25641. Establish a particular constant function as a function. (Contributed by Scott Fenton, 29-Jun-2013.)
((3...𝑁) × {0}):(3...𝑁)⟶ℝ

Theoremaxlowdimlem2 25623 Lemma for axlowdim 25641. Show that two sets are disjoint. (Contributed by Scott Fenton, 29-Jun-2013.)
((1...2) ∩ (3...𝑁)) = ∅

Theoremaxlowdimlem3 25624 Lemma for axlowdim 25641. 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)))

Theoremaxlowdimlem4 25625 Lemma for axlowdim 25641. Set up a particular constant function. (Contributed by Scott Fenton, 17-Apr-2013.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       {⟨1, 𝐴⟩, ⟨2, 𝐵⟩}:(1...2)⟶ℝ

Theoremaxlowdimlem5 25626 Lemma for axlowdim 25641. 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))

Theoremaxlowdimlem6 25627 Lemma for axlowdim 25641. 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⟩))

Theoremaxlowdimlem7 25628 Lemma for axlowdim 25641. 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))

Theoremaxlowdimlem8 25629 Lemma for axlowdim 25641. Calculate the value of 𝑃 at three. (Contributed by Scott Fenton, 21-Apr-2013.)
𝑃 = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))       (𝑃‘3) = -1

Theoremaxlowdimlem9 25630 Lemma for axlowdim 25641. 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) → (𝑃𝐾) = 0)

Theoremaxlowdimlem10 25631 Lemma for axlowdim 25641. 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))

Theoremaxlowdimlem11 25632 Lemma for axlowdim 25641. 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

Theoremaxlowdimlem12 25633 Lemma for axlowdim 25641. 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)) → (𝑄𝐾) = 0)

Theoremaxlowdimlem13 25634 Lemma for axlowdim 25641. 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))) → 𝑃𝑄)

Theoremaxlowdimlem14 25635 Lemma for axlowdim 25641. Take two possible 𝑄 from axlowdimlem10 25631. 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 = 𝑅𝐼 = 𝐽))

Theoremaxlowdimlem15 25636* Lemma for axlowdim 25641. 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))

Theoremaxlowdimlem16 25637* Lemma for axlowdim 25641. 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)((𝑃𝑖)↑2) = Σ𝑖 ∈ (3...𝑁)((𝑄𝑖)↑2))

Theoremaxlowdimlem17 25638 Lemma for axlowdim 25641. 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⟩)

Theoremaxlowdim1 25639* 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 25640. (Contributed by Scott Fenton, 22-Apr-2013.)
(𝑁 ∈ ℕ → ∃𝑥 ∈ (𝔼‘𝑁)∃𝑦 ∈ (𝔼‘𝑁)𝑥𝑦)

Theoremaxlowdim2 25640* 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⟩))

Theoremaxlowdim 25641* 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) → ∃𝑝𝑥 ∈ (𝔼‘𝑁)∃𝑦 ∈ (𝔼‘𝑁)∃𝑧 ∈ (𝔼‘𝑁)(𝑝:(1...(𝑁 − 1))–1-1→(𝔼‘𝑁) ∧ ∀𝑖 ∈ (2...(𝑁 − 1))(⟨(𝑝‘1), 𝑥⟩Cgr⟨(𝑝𝑖), 𝑥⟩ ∧ ⟨(𝑝‘1), 𝑦⟩Cgr⟨(𝑝𝑖), 𝑦⟩ ∧ ⟨(𝑝‘1), 𝑧⟩Cgr⟨(𝑝𝑖), 𝑧⟩) ∧ ¬ (𝑥 Btwn ⟨𝑦, 𝑧⟩ ∨ 𝑦 Btwn ⟨𝑧, 𝑥⟩ ∨ 𝑧 Btwn ⟨𝑥, 𝑦⟩)))

