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Theorem List for Metamath Proof Explorer - 26201-26300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremleagne4 26201 Deduce inequality from the less-than angle relation. (Contributed by Thierry Arnoux, 25-Feb-2023.)
𝑃 = (Base‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(≤𝐺)⟨“𝐷𝐸𝐹”⟩)       (𝜑𝐹𝐸)

Theoremcgrg3col4 26202* Lemma 11.28 of [Schwabhauser] p. 102. Extend a congruence of three points with a fourth colinear point. (Contributed by Thierry Arnoux, 8-Oct-2020.)
𝑃 = (Base‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝑋𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))       (𝜑 → ∃𝑦𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)

15.2.18  Congruence Theorems

Theoremtgsas1 26203 First congruence theorem: SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then third sides are equal in length. Theorem 11.49 of [Schwabhauser] p. 107. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))       (𝜑 → (𝐶 𝐴) = (𝐹 𝐷))

Theoremtgsas 26204 First congruence theorem: SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent. Theorem 11.49 of [Schwabhauser] p. 107. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))       (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)

Theoremtgsas2 26205 First congruence theorem: SAS. Theorem 11.49 of [Schwabhauser] p. 107. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))    &   (𝜑𝐴𝐶)       (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩)

Theoremtgsas3 26206 First congruence theorem: SAS. Theorem 11.49 of [Schwabhauser] p. 107. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))    &   (𝜑𝐴𝐶)       (𝜑 → ⟨“𝐵𝐶𝐴”⟩(cgrA‘𝐺)⟨“𝐸𝐹𝐷”⟩)

Theoremtgasa1 26207 Second congruence theorem: ASA. (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent. Theorem 11.50 of [Schwabhauser] p. 108. (Contributed by Thierry Arnoux, 15-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩)       (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))

Theoremtgasa 26208 Second congruence theorem: ASA. (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent. Theorem 11.50 of [Schwabhauser] p. 108. (Contributed by Thierry Arnoux, 15-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)

Theoremtgsss1 26209 Third congruence theorem: SSS (Side-Side-Side): If the three pairs of sides of two triangles are equal in length, then the triangles are congruent. Theorem 11.51 of [Schwabhauser] p. 109. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))    &   (𝜑 → (𝐶 𝐴) = (𝐹 𝐷))    &   (𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)    &   (𝜑𝐶𝐴)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)

Theoremtgsss2 26210 Third congruence theorem: SSS. Theorem 11.51 of [Schwabhauser] p. 109. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))    &   (𝜑 → (𝐶 𝐴) = (𝐹 𝐷))    &   (𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)    &   (𝜑𝐶𝐴)       (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩)

Theoremtgsss3 26211 Third congruence theorem: SSS. Theorem 11.51 of [Schwabhauser] p. 109. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))    &   (𝜑 → (𝐶 𝐴) = (𝐹 𝐷))    &   (𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)    &   (𝜑𝐶𝐴)       (𝜑 → ⟨“𝐵𝐶𝐴”⟩(cgrA‘𝐺)⟨“𝐸𝐹𝐷”⟩)

Theoremdfcgrg2 26212 Congruence for two triangles can also be defined as congruence of sides and angles (6 parts). This is often the actual textbook definition of triangle congruence, see for example https://en.wikipedia.org/wiki/Congruence_(geometry)#Congruence_of_triangles With this definition, the "SSS" congruence theorem has an additional part, namely, that triangle congruence implies congruence of the sides (which means equality of the lengths). Because our development of elementary geometry strives to closely follow Schwabhaeuser's, our original definition of shape congruence, df-cgrg 25862, already covers that part: see trgcgr 25867. This theorem is also named "CPCTC", which stands for "Corresponding Parts of Congruent Triangles are Congruent", see https://en.wikipedia.org/wiki/Congruence_(geometry)#CPCTC (Contributed by Thierry Arnoux, 18-Jan-2023.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)    &   (𝜑𝐶𝐴)       (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ (((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷)) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩ ∧ ⟨“𝐵𝐶𝐴”⟩(cgrA‘𝐺)⟨“𝐸𝐹𝐷”⟩))))

Theoremisoas 26213 Congruence theorem for isocele triangles: if two angles of a triangle are congruent, then the corresponding sides also are. (Contributed by Thierry Arnoux, 5-Oct-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐵”⟩)       (𝜑 → (𝐴 𝐵) = (𝐴 𝐶))

