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Type | Label | Description |
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Statement | ||
Theorem | mircgrextend 26001 | Link congruence over a pair of mirror points. cf tgcgrextend 25804. (Contributed by Thierry Arnoux, 4-Oct-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ ∼ = (cgrG‘𝐺) & ⊢ 𝑀 = (𝑆‘𝐵) & ⊢ 𝑁 = (𝑆‘𝑌) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → (𝐴 − 𝐵) = (𝑋 − 𝑌)) ⇒ ⊢ (𝜑 → (𝐴 − (𝑀‘𝐴)) = (𝑋 − (𝑁‘𝑋))) | ||
Theorem | mirtrcgr 26002 | Point inversion of one point of a triangle around another point preserves triangle congruence. (Contributed by Thierry Arnoux, 4-Oct-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ ∼ = (cgrG‘𝐺) & ⊢ 𝑀 = (𝑆‘𝐵) & ⊢ 𝑁 = (𝑆‘𝑌) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝑋𝑌𝑍”〉) ⇒ ⊢ (𝜑 → 〈“(𝑀‘𝐴)𝐵𝐶”〉 ∼ 〈“(𝑁‘𝑋)𝑌𝑍”〉) | ||
Theorem | mirauto 26003 | Point inversion preserves point inversion. (Contributed by Thierry Arnoux, 30-Jul-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝑀 = (𝑆‘𝑇) & ⊢ 𝑋 = (𝑀‘𝐴) & ⊢ 𝑌 = (𝑀‘𝐵) & ⊢ 𝑍 = (𝑀‘𝐶) & ⊢ (𝜑 → 𝑇 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → ((𝑆‘𝐴)‘𝐵) = 𝐶) ⇒ ⊢ (𝜑 → ((𝑆‘𝑋)‘𝑌) = 𝑍) | ||
Theorem | miduniq 26004 | Uniqueness of the middle point, expressed with point inversion. Theorem 7.17 of [Schwabhauser] p. 51. (Contributed by Thierry Arnoux, 30-Jul-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → ((𝑆‘𝐴)‘𝑋) = 𝑌) & ⊢ (𝜑 → ((𝑆‘𝐵)‘𝑋) = 𝑌) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | miduniq1 26005 | Uniqueness of the middle point, expressed with point inversion. Theorem 7.18 of [Schwabhauser] p. 52. (Contributed by Thierry Arnoux, 30-Jul-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → ((𝑆‘𝐴)‘𝑋) = ((𝑆‘𝐵)‘𝑋)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | miduniq2 26006 | If two point inversions commute, they are identical. Theorem 7.19 of [Schwabhauser] p. 52. (Contributed by Thierry Arnoux, 30-Jul-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → ((𝑆‘𝐴)‘((𝑆‘𝐵)‘𝑋)) = ((𝑆‘𝐵)‘((𝑆‘𝐴)‘𝑋))) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | colmid 26007 | Colinearity and equidistance implies midpoint. Theorem 7.20 of [Schwabhauser] p. 52. (Contributed by Thierry Arnoux, 30-Jul-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝑀 = (𝑆‘𝑋) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → (𝑋 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)) & ⊢ (𝜑 → (𝑋 − 𝐴) = (𝑋 − 𝐵)) ⇒ ⊢ (𝜑 → (𝐵 = (𝑀‘𝐴) ∨ 𝐴 = 𝐵)) | ||
Theorem | symquadlem 26008 | Lemma of the symetrial quadrilateral. The diagonals of quadrilaterals with congruent opposing sides intersect at their middle point. In Euclidean geometry, such quadrilaterals are called parallelograms, as opposing sides are parallel. However, this is not necessarily true in the case of absolute geometry. Lemma 7.21 of [Schwabhauser] p. 52. (Contributed by Thierry Arnoux, 6-Aug-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝑀 = (𝑆‘𝑋) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶)) & ⊢ (𝜑 → 𝐵 ≠ 𝐷) & ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷)) & ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐷 − 𝐴)) & ⊢ (𝜑 → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶)) & ⊢ (𝜑 → (𝑋 ∈ (𝐵𝐿𝐷) ∨ 𝐵 = 𝐷)) ⇒ ⊢ (𝜑 → 𝐴 = (𝑀‘𝐶)) | ||
Theorem | krippenlem 26009 | Lemma for krippen 26010. We can assume krippen.7 "without loss of generality" (Contributed by Thierry Arnoux, 12-Aug-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝑀 = (𝑆‘𝑋) & ⊢ 𝑁 = (𝑆‘𝑌) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐸)) & ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐹)) & ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐶 − 𝐵)) & ⊢ (𝜑 → (𝐶 − 𝐸) = (𝐶 − 𝐹)) & ⊢ (𝜑 → 𝐵 = (𝑀‘𝐴)) & ⊢ (𝜑 → 𝐹 = (𝑁‘𝐸)) & ⊢ ≤ = (≤G‘𝐺) & ⊢ (𝜑 → (𝐶 − 𝐴) ≤ (𝐶 − 𝐸)) ⇒ ⊢ (𝜑 → 𝐶 ∈ (𝑋𝐼𝑌)) | ||