Theoremaxeuclidlem 25642* Lemma for axeuclid 25643. 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) · (𝐴𝑖)) + (𝑃 · (𝑇𝑖))) = (((1 − 𝑄) · (𝐵𝑖)) + (𝑄 · (𝐶𝑖)))) → ∃𝑥 ∈ (𝔼‘𝑁)∃𝑦 ∈ (𝔼‘𝑁)∃𝑟 ∈ (0[,]1)∃𝑠 ∈ (0[,]1)∃𝑢 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)((𝐵𝑖) = (((1 − 𝑟) · (𝐴𝑖)) + (𝑟 · (𝑥𝑖))) ∧ (𝐶𝑖) = (((1 − 𝑠) · (𝐴𝑖)) + (𝑠 · (𝑦𝑖))) ∧ (𝑇𝑖) = (((1 − 𝑢) · (𝑥𝑖)) + (𝑢 · (𝑦𝑖)))))

Theoremaxeuclid 25643* 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⟩ ∧ 𝐴𝐷) → ∃𝑥 ∈ (𝔼‘𝑁)∃𝑦 ∈ (𝔼‘𝑁)(𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ 𝐶 Btwn ⟨𝐴, 𝑦⟩ ∧ 𝑇 Btwn ⟨𝑥, 𝑦⟩)))

Theoremaxcontlem1 25644* Lemma for axcont 25656. Change bound variables for later use. (Contributed by Scott Fenton, 20-Jun-2013.)
𝐹 = {⟨𝑥, 𝑡⟩ ∣ (𝑥𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑡) · (𝑍𝑖)) + (𝑡 · (𝑈𝑖)))))}       𝐹 = {⟨𝑦, 𝑠⟩ ∣ (𝑦𝐷 ∧ (𝑠 ∈ (0[,)+∞) ∧ ∀𝑗 ∈ (1...𝑁)(𝑦𝑗) = (((1 − 𝑠) · (𝑍𝑗)) + (𝑠 · (𝑈𝑗)))))}

Theoremaxcontlem2 25645* Lemma for axcont 25656. The idea here is to set up a mapping 𝐹 that will allow us to transfer dedekind 10079 to two sets of points. Here, we set up 𝐹 and show its domain and range. (Contributed by Scott Fenton, 17-Jun-2013.)
𝐷 = {𝑝 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn ⟨𝑍, 𝑝⟩ ∨ 𝑝 Btwn ⟨𝑍, 𝑈⟩)}    &   𝐹 = {⟨𝑥, 𝑡⟩ ∣ (𝑥𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑡) · (𝑍𝑖)) + (𝑡 · (𝑈𝑖)))))}       (((𝑁 ∈ ℕ ∧ 𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ (𝔼‘𝑁)) ∧ 𝑍𝑈) → 𝐹:𝐷1-1-onto→(0[,)+∞))

Theoremaxcontlem3 25646* Lemma for axcont 25656. 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) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ (𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈𝐴𝑍𝑈)) → 𝐵𝐷)

Theoremaxcontlem4 25647* Lemma for axcont 25656. 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) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈𝐴𝐵 ≠ ∅) ∧ 𝑍𝑈)) → 𝐴𝐷)

Theoremaxcontlem5 25648* Lemma for axcont 25656. 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⟩ ∣ (𝑥𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑡) · (𝑍𝑖)) + (𝑡 · (𝑈𝑖)))))}       ((((𝑁 ∈ ℕ ∧ 𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ (𝔼‘𝑁)) ∧ 𝑍𝑈) ∧ 𝑃𝐷) → ((𝐹𝑃) = 𝑇 ↔ (𝑇 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑃𝑖) = (((1 − 𝑇) · (𝑍𝑖)) + (𝑇 · (𝑈𝑖))))))

Theoremaxcontlem6 25649* Lemma for axcont 25656. 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⟩ ∣ (𝑥𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑡) · (𝑍𝑖)) + (𝑡 · (𝑈𝑖)))))}       ((((𝑁 ∈ ℕ ∧ 𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ (𝔼‘𝑁)) ∧ 𝑍𝑈) ∧ 𝑃𝐷) → ((𝐹𝑃) ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑃𝑖) = (((1 − (𝐹𝑃)) · (𝑍𝑖)) + ((𝐹𝑃) · (𝑈𝑖)))))

Theoremaxcontlem7 25650* Lemma for axcont 25656. 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⟩ ∣ (𝑥𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑡) · (𝑍𝑖)) + (𝑡 · (𝑈𝑖)))))}       ((((𝑁 ∈ ℕ ∧ 𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ (𝔼‘𝑁)) ∧ 𝑍𝑈) ∧ (𝑃𝐷𝑄𝐷)) → (𝑃 Btwn ⟨𝑍, 𝑄⟩ ↔ (𝐹𝑃) ≤ (𝐹𝑄)))