15.2.19  Equilateral triangles

Syntaxceqlg 26214 Declare the class of equilateral triangles.
class eqltrG

Definitiondf-eqlg 26215* Define the class of equilateral triangles. (Contributed by Thierry Arnoux, 27-Nov-2019.)
eqltrG = (𝑔 ∈ V ↦ {𝑥 ∈ ((Base‘𝑔) ↑𝑚 (0..^3)) ∣ 𝑥(cgrG‘𝑔)⟨“(𝑥‘1)(𝑥‘2)(𝑥‘0)”⟩})

Theoremiseqlg 26216 Property of a triangle being equilateral. (Contributed by Thierry Arnoux, 5-Oct-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)       (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ∈ (eqltrG‘𝐺) ↔ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐵𝐶𝐴”⟩))

Theoremiseqlgd 26217 Condition for a triangle to be equilateral. (Contributed by Thierry Arnoux, 5-Oct-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐵 𝐶))    &   (𝜑 → (𝐵 𝐶) = (𝐶 𝐴))    &   (𝜑 → (𝐶 𝐴) = (𝐴 𝐵))       (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (eqltrG‘𝐺))

15.3  Properties of geometries

15.3.1  Isomorphisms between geometries

Theoremf1otrgds 26218* Convenient lemma for f1otrg 26220. (Contributed by Thierry Arnoux, 19-Mar-2019.)
𝑃 = (Base‘𝐺)    &   𝐷 = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐵 = (Base‘𝐻)    &   𝐸 = (dist‘𝐻)    &   𝐽 = (Itv‘𝐻)    &   (𝜑𝐹:𝐵1-1-onto𝑃)    &   ((𝜑 ∧ (𝑒𝐵𝑓𝐵)) → (𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)))    &   ((𝜑 ∧ (𝑒𝐵𝑓𝐵𝑔𝐵)) → (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑒)𝐼(𝐹𝑓))))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐸𝑌) = ((𝐹𝑋)𝐷(𝐹𝑌)))

Theoremf1otrgitv 26219* Convenient lemma for f1otrg 26220. (Contributed by Thierry Arnoux, 19-Mar-2019.)
𝑃 = (Base‘𝐺)    &   𝐷 = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐵 = (Base‘𝐻)    &   𝐸 = (dist‘𝐻)    &   𝐽 = (Itv‘𝐻)    &   (𝜑𝐹:𝐵1-1-onto𝑃)    &   ((𝜑 ∧ (𝑒𝐵𝑓𝐵)) → (𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)))    &   ((𝜑 ∧ (𝑒𝐵𝑓𝐵𝑔𝐵)) → (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑒)𝐼(𝐹𝑓))))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → (𝑍 ∈ (𝑋𝐽𝑌) ↔ (𝐹𝑍) ∈ ((𝐹𝑋)𝐼(𝐹𝑌))))

Theoremf1otrg 26220* A bijection between bases which conserves distances and intervals conserves also geometries. (Contributed by Thierry Arnoux, 23-Mar-2019.)
𝑃 = (Base‘𝐺)    &   𝐷 = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐵 = (Base‘𝐻)    &   𝐸 = (dist‘𝐻)    &   𝐽 = (Itv‘𝐻)    &   (𝜑𝐹:𝐵1-1-onto𝑃)    &   ((𝜑 ∧ (𝑒𝐵𝑓𝐵)) → (𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)))    &   ((𝜑 ∧ (𝑒𝐵𝑓𝐵𝑔𝐵)) → (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑒)𝐼(𝐹𝑓))))    &   (𝜑𝐻𝑉)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑 → (LineG‘𝐻) = (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐽𝑦) ∨ 𝑥 ∈ (𝑧𝐽𝑦) ∨ 𝑦 ∈ (𝑥𝐽𝑧))}))       (𝜑𝐻 ∈ TarskiG)

Theoremf1otrge 26221* A bijection between bases which conserves distances and intervals conserves also the property of being a Euclidean geometry. (Contributed by Thierry Arnoux, 23-Mar-2019.)
𝑃 = (Base‘𝐺)    &   𝐷 = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐵 = (Base‘𝐻)    &   𝐸 = (dist‘𝐻)    &   𝐽 = (Itv‘𝐻)    &   (𝜑𝐹:𝐵1-1-onto𝑃)    &   ((𝜑 ∧ (𝑒𝐵𝑓𝐵)) → (𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)))    &   ((𝜑 ∧ (𝑒𝐵𝑓𝐵𝑔𝐵)) → (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑒)𝐼(𝐹𝑓))))    &   (𝜑𝐻𝑉)    &   (𝜑𝐺 ∈ TarskiGE)       (𝜑𝐻 ∈ TarskiGE)

