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Theorem List for Metamath Proof Explorer - 25501-25600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremax5seglem3 25501* Lemma for ax5seg 25508. 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 25502* Lemma for ax5seg 25508. 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 25503* Lemma for ax5seg 25508. 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 25504* Lemma for ax5seg 25508. 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 25505 Lemma for ax5seg 25508. 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 25506 Lemma for ax5seg 25508. Use the weak deduction theorem to eliminate the hypotheses from ax5seglem7 25505. (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 25507* Lemma for ax5seg 25508. Take the calculation in ax5seglem8 25506 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 25508 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 25509 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 25510* Lemma for axpasch 25511. 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 25511* 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 25512 Lemma for axlowdim 25531. Establish a particular constant function as a function. (Contributed by Scott Fenton, 29-Jun-2013.)
((3...𝑁) × {0}):(3...𝑁)⟶ℝ

Theoremaxlowdimlem2 25513 Lemma for axlowdim 25531. Show that two sets are disjoint. (Contributed by Scott Fenton, 29-Jun-2013.)
((1...2) ∩ (3...𝑁)) = ∅

Theoremaxlowdimlem3 25514 Lemma for axlowdim 25531. 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 25515 Lemma for axlowdim 25531. 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 25516 Lemma for axlowdim 25531. 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 25517 Lemma for axlowdim 25531. 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 25518 Lemma for axlowdim 25531. 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 25519 Lemma for axlowdim 25531. 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 25520 Lemma for axlowdim 25531. 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 25521 Lemma for axlowdim 25531. 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 25522 Lemma for axlowdim 25531. 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 25523 Lemma for axlowdim 25531. 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 25524 Lemma for axlowdim 25531. 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 25525 Lemma for axlowdim 25531. Take two possible 𝑄 from axlowdimlem10 25521. 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 25526* Lemma for axlowdim 25531. 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 25527* Lemma for axlowdim 25531. 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 25528 Lemma for axlowdim 25531. 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 25529* 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 25530. (Contributed by Scott Fenton, 22-Apr-2013.)
(𝑁 ∈ ℕ → ∃𝑥 ∈ (𝔼‘𝑁)∃𝑦 ∈ (𝔼‘𝑁)𝑥𝑦)

Theoremaxlowdim2 25530* 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 25531* 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 25532* Lemma for axeuclid 25533. 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 25533* 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 25534* Lemma for axcont 25546. 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 25535* Lemma for axcont 25546. The idea here is to set up a mapping 𝐹 that will allow us to transfer dedekind 9951 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 25536* Lemma for axcont 25546. 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 25537* Lemma for axcont 25546. 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 25538* Lemma for axcont 25546. 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 25539* Lemma for axcont 25546. 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 25540* Lemma for axcont 25546. 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 25541* Lemma for axcont 25546. 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 25542* Lemma for axcont 25546. 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 25543* Lemma for axcont 25546. 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 25544* Lemma for axcont 25546. Eliminate the hypotheses from axcontlem10 25543. (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 25545* Lemma for axcont 25546. 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 25546* 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 25547 Extends class notation with the Tarski geometry structure for 𝔼↑𝑁.
class EEG

Definitiondf-eeng 25548* 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 25549* 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 25550 The Euclidean geometry as a structure. (Contributed by Thierry Arnoux, 15-Mar-2019.)
(𝑁 ∈ ℕ → (EEG‘𝑁) Struct ⟨1, 17⟩)

Theoremeengbas 25551 The Base of the Euclidean geometry. (Contributed by Thierry Arnoux, 15-Mar-2019.)
(𝑁 ∈ ℕ → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))

Theoremebtwntg 25552 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 25553 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 25554* 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 25555 The geometry structure for 𝔼↑𝑁 is a Tarski geometry. (Contributed by Thierry Arnoux, 15-Mar-2019.)
(𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiG)

Theoremeengtrkge 25556 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

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 kinds of graphs:

TermReferenceDefinitionRemarks
Undirected hypergraph df-uhgra 25559 an ordered pair 𝑉, 𝐸 of a set 𝑉 and a function 𝐸 into the powerset of 𝑉 (ran 𝐸 ⊆ (𝒫 𝑉)).
An element of 𝑉 is called "vertex", an element of ran 𝐸 is called "edge", the function 𝐸 is called the "edge-function" .
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 (18-Jan-2020) such a hypergraph is called "non-simple hypergraph", "multiple hypergraph" or "multi-hypergraphs". According to Wikipedia "Incidence structure", see https://en.wikipedia.org/wiki/Incidence_structure (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".