Theoremaxcontlem8 25651* Lemma for axcont 25656. 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⟩ ∣ (𝑥𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑡) · (𝑍𝑖)) + (𝑡 · (𝑈𝑖)))))}       ((((𝑁 ∈ ℕ ∧ 𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ (𝔼‘𝑁)) ∧ 𝑍𝑈) ∧ (𝑃𝐷𝑄𝐷𝑅𝐷)) → (((𝐹𝑃) ≤ (𝐹𝑄) ∧ (𝐹𝑄) ≤ (𝐹𝑅)) → 𝑄 Btwn ⟨𝑃, 𝑅⟩))

Theoremaxcontlem9 25652* Lemma for axcont 25656. 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⟩ ∣ (𝑥𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑡) · (𝑍𝑖)) + (𝑡 · (𝑈𝑖)))))}       (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈𝐴𝐵 ≠ ∅) ∧ 𝑍𝑈)) → ∀𝑛 ∈ (𝐹𝐴)∀𝑚 ∈ (𝐹𝐵)𝑛𝑚)

Theoremaxcontlem10 25653* Lemma for axcont 25656. 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⟩ ∣ (𝑥𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑡) · (𝑍𝑖)) + (𝑡 · (𝑈𝑖)))))}       (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈𝐴𝐵 ≠ ∅) ∧ 𝑍𝑈)) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥𝐴𝑦𝐵 𝑏 Btwn ⟨𝑥, 𝑦⟩)

Theoremaxcontlem11 25654* Lemma for axcont 25656. Eliminate the hypotheses from axcontlem10 25653. (Contributed by Scott Fenton, 20-Jun-2013.)
(((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈𝐴𝐵 ≠ ∅) ∧ 𝑍𝑈)) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥𝐴𝑦𝐵 𝑏 Btwn ⟨𝑥, 𝑦⟩)

Theoremaxcontlem12 25655* Lemma for axcont 25656. Eliminate the trivial cases from the previous lemmas. (Contributed by Scott Fenton, 20-Jun-2013.)
(((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ 𝑍 ∈ (𝔼‘𝑁)) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥𝐴𝑦𝐵 𝑏 Btwn ⟨𝑥, 𝑦⟩)

Theoremaxcont 25656* 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)∀𝑥𝐴𝑦𝐵 𝑥 Btwn ⟨𝑎, 𝑦⟩)) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥𝐴𝑦𝐵 𝑏 Btwn ⟨𝑥, 𝑦⟩)

15.4.2.3  EE^n fulfills Tarski's Axioms

Syntaxceeng 25657 Extends class notation with the Tarski geometry structure for 𝔼↑𝑁.
class EEG

Definitiondf-eeng 25658* 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)(((𝑥𝑖) − (𝑦𝑖))↑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⟩)})⟩}))

Theoremeengv 25659* 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)(((𝑥𝑖) − (𝑦𝑖))↑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⟩)})⟩}))

Theoremeengstr 25660 The Euclidean geometry as a structure. (Contributed by Thierry Arnoux, 15-Mar-2019.)
(𝑁 ∈ ℕ → (EEG‘𝑁) Struct ⟨1, 17⟩)

Theoremeengbas 25661 The Base of the Euclidean geometry. (Contributed by Thierry Arnoux, 15-Mar-2019.)
(𝑁 ∈ ℕ → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))

Theoremebtwntg 25662 The betweenness relation used in the Tarski structure for the Euclidean geometry is the same as Btwn. (Contributed by Thierry Arnoux, 15-Mar-2019.)
(𝜑𝑁 ∈ ℕ)    &   𝑃 = (Base‘(EEG‘𝑁))    &   𝐼 = (Itv‘(EEG‘𝑁))    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)       (𝜑 → (𝑍 Btwn ⟨𝑋, 𝑌⟩ ↔ 𝑍 ∈ (𝑋𝐼𝑌)))

Theoremecgrtg 25663 The congruence relation used in the Tarski structure for the Euclidean geometry is the same as Cgr. (Contributed by Thierry Arnoux, 15-Mar-2019.)
(𝜑𝑁 ∈ ℕ)    &   𝑃 = (Base‘(EEG‘𝑁))    &    = (dist‘(EEG‘𝑁))    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)       (𝜑 → (⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩ ↔ (𝐴 𝐵) = (𝐶 𝐷)))