15.4  Geometry in Hilbert spaces

Syntaxcttg 26222 Function to convert an algebraic structure to a Tarski geometry.
class toTG

Definitiondf-ttg 26223* Define a function converting a subcomplex Hilbert space to a Tarski Geometry. It does so by equipping the structure with a betweenness operation. Note that because the scalar product is applied over the interval (0[,]1), only spaces whose scalar field is a superset of that interval can be considered. (Contributed by Thierry Arnoux, 24-Mar-2019.)
toTG = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ ∃𝑘 ∈ (0[,]1)(𝑧(-g𝑤)𝑥) = (𝑘( ·𝑠𝑤)(𝑦(-g𝑤)𝑥))}) / 𝑖((𝑤 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩))

Theoremttgval 26224* Define a function to augment a subcomplex Hilbert space with betweenness and a line definition. (Contributed by Thierry Arnoux, 25-Mar-2019.)
𝐺 = (toTG‘𝐻)    &   𝐵 = (Base‘𝐻)    &    = (-g𝐻)    &    · = ( ·𝑠𝐻)    &   𝐼 = (Itv‘𝐺)       (𝐻𝑉 → (𝐺 = ((𝐻 sSet ⟨(Itv‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})⟩) ∧ 𝐼 = (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})))

Theoremttglem 26225 Lemma for ttgbas 26226 and ttgvsca 26229. (Contributed by Thierry Arnoux, 15-Apr-2019.)
𝐺 = (toTG‘𝐻)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   𝑁 < 16       (𝐸𝐻) = (𝐸𝐺)

Theoremttgbas 26226 The base set of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.)
𝐺 = (toTG‘𝐻)    &   𝐵 = (Base‘𝐻)       𝐵 = (Base‘𝐺)

Theoremttgplusg 26227 The addition operation of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.)
𝐺 = (toTG‘𝐻)    &    + = (+g𝐻)        + = (+g𝐺)

Theoremttgsub 26228 The subtraction operation of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.)
𝐺 = (toTG‘𝐻)    &    = (-g𝐻)        = (-g𝐺)

Theoremttgvsca 26229 The scalar product of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.)
𝐺 = (toTG‘𝐻)    &    · = ( ·𝑠𝐻)        · = ( ·𝑠𝐺)

Theoremttgds 26230 The metric of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.)
𝐺 = (toTG‘𝐻)    &   𝐷 = (dist‘𝐻)       𝐷 = (dist‘𝐺)

Theoremttgitvval 26231* Betweenness for a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.)
𝐺 = (toTG‘𝐻)    &   𝐼 = (Itv‘𝐺)    &   𝑃 = (Base‘𝐻)    &    = (-g𝐻)    &    · = ( ·𝑠𝐻)       ((𝐻𝑉𝑋𝑃𝑌𝑃) → (𝑋𝐼𝑌) = {𝑧𝑃 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑋) = (𝑘 · (𝑌 𝑋))})

Theoremttgelitv 26232* Betweenness for a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.)
𝐺 = (toTG‘𝐻)    &   𝐼 = (Itv‘𝐺)    &   𝑃 = (Base‘𝐻)    &    = (-g𝐻)    &    · = ( ·𝑠𝐻)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝐻𝑉)    &   (𝜑𝑍𝑃)       (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ↔ ∃𝑘 ∈ (0[,]1)(𝑍 𝑋) = (𝑘 · (𝑌 𝑋))))

Theoremttgbtwnid 26233 Any subcomplex module equipped with the betweenness operation fulfills the identity of betweenness (Axiom A6). (Contributed by Thierry Arnoux, 26-Mar-2019.)
𝐺 = (toTG‘𝐻)    &   𝐼 = (Itv‘𝐺)    &   𝑃 = (Base‘𝐻)    &    = (-g𝐻)    &    · = ( ·𝑠𝐻)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   𝑅 = (Base‘(Scalar‘𝐻))    &   (𝜑 → (0[,]1) ⊆ 𝑅)    &   (𝜑𝐻 ∈ ℂMod)    &   (𝜑𝑌 ∈ (𝑋𝐼𝑋))       (𝜑𝑋 = 𝑌)