If a graph is represented by a class variable, e.g. 𝐺, the edges of this graph are often represented by the function value (Edges‘𝐺). If the graph is given as pair 𝑉, 𝐸, however, (Edges‘⟨𝑉, 𝐸⟩) or preferably (𝑉Edges𝐸) is only used to talk about edges more explicitly. Otherwise, ran 𝐸 is used, because this is much shorter.
Notice that by using (Edges‘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 graphs.
For the set of vertices, a function "Vertices" could have been defined analogously. But "( Vertices ` G )" would have been exactly the same as (1st𝐺), so the latter is used to denote the set of vertices if the graph is represented by a class variable.
Undirected simple hypergraph df-ushgra 25560 an ordered pair 𝑉, 𝐸 of a set 𝑉 and a one-to-one function 𝐸 into the powerset of 𝑉 (ran 𝐸 ⊆ (𝒫 𝑉)). See also Wikipedia "Hypergraph", https://en.wikipedia.org/wiki/Hypergraph (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 multigraph needs not to be a simple hypergraph.
According to [Berge] p. 1, "A simple hypergraph (or "Sperner family") is a hypergraph H = { E1, E2, ..., Em } such that Ei C_ Ej => i = j". By this definition, a simple hypergraph cannot contain the edges E1 = { v1 , v2 } and E2 = { v1, v2, v3 }, because E1 C_ E2, but 1 =/= 2.
Undirected multigraph df-umgra 25580 a graph 𝑉, 𝐸 such that 𝐸 is a function into the set of (proper or not proper) unordered pairs of 𝑉.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 4230).
According to the definition 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 according to [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.".
Undirected simple graph with loops df-uslgra 25599 a graph 𝑉, 𝐸 such that 𝐸 is a one-to-one function into the set of (proper or not proper) unordered pairs of 𝑉.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 without loops (in short "simple graph") df-usgra 25600 a graph 𝑉, 𝐸 such that 𝐸 is a one-to-one function into the set of (proper) unordered pairs of 𝑉.An ordered pair 𝑉, 𝐸 of two distinct sets 𝑉 and 𝐸 (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 ausisusgra 25622.
Finite graph---a graph 𝑉, 𝐸 with finite sets 𝑉 and 𝐸.See definitions in [Bollobas] p. 1 or [Diestel] p. 2.
In simple graphs, 𝐸 is finite if 𝑉 is finite, see usgrafis 25682. The number of edges is limited by (𝑛C2) (or "𝑛 choose 2") with 𝑛 = (#‘𝑉), see usgramaxsize 25753. Analogously, the number of edges of an undirected simple graph with loops is limited by ((𝑛 + 1)C2). In multigraphs, however, 𝐸 can be infinite although 𝑉 is finite.
Graph of finite size---a graph 𝑉, 𝐸 with finite set 𝐸, i.e. with a finite number of edges.A graph can be of finite size although 𝑉 is infinite.