Theoremelntg 25664* 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)) = (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))

Theoremeengtrkg 25665 The geometry structure for 𝔼↑𝑁 is a Tarski geometry. (Contributed by Thierry Arnoux, 15-Mar-2019.)
(𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiG)

Theoremeengtrkge 25666 The geometry structure for 𝔼↑𝑁 is a Euclidean geometry. (Contributed by Thierry Arnoux, 15-Mar-2019.)
(𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiGE)

PART 16  GRAPH THEORY (UNDER CONSTRUCTION)

16.1  Undirected graphs - preliminaries

16.1.1  The edge function extractor for extensible structures

Syntaxcedgf 25667 Extend class notation with an edge function.
class .ef

Definitiondf-edgf 25668 Define the edge function (indexed edges) of a graph. (Contributed by AV, 18-Jan-2020.)
.ef = Slot 18

Theoremedgfndxnn 25669 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.)
(.ef‘ndx) ∈ ℕ

Theoremedgfndxid 25670 The value of the edge function extractor is the value of the corresponding slot of the structure. (Contributed by AV, 21-Sep-2020.)
(𝐺𝑉 → (.ef‘𝐺) = (𝐺‘(.ef‘ndx)))

Theorembaseltedgf 25671 The index value of the Base slot is less than the index value of the .ef slot. (Contributed by AV, 21-Sep-2020.)
(Base‘ndx) < (.ef‘ndx)

Theoremslotsbaseefdif 25672 The slots Base and .ef are different. (Contributed by AV, 21-Sep-2020.)
(Base‘ndx) ≠ (.ef‘ndx)

16.1.2  Vertices and 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-/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 consistes 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 15697.

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 25675 and df-iedg 25676), 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) = (1st𝐺) (see opvtxval 25680) and the set of (indexed) edges is (iEdg‘𝐺) = (2nd𝐺) (see opiedgval 25683), or if 𝐺 is given as ordered pair 𝐺 = ⟨𝑉, 𝐸, the set of vertices is (Vtx‘𝐺) = 𝑉 (see opvtxfv 25681) and the set of (indexed) edges is (iEdg‘𝐺) = 𝐸 (see opiedgfv 25684).

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 25695) and the set of (indexed) edges is (iEdg‘𝐺) = (.ef‘𝐺) (see funiedgval 25696), 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 25704) and the set of (indexed) edges is (iEdg‘𝐺) = 𝐸 (see struct2griedg 25705).

These two representations are convertible, see graop 25706 and grastruct 25707: 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 25708 and gropeld 25710 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 25709 and grstructeld 25711 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 25712 and snstriedgval 25713. Analogously, the empty set can be used to represent the null graph, see vtxval0 25714 and iedgval0 25715, 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 25720 and iedgvalprc 25721.

Other classes should not be used to represent graphs, because there could be a degenerated behavior of the vertex set and (indexed) edge functions, see vtxvalsnop 25716 resp. iedgvalsnop 25717, and vtxval3sn 25718 resp. iedgval3sn 25719.

16.1.2.1  Definitions and basic properties

Syntaxcvtx 25673 Extend class notation with the vertices of "graphs".
class Vtx

Syntaxciedg 25674 Extend class notation with the indexed edges of "graphs".
class iEdg

Definitiondf-vtx 25675 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), (1st𝑔), (Base‘𝑔)))

Definitiondf-iedg 25676 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), (2nd𝑔), (.ef‘𝑔)))

Theoremvtxval 25677 The set of vertices of a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.)
(𝐺𝑉 → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)))

Theoremiedgval 25678 The set of indexed edges of a graph. (Contributed by AV, 21-Sep-2020.)
(𝐺𝑉 → (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺)))

Theorem1vgrex 25679 A graph with at least one vertex is a set. (Contributed by AV, 2-Mar-2021.)
𝑉 = (Vtx‘𝐺)       (𝑁𝑉𝐺 ∈ V)

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

Theoremopvtxval 25680 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) = (1st𝐺))

Theoremopvtxfv 25681 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.)
((𝑉𝑋𝐸𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)