Theoremttgcontlem1 26234 Lemma for % ttgcont . (Contributed by Thierry Arnoux, 24-May-2019.)
𝐺 = (toTG‘𝐻)    &   𝐼 = (Itv‘𝐺)    &   𝑃 = (Base‘𝐻)    &    = (-g𝐻)    &    · = ( ·𝑠𝐻)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   𝑅 = (Base‘(Scalar‘𝐻))    &   (𝜑 → (0[,]1) ⊆ 𝑅)    &    + = (+g𝐻)    &   (𝜑𝐻 ∈ ℂVec)    &   (𝜑𝐴𝑃)    &   (𝜑𝑁𝑃)    &   (𝜑𝑀 ≠ 0)    &   (𝜑𝐾 ≠ 0)    &   (𝜑𝐾 ≠ 1)    &   (𝜑𝐿𝑀)    &   (𝜑𝐿 ≤ (𝑀 / 𝐾))    &   (𝜑𝐿 ∈ (0[,]1))    &   (𝜑𝐾 ∈ (0[,]1))    &   (𝜑𝑀 ∈ (0[,]𝐿))    &   (𝜑 → (𝑋 𝐴) = (𝐾 · (𝑌 𝐴)))    &   (𝜑 → (𝑋 𝐴) = (𝑀 · (𝑁 𝐴)))    &   (𝜑𝐵 = (𝐴 + (𝐿 · (𝑁 𝐴))))       (𝜑𝐵 ∈ (𝑋𝐼𝑌))

Theoremxmstrkgc 26235 Any metric space fulfills Tarski's geometry axioms of congruence. (Contributed by Thierry Arnoux, 13-Mar-2019.)
(𝐺 ∈ ∞MetSp → 𝐺 ∈ TarskiGC)

15.4.1  Geometry in the complex plane

Theoremcchhllem 26236* Lemma for chlbas and chlvsca . (Contributed by Thierry Arnoux, 15-Apr-2019.)
𝐶 = (((subringAlg ‘ℂfld)‘ℝ) sSet ⟨(·𝑖‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · (∗‘𝑦)))⟩)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   (𝑁 < 5 ∨ 8 < 𝑁)       (𝐸‘ℂfld) = (𝐸𝐶)

15.4.2  Geometry in Euclidean spaces

15.4.2.1  Definition of the Euclidean space

Syntaxcee 26237 Declare the syntax for the Euclidean space generator.
class 𝔼

Syntaxcbtwn 26238 Declare the syntax for the Euclidean betweenness predicate.
class Btwn

Syntaxccgr 26239 Declare the syntax for the Euclidean congruence predicate.
class Cgr

Definitiondf-ee 26240 Define the Euclidean space generator. For details, see elee 26243. (Contributed by Scott Fenton, 3-Jun-2013.)
𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑𝑚 (1...𝑛)))

Definitiondf-btwn 26241* Define the Euclidean betweenness predicate. For details, see brbtwn 26248. (Contributed by Scott Fenton, 3-Jun-2013.)
Btwn = {⟨⟨𝑥, 𝑧⟩, 𝑦⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑦𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑧𝑖))))}

Definitiondf-cgr 26242* Define the Euclidean congruence predicate. For details, see brcgr 26249. (Contributed by Scott Fenton, 3-Jun-2013.)
Cgr = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑦 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))) ∧ Σ𝑖 ∈ (1...𝑛)((((1st𝑥)‘𝑖) − ((2nd𝑥)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st𝑦)‘𝑖) − ((2nd𝑦)‘𝑖))↑2))}

Theoremelee 26243 Membership in a Euclidean space. We define Euclidean space here using Cartesian coordinates over 𝑁 space. We later abstract away from this using Tarski's geometry axioms, so this exact definition is unimportant. (Contributed by Scott Fenton, 3-Jun-2013.)
(𝑁 ∈ ℕ → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴:(1...𝑁)⟶ℝ))

Theoremmptelee 26244* A condition for a mapping to be an element of a Euclidean space. (Contributed by Scott Fenton, 7-Jun-2013.)
(𝑁 ∈ ℕ → ((𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) ∈ (𝔼‘𝑁) ↔ ∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ))

Theoremeleenn 26245 If 𝐴 is in (𝔼‘𝑁), then 𝑁 is a natural. (Contributed by Scott Fenton, 1-Jul-2013.)
(𝐴 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ)