Terms and properties of graphs:
TermReferenceDefinitionRemarks
Edge joining (two) vertices --- An edge 𝑒 ∈ ran 𝐸 "joins" the vertices v1, v2, ... vn (𝑛 ∈ ℕ) if 𝑒 = { v1, v2, ... vn }. If 𝑛 = 1, 𝑒 = { v1 } is a "loop", if 𝑛 = 2, 𝑒 = { v1 , v2 } 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 ∈ ran 𝐸 joins the vertices v1, v2, ... vn (𝑛 ∈ ℕ), then the vertices v1, v2, ... vn are called the "endvertices" of the edge 𝑒.
(Two) Adjacent vertices see definition in Section I.1 in [Bollobas] p. 1/2. The vertices v1, v2, ... vn (𝑛 ∈ ℕ) are "adjacent" if there is an edge e = { v1, v2, ... vn } 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 ∈ ran 𝐸 is "ending" at a vertex 𝑣 if the vertex is an endvertex of the edge: 𝑣𝑒. In other words, the vertex 𝑣 is incident with the edge 𝑒.
(Two) Adjacent edges The edges e0, e1, ... en (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 𝑉, 𝐸 is the number of vertices in the graph ((#‘𝑉)).
Size of a graph see definition in Section I.1 in [Bollobas] p. 3 the "size" of a graph 𝑉, 𝐸 is the number of edges in the graph ((#‘𝐸)). Or, for simple graphs 𝐺: (#‘(Edges‘𝐺))).
Neighborhood of a vertex df-nbgra 25687 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}.
Degree of a vertex df-vdgr 26159 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 usgravd0nedg 26183 A vertex is called "isolated" if it is not an endvertex of any edge, thus having degree 0.
Universal vertex df-uvtx 25689 A vertex is called "universal" if it is connected with every other vertex of the graph by an edge, thus having degree (#‘𝑉).

Special kinds of graphs:
TermReferenceDefinitionRemarks
Complete graph df-cusgra 25688 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 cusgrasize 25744.
Empty graph umgra0 25592 and usgra0 25637 A graph is called "empty" if it has no edges.
Null graph usgra0v 25638 A graph is called the "null graph" if it has no vertices (and therefore also no edges).
Trivial graph usgra1v 25657 A graph is called the "trivial graph" if it has only one vertex and no edges.
Connected graph df-conngra 25936 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.

16.1  Undirected graphs - basics

16.1.1  Undirected hypergraphs

Syntaxcuhg 25557 Extend class notation with undirected hypergraphs.
class UHGrph

Syntaxcushg 25558 Extend class notation with undirected simple hypergraphs.
class USHGrph

Definitiondf-uhgra 25559* Define the class of all undirected hypergraphs. An undirected hypergraph is a pair of a set and a function into the powerset of this set (the empty set excluded). (Contributed by Alexander van der Vekens, 26-Dec-2017.)
UHGrph = {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})}

Definitiondf-ushgra 25560* Define the class of all undirected simple hypergraphs. An undirected simple hypergraph is a pair of a set and an injective (one-to-one) function into the powerset of this set (the empty set excluded). (Contributed by AV, 19-Jan-2020.)
USHGrph = {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})}

Theoremreluhgra 25561 The class of all undirected hypergraphs is a relation. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
Rel UHGrph

Theoremrelushgra 25562 The class of all undirected simple hypergraphs is a relation. (Contributed by AV, 19-Jan-2020.)
Rel USHGrph

Theoremuhgrav 25563 The classes of vertices and edges of an undirected hypergraph are sets. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
(𝑉 UHGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))

Theoremuhgraopelvv 25564 An undirected hypergraph is a member in the universal class of ordered pairs. (Contributed by AV, 3-Jan-2020.)
(𝐺 ∈ UHGrph → 𝐺 ∈ (V × V))

Theoremisuhgra 25565 The property of being an undirected hypergraph. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
((𝑉𝑊𝐸𝑋) → (𝑉 UHGrph 𝐸𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))

Theoremuhgraf 25566 The edge function of an undirected hypergraph is a function into the power set of the set of vertices. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
(𝑉 UHGrph 𝐸𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))

Theoremuhgrafun 25567 The edge function of an undirected hypergraph is a function. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
(𝑉 UHGrph 𝐸 → Fun 𝐸)

Theoremisushgra 25568 The property of being an undirected simple hypergraph. (Contributed by AV, 3-Jan-2020.)
((𝑉𝑊𝐸𝑋) → (𝑉 USHGrph 𝐸𝐸:dom 𝐸1-1→(𝒫 𝑉 ∖ {∅})))