Theoremopvtxov 25682 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.)
((𝑉𝑋𝐸𝑌) → (𝑉Vtx𝐸) = 𝑉)

Theoremopiedgval 25683 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) = (2nd𝐺))

Theoremopiedgfv 25684 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.)
((𝑉𝑋𝐸𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)

Theoremopiedgov 25685 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.)
((𝑉𝑋𝐸𝑌) → (𝑉iEdg𝐸) = 𝐸)

Theoremopvtxfvi 25686 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

Theoremopiedgfvi 25687 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

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

Theoremfunvtxdm2val 25688 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.)
𝐴 ∈ V    &   𝐵 ∈ V       (((𝐺𝑉 ∧ Fun (𝐺 ∖ {∅})) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺))

Theoremfuniedgdm2val 25689 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.)
𝐴 ∈ V    &   𝐵 ∈ V       (((𝐺𝑉 ∧ Fun (𝐺 ∖ {∅})) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → (iEdg‘𝐺) = (.ef‘𝐺))

Theoremfunvtxval0 25690 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.)
𝑆 ∈ V       (((𝐺𝑉 ∧ Fun (𝐺 ∖ {∅})) ∧ 𝑆 ≠ (Base‘ndx) ∧ {(Base‘ndx), 𝑆} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺))

Theoremfunvtxdmge2val 25691 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.)
((𝐺𝑉 ∧ Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (#‘dom 𝐺)) → (Vtx‘𝐺) = (Base‘𝐺))

Theoremfuniedgdmge2val 25692 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.)
((𝐺𝑉 ∧ Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (#‘dom 𝐺)) → (iEdg‘𝐺) = (.ef‘𝐺))

Theorembasvtxval 25693 The set of vertices of a graph represented as an extensible structure with the set of vertices as base set. (Contributed by AV, 14-Oct-2020.)
(𝜑𝐺𝑋)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → 2 ≤ (#‘dom 𝐺))    &   (𝜑𝑉𝑌)    &   (𝜑 → ⟨(Base‘ndx), 𝑉⟩ ∈ 𝐺)       (𝜑 → (Vtx‘𝐺) = 𝑉)

Theoremedgfiedgval 25694 The set of indexed edges of a graph represented as an extensible structure with the indexed edges in the slot for edge functions. (Contributed by AV, 14-Oct-2020.)
(𝜑𝐺𝑋)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → 2 ≤ (#‘dom 𝐺))    &   (𝜑𝐸𝑌)    &   (𝜑 → ⟨(.ef‘ndx), 𝐸⟩ ∈ 𝐺)       (𝜑 → (iEdg‘𝐺) = 𝐸)

Theoremfunvtxval 25695 The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.)
((𝐺𝑉 ∧ Fun (𝐺 ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺))

Theoremfuniedgval 25696 The set of indexed edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 21-Sep-2020.) (Revised by AV, 7-Jun-2021.)
((𝐺𝑉 ∧ Fun (𝐺 ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom 𝐺) → (iEdg‘𝐺) = (.ef‘𝐺))

Theoremstructvtxvallem 25697 Lemma for structvtxval 25698 and structiedg0val 25699. (Contributed by AV, 23-Sep-2020.)
𝑆 ∈ ℕ    &   (Base‘ndx) < 𝑆    &   𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨𝑆, 𝐸⟩}       ((𝑉𝑋𝐸𝑌) → (𝐺 ∈ V ∧ Fun 𝐺 ∧ {(Base‘ndx), 𝑆} ⊆ dom 𝐺))

Theoremstructvtxval 25698 The set of vertices of an extensible structure with a base set and another slot. (Contributed by AV, 23-Sep-2020.)
𝑆 ∈ ℕ    &   (Base‘ndx) < 𝑆    &   𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨𝑆, 𝐸⟩}       ((𝑉𝑋𝐸𝑌) → (Vtx‘𝐺) = 𝑉)

Theoremstructiedg0val 25699 The set of indexed edges of an extensible structure with a base set and another slot not being the slot for edge functions is empty. (Contributed by AV, 23-Sep-2020.)
𝑆 ∈ ℕ    &   (Base‘ndx) < 𝑆    &   𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨𝑆, 𝐸⟩}       ((𝑉𝑋𝐸𝑌𝑆 ≠ (.ef‘ndx)) → (iEdg‘𝐺) = ∅)