Theoremeleei 26246 The forward direction of elee 26243. (Contributed by Scott Fenton, 1-Jul-2013.)
(𝐴 ∈ (𝔼‘𝑁) → 𝐴:(1...𝑁)⟶ℝ)

Theoremeedimeq 26247 A point belongs to at most one Euclidean space. (Contributed by Scott Fenton, 1-Jul-2013.)
((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑀)) → 𝑁 = 𝑀)

Theorembrbtwn 26248* The binary relation form of the betweenness predicate. The statement 𝐴 Btwn ⟨𝐵, 𝐶 should be informally read as "𝐴 lies on a line segment between 𝐵 and 𝐶. This exact definition is abstracted away by Tarski's geometry axioms later on. (Contributed by Scott Fenton, 3-Jun-2013.)
((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) → (𝐴 Btwn ⟨𝐵, 𝐶⟩ ↔ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝐴𝑖) = (((1 − 𝑡) · (𝐵𝑖)) + (𝑡 · (𝐶𝑖)))))

Theorembrcgr 26249* The binary relation form of the congruence predicate. The statement 𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷 should be read informally as "the 𝑁 dimensional point 𝐴 is as far from 𝐵 as 𝐶 is from 𝐷, or "the line segment 𝐴𝐵 is congruent to the line segment 𝐶𝐷. This particular definition is encapsulated by Tarski's axioms later on. (Contributed by Scott Fenton, 3-Jun-2013.)
(((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩ ↔ Σ𝑖 ∈ (1...𝑁)(((𝐴𝑖) − (𝐵𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶𝑖) − (𝐷𝑖))↑2)))

Theoremfveere 26250 The function value of a point is a real. (Contributed by Scott Fenton, 10-Jun-2013.)
((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐼 ∈ (1...𝑁)) → (𝐴𝐼) ∈ ℝ)

Theoremfveecn 26251 The function value of a point is a complex. (Contributed by Scott Fenton, 10-Jun-2013.)
((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐼 ∈ (1...𝑁)) → (𝐴𝐼) ∈ ℂ)

Theoremeqeefv 26252* Two points are equal iff they agree in all dimensions. (Contributed by Scott Fenton, 10-Jun-2013.)
((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (𝐴 = 𝐵 ↔ ∀𝑖 ∈ (1...𝑁)(𝐴𝑖) = (𝐵𝑖)))

Theoremeqeelen 26253* Two points are equal iff the square of the distance between them is zero. (Contributed by Scott Fenton, 10-Jun-2013.) (Revised by Mario Carneiro, 22-May-2014.)
((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (𝐴 = 𝐵 ↔ Σ𝑖 ∈ (1...𝑁)(((𝐴𝑖) − (𝐵𝑖))↑2) = 0))

Theorembrbtwn2 26254* Alternate characterization of betweenness, with no existential quantifiers. (Contributed by Scott Fenton, 24-Jun-2013.)
((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) → (𝐴 Btwn ⟨𝐵, 𝐶⟩ ↔ (∀𝑖 ∈ (1...𝑁)(((𝐵𝑖) − (𝐴𝑖)) · ((𝐶𝑖) − (𝐴𝑖))) ≤ 0 ∧ ∀𝑖 ∈ (1...𝑁)∀𝑗 ∈ (1...𝑁)(((𝐵𝑖) − (𝐴𝑖)) · ((𝐶𝑗) − (𝐴𝑗))) = (((𝐵𝑗) − (𝐴𝑗)) · ((𝐶𝑖) − (𝐴𝑖))))))

Theoremcolinearalglem1 26255 Lemma for colinearalg 26259. Expand out a multiplication. (Contributed by Scott Fenton, 24-Jun-2013.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → (((𝐵𝐴) · (𝐹𝐷)) = ((𝐸𝐷) · (𝐶𝐴)) ↔ ((𝐵 · 𝐹) − ((𝐴 · 𝐹) + (𝐵 · 𝐷))) = ((𝐶 · 𝐸) − ((𝐴 · 𝐸) + (𝐶 · 𝐷)))))