Theoremushgraf 25569 The edge function of an undirected simple hypergraph is a function into the power set of the set of vertices. (Contributed by AV, 19-Jan-2020.)
(𝑉 USHGrph 𝐸𝐸:dom 𝐸1-1→(𝒫 𝑉 ∖ {∅}))

Theoremushgrauhgra 25570 An undirected simple hypergraph is an undirected hypergraph. (Contributed by AV, 19-Jan-2020.)
(𝑉 USHGrph 𝐸𝑉 UHGrph 𝐸)

Theoremuhgraop 25571 The property of being an undirected hypergraph represented as an ordered pair. The representation as an ordered pair is the usual representation of a graph, see section I.1 of [Bollobas] p. 1. (Contributed by AV, 1-Jan-2020.)
((𝑉𝑊𝐸𝑋) → (⟨𝑉, 𝐸⟩ ∈ UHGrph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))

Theoremuhgrac 25572 The property of being an undirected hypergraph represented by a class. This representation is useful if the set of vertices and the edge function is/needs not to be known. (Contributed by AV, 1-Jan-2020.)
(𝐺 ∈ UHGrph → (2nd𝐺):dom (2nd𝐺)⟶(𝒫 (1st𝐺) ∖ {∅}))

Theoremuhgrass 25573 An edge is a subset of vertices, analogous to umgrass 25586. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
((𝑉 UHGrph 𝐸𝐹 ∈ dom 𝐸) → (𝐸𝐹) ⊆ 𝑉)

Theoremuhgraeq12d 25574 Equality of hypergraphs. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → (𝑉 UHGrph 𝐸𝑊 UHGrph 𝐹))

Theoremuhgrares 25575 A subgraph of a hypergraph (formed by removing some edges from the original graph) is a hypergraph, analogous to umgrares 25591. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
(𝑉 UHGrph 𝐸𝑉 UHGrph (𝐸𝐴))

Theoremuhgra0 25576 The empty graph, with vertices but no edges, is a hypergraph, analogous to umgra0 25592. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
(𝑉𝑊𝑉 UHGrph ∅)

Theoremuhgra0v 25577 The null graph, with no vertices, is a hypergraph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
(∅ UHGrph 𝐸𝐸 = ∅)

Theoremuhgraun 25578 The union of two (undirected) hypergraphs (with the same vertex set): If 𝑉, 𝐸 and 𝑉, 𝐹 are hypergraphs, then 𝑉, 𝐸𝐹 is a hypergraph (the vertex set stays the same, but the edges from both graphs are kept, possibly resulting in two edges between two vertices), analogous to umgraun 25595. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
(𝜑𝐸 Fn 𝐴)    &   (𝜑𝐹 Fn 𝐵)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝑉 UHGrph 𝐸)    &   (𝜑𝑉 UHGrph 𝐹)       (𝜑𝑉 UHGrph (𝐸𝐹))

16.1.2  Undirected multigraphs

Syntaxcumg 25579 Extend class notation with undirected multigraphs.
class UMGrph

Definitiondf-umgra 25580* Define the class of all undirected multigraphs. A multigraph is a pair 𝑉, 𝐸 where 𝐸 is a function into subsets of 𝑉 of cardinality one or two, representing the two vertices incident to the edge, or the one vertex if the edge is a loop. (Contributed by Mario Carneiro, 11-Mar-2015.)
UMGrph = {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}}

Theoremrelumgra 25581 The class of all undirected multigraphs is a relation. (Contributed by Mario Carneiro, 11-Mar-2015.)
Rel UMGrph

Theoremisumgra 25582* The property of being an undirected multigraph. (Contributed by Mario Carneiro, 11-Mar-2015.)
((𝑉𝑊𝐸𝑋) → (𝑉 UMGrph 𝐸𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))

Theoremwrdumgra 25583* The property of being an undirected multigraph, expressing the edges as "words". (Contributed by Mario Carneiro, 11-Mar-2015.)
((𝑉𝑊𝐸 ∈ Word 𝑋) → (𝑉 UMGrph 𝐸𝐸 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))

Theoremumgraf2 25584* The edge function of an undirected multigraph is a function into unordered pairs of vertices. Version of umgraf 25585 without explicitly specified domain of the edge function. (Contributed by Mario Carneiro, 12-Mar-2015.)
(𝑉 UMGrph 𝐸𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})

Theoremumgraf 25585* The edge function of an undirected multigraph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
((𝑉 UMGrph 𝐸𝐸 Fn 𝐴) → 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})

Theoremumgrass 25586 An edge is a subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
((𝑉 UMGrph 𝐸𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ⊆ 𝑉)

Theoremumgran0 25587 An edge is a nonempty subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
((𝑉 UMGrph 𝐸𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ≠ ∅)

Theoremumgrale 25588 An edge has at most two ends. (Contributed by Mario Carneiro, 11-Mar-2015.)
((𝑉 UMGrph 𝐸𝐸 Fn 𝐴𝐹𝐴) → (#‘(𝐸𝐹)) ≤ 2)

Theoremumgrafi 25589 An edge is a finite subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
((𝑉 UMGrph 𝐸𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ∈ Fin)

Theoremumgraex 25590* An edge is an unordered pair of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
((𝑉 UMGrph 𝐸𝐸 Fn 𝐴𝐹𝐴) → ∃𝑥𝑉𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦})

Theoremumgrares 25591 A subgraph of a graph (formed by removing some edges from the original graph) is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
(𝑉 UMGrph 𝐸𝑉 UMGrph (𝐸𝐴))

Theoremumgra0 25592 The empty graph, with vertices but no edges, is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
(𝑉𝑊𝑉 UMGrph ∅)

Theoremumgra1 25593 The graph with one edge. (Contributed by Mario Carneiro, 12-Mar-2015.)
(((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) → 𝑉 UMGrph {⟨𝐴, {𝐵, 𝐶}⟩})

Theoremumisuhgra 25594 An undirected multigraph is an undirected hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
(𝑉 UMGrph 𝐸𝑉 UHGrph 𝐸)

Theoremumgraun 25595 The union of two (undirected) multigraphs (with the same vertex set): If 𝑉, 𝐸 and 𝑉, 𝐹 are graphs, then 𝑉, 𝐸𝐹 is a graph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.)
(𝜑𝐸 Fn 𝐴)    &   (𝜑𝐹 Fn 𝐵)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝑉 UMGrph 𝐸)    &   (𝜑𝑉 UMGrph 𝐹)       (𝜑𝑉 UMGrph (𝐸𝐹))

16.1.3  Undirected simple graphs

16.1.3.1  Undirected simple graphs - basics

Syntaxcuslg 25596 Extend class notation with undirected (simple) graphs with loops.
class USLGrph

Syntaxcusg 25597 Extend class notation with undirected (simple) graphs (without loops).
class USGrph

Syntaxcedg 25598 Extend class notation with the set of edges (of an undirected simple graph).
class Edges

Definitiondf-uslgra 25599* Define the class of all undirected simple graphs with loops. An undirected simple graph with loops is a special undirected multigraph 𝑉, 𝐸 where 𝐸 is an injective (one-to-one) function into subsets of 𝑉 of cardinality one or two, representing the two vertices incident to the edge, or the one vertex if the edge is a loop. In contrast to a multigraph, there is at most one edge between two vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
USLGrph = {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}}

Definitiondf-usgra 25600* Define the class of all undirected simple graphs without loops. An undirected simple graph without loops is a special undirected simple graph 𝑉, 𝐸 where 𝐸 is an injective (one-to-one) function into subsets of 𝑉 of cardinality two, representing the two vertices incident to the edge. Such graphs are usually simply called "undirected graphs", so if only the term "undirected graph" is used, an undirected simple graph without loops is meant. Therefore, an undirected graph has no loops (edges a vertex to itself). (Contributed by Alexander van der Vekens, 10-Aug-2017.)
USGrph = {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}}

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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