Theoremcolinearalglem2 26256* Lemma for colinearalg 26259. Translate between two forms of the colinearity condition. (Contributed by Scott Fenton, 24-Jun-2013.)
((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) → (∀𝑖 ∈ (1...𝑁)∀𝑗 ∈ (1...𝑁)(((𝐵𝑖) − (𝐴𝑖)) · ((𝐶𝑗) − (𝐴𝑗))) = (((𝐵𝑗) − (𝐴𝑗)) · ((𝐶𝑖) − (𝐴𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)∀𝑗 ∈ (1...𝑁)(((𝐶𝑖) − (𝐵𝑖)) · ((𝐴𝑗) − (𝐵𝑗))) = (((𝐶𝑗) − (𝐵𝑗)) · ((𝐴𝑖) − (𝐵𝑖)))))

Theoremcolinearalglem3 26257* Lemma for colinearalg 26259. Translate between two forms of the colinearity condition. (Contributed by Scott Fenton, 24-Jun-2013.)
((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) → (∀𝑖 ∈ (1...𝑁)∀𝑗 ∈ (1...𝑁)(((𝐵𝑖) − (𝐴𝑖)) · ((𝐶𝑗) − (𝐴𝑗))) = (((𝐵𝑗) − (𝐴𝑗)) · ((𝐶𝑖) − (𝐴𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)∀𝑗 ∈ (1...𝑁)(((𝐴𝑖) − (𝐶𝑖)) · ((𝐵𝑗) − (𝐶𝑗))) = (((𝐴𝑗) − (𝐶𝑗)) · ((𝐵𝑖) − (𝐶𝑖)))))

Theoremcolinearalglem4 26258* Lemma for colinearalg 26259. Prove a disjunction that will be needed in the final proof. (Contributed by Scott Fenton, 27-Jun-2013.)
(((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ 𝐾 ∈ ℝ) → (∀𝑖 ∈ (1...𝑁)((((𝐾 · ((𝐶𝑖) − (𝐴𝑖))) + (𝐴𝑖)) − (𝐴𝑖)) · ((𝐶𝑖) − (𝐴𝑖))) ≤ 0 ∨ ∀𝑖 ∈ (1...𝑁)(((𝐶𝑖) − ((𝐾 · ((𝐶𝑖) − (𝐴𝑖))) + (𝐴𝑖))) · ((𝐴𝑖) − ((𝐾 · ((𝐶𝑖) − (𝐴𝑖))) + (𝐴𝑖)))) ≤ 0 ∨ ∀𝑖 ∈ (1...𝑁)(((𝐴𝑖) − (𝐶𝑖)) · (((𝐾 · ((𝐶𝑖) − (𝐴𝑖))) + (𝐴𝑖)) − (𝐶𝑖))) ≤ 0))

Theoremcolinearalg 26259* An algebraic characterization of colinearity. Note the similarity to brbtwn2 26254. (Contributed by Scott Fenton, 24-Jun-2013.)
((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) → ((𝐴 Btwn ⟨𝐵, 𝐶⟩ ∨ 𝐵 Btwn ⟨𝐶, 𝐴⟩ ∨ 𝐶 Btwn ⟨𝐴, 𝐵⟩) ↔ ∀𝑖 ∈ (1...𝑁)∀𝑗 ∈ (1...𝑁)(((𝐵𝑖) − (𝐴𝑖)) · ((𝐶𝑗) − (𝐴𝑗))) = (((𝐵𝑗) − (𝐴𝑗)) · ((𝐶𝑖) − (𝐴𝑖)))))

Theoremeleesub 26260* Membership of a subtraction mapping in a Euclidean space. (Contributed by Scott Fenton, 17-Jul-2013.)
𝐶 = (𝑖 ∈ (1...𝑁) ↦ ((𝐴𝑖) − (𝐵𝑖)))       ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝐶 ∈ (𝔼‘𝑁))

Theoremeleesubd 26261* Membership of a subtraction mapping in a Euclidean space. Deduction form of eleesub 26260. (Contributed by Scott Fenton, 17-Jul-2013.)
(𝜑𝐶 = (𝑖 ∈ (1...𝑁) ↦ ((𝐴𝑖) − (𝐵𝑖))))       ((𝜑𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝐶 ∈ (𝔼‘𝑁))

15.4.2.2  Tarski's axioms for geometry for the Euclidean space

Theoremaxdimuniq 26262 The unique dimension axiom. If a point is in 𝑁 dimensional space and in 𝑀 dimensional space, then 𝑁 = 𝑀. This axiom is not traditionally presented with Tarski's axioms, but we require it here as we are considering spaces in arbitrary dimensions. (Contributed by Scott Fenton, 24-Sep-2013.)
(((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁)) ∧ (𝑀 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑀))) → 𝑁 = 𝑀)

Theoremaxcgrrflx 26263 𝐴 is as far from 𝐵 as 𝐵 is from 𝐴. Axiom A1 of [Schwabhauser] p. 10. (Contributed by Scott Fenton, 3-Jun-2013.)
((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ⟨𝐴, 𝐵⟩Cgr⟨𝐵, 𝐴⟩)

Theoremaxcgrtr 26264 Congruence is transitive. Axiom A2 of [Schwabhauser] p. 10. (Contributed by Scott Fenton, 3-Jun-2013.)
((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → ((⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩) → ⟨𝐶, 𝐷⟩Cgr⟨𝐸, 𝐹⟩))

Theoremaxcgrid 26265 If there is no distance between 𝐴 and 𝐵, then 𝐴 = 𝐵. Axiom A3 of [Schwabhauser] p. 10. (Contributed by Scott Fenton, 3-Jun-2013.)
((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐶⟩ → 𝐴 = 𝐵))

Theoremaxsegconlem1 26266* Lemma for axsegcon 26276. Handle the degenerate case. (Contributed by Scott Fenton, 7-Jun-2013.)
((𝐴 = 𝐵 ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)))) → ∃𝑥 ∈ (𝔼‘𝑁)∃𝑡 ∈ (0[,]1)(∀𝑖 ∈ (1...𝑁)(𝐵𝑖) = (((1 − 𝑡) · (𝐴𝑖)) + (𝑡 · (𝑥𝑖))) ∧ Σ𝑖 ∈ (1...𝑁)(((𝐵𝑖) − (𝑥𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶𝑖) − (𝐷𝑖))↑2)))

Theoremaxsegconlem2 26267* Lemma for axsegcon 26276. 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)(((𝐴𝑝) − (𝐵𝑝))↑2)       ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝑆 ∈ ℝ)

Theoremaxsegconlem3 26268* Lemma for axsegcon 26276. Show that the square of 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)

Theoremaxsegconlem4 26269* Lemma for axsegcon 26276. Show that the distance between two points is a real number. (Contributed by Scott Fenton, 17-Sep-2013.)
𝑆 = Σ𝑝 ∈ (1...𝑁)(((𝐴𝑝) − (𝐵𝑝))↑2)       ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (√‘𝑆) ∈ ℝ)

Theoremaxsegconlem5 26270* Lemma for axsegcon 26276. 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 26271* Lemma for axsegcon 26276. 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 26272* Lemma for axsegcon 26276. 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 26273* Lemma for axsegcon 26276. 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 26274* Lemma for axsegcon 26276. 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 26275* Lemma for axsegcon 26276. Show that the scaling constant from axsegconlem7 26272 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 26276* 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 26277* Lemma for ax5seg 26287. 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 26278* Lemma for ax5seg 26287. 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 26279 Lemma for ax5seg 26287. (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 26280* Lemma for ax5seg 26287. 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 26281* Lemma for ax5seg 26287. 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 26282* Lemma for ax5seg 26287. 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 26283* Lemma for ax5seg 26287. 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 26284 Lemma for ax5seg 26287. 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 26285 Lemma for ax5seg 26287. Use the weak deduction theorem to eliminate the hypotheses from ax5seglem7 26284. (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 26286* Lemma for ax5seg 26287. Take the calculation in ax5seglem8 26285 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 26287 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 26288 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 26289* Lemma for axpasch 26290. 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 26290* 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 26291 Lemma for axlowdim 26310. Establish a particular constant function as a function. (Contributed by Scott Fenton, 29-Jun-2013.)
((3...𝑁) × {0}):(3...𝑁)⟶ℝ

Theoremaxlowdimlem2 26292 Lemma for axlowdim 26310. Show that two sets are disjoint. (Contributed by Scott Fenton, 29-Jun-2013.)
((1...2) ∩ (3...𝑁)) = ∅

Theoremaxlowdimlem3 26293 Lemma for axlowdim 26310. 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 26294 Lemma for axlowdim 26310. 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 26295 Lemma for axlowdim 26310. 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 26296 Lemma for axlowdim 26310. 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 26297 Lemma for axlowdim 26310. 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 26298 Lemma for axlowdim 26310. 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 26299 Lemma for axlowdim 26310. 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 26300 Lemma for axlowdim 26310. 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))

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