Theorem | krippen 26010 | Krippenlemma (German for crib's lemma) Lemma 7.22 of [Schwabhauser] p. 53. proven by Gupta 1965 as Theorem 3.45. (Contributed by Thierry Arnoux, 12-Aug-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝑀 = (𝑆‘𝑋) & ⊢ 𝑁 = (𝑆‘𝑌) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐸)) & ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐹)) & ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐶 − 𝐵)) & ⊢ (𝜑 → (𝐶 − 𝐸) = (𝐶 − 𝐹)) & ⊢ (𝜑 → 𝐵 = (𝑀‘𝐴)) & ⊢ (𝜑 → 𝐹 = (𝑁‘𝐸)) ⇒ ⊢ (𝜑 → 𝐶 ∈ (𝑋𝐼𝑌)) | ||
Theorem | midexlem 26011* | Lemma for the existence of a middle point. Lemma 7.25 of [Schwabhauser] p. 55. This proof of the existence of a midpoint requires the existence of a third point 𝐶 equidistant to 𝐴 and 𝐵 This condition will be removed later. Because the operation notation (𝐴(midG‘𝐺)𝐵) for a midpoint implies its uniqueness, it cannot be used until uniqueness is proven, and until then, an equivalent mirror point notation 𝐵 = (𝑀‘𝐴) has to be used. See mideu 26054 for the existence and uniqueness of the midpoint. (Contributed by Thierry Arnoux, 25-Aug-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝑀 = (𝑆‘𝑥) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐶 − 𝐵)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 𝐵 = (𝑀‘𝐴)) | ||
Syntax | crag 26012 | Declare the constant for the class of right angles. |
class ∟G | ||
Definition | df-rag 26013* | Define the class of right angles. Definition 8.1 of [Schwabhauser] p. 57. See israg 26016. (Contributed by Thierry Arnoux, 25-Aug-2019.) |
⊢ ∟G = (𝑔 ∈ V ↦ {𝑤 ∈ Word (Base‘𝑔) ∣ ((♯‘𝑤) = 3 ∧ ((𝑤‘0)(dist‘𝑔)(𝑤‘2)) = ((𝑤‘0)(dist‘𝑔)(((pInvG‘𝑔)‘(𝑤‘1))‘(𝑤‘2))))}) | ||
Syntax | cperpg 26014 | Declare the constant for the perpendicular relation. |
class ⟂G | ||
Definition | df-perpg 26015* | Define the "perpendicular" relation. Definition 8.11 of [Schwabhauser] p. 59. See isperp 26031. (Contributed by Thierry Arnoux, 8-Sep-2019.) |
⊢ ⟂G = (𝑔 ∈ V ↦ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝑔))}) | ||
Theorem | israg 26016 | Property for 3 points A, B, C to form a right angle. Definition 8.1 of [Schwabhauser] p. 57. (Contributed by Thierry Arnoux, 25-Aug-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) ⇒ ⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺) ↔ (𝐴 − 𝐶) = (𝐴 − ((𝑆‘𝐵)‘𝐶)))) | ||
Theorem | ragcom 26017 | Commutative rule for right angles. Theorem 8.2 of [Schwabhauser] p. 57. (Contributed by Thierry Arnoux, 25-Aug-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) ⇒ ⊢ (𝜑 → 〈“𝐶𝐵𝐴”〉 ∈ (∟G‘𝐺)) | ||
Theorem | ragcol 26018 | The right angle property is independent of the choice of point on one side. Theorem 8.3 of [Schwabhauser] p. 58. (Contributed by Thierry Arnoux, 25-Aug-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → (𝐴 ∈ (𝐵𝐿𝐷) ∨ 𝐵 = 𝐷)) ⇒ ⊢ (𝜑 → 〈“𝐷𝐵𝐶”〉 ∈ (∟G‘𝐺)) | ||
Theorem | ragmir 26019 | Right angle property is preserved by point inversion. Theorem 8.4 of [Schwabhauser] p. 58. (Contributed by Thierry Arnoux, 25-Aug-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) ⇒ ⊢ (𝜑 → 〈“𝐴𝐵((𝑆‘𝐵)‘𝐶)”〉 ∈ (∟G‘𝐺)) | ||
Theorem | mirrag 26020 | Right angle is conserved by point inversion. (Contributed by Thierry Arnoux, 3-Nov-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) & ⊢ 𝑀 = (𝑆‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) ⇒ ⊢ (𝜑 → 〈“(𝑀‘𝐴)(𝑀‘𝐵)(𝑀‘𝐶)”〉 ∈ (∟G‘𝐺)) | ||
Theorem | ragtrivb 26021 | Trivial right angle. Theorem 8.5 of [Schwabhauser] p. 58. (Contributed by Thierry Arnoux, 25-Aug-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) ⇒ ⊢ (𝜑 → 〈“𝐴𝐵𝐵”〉 ∈ (∟G‘𝐺)) | ||
Theorem | ragflat2 26022 | Deduce equality from two right angles. Theorem 8.6 of [Schwabhauser] p. 58. (Contributed by Thierry Arnoux, 3-Sep-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) & ⊢ (𝜑 → 〈“𝐷𝐵𝐶”〉 ∈ (∟G‘𝐺)) & ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷)) ⇒ ⊢ (𝜑 → 𝐵 = 𝐶) | ||
Theorem | ragflat 26023 | Deduce equality from two right angles. Theorem 8.7 of [Schwabhauser] p. 58. (Contributed by Thierry Arnoux, 3-Sep-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) & ⊢ (𝜑 → 〈“𝐴𝐶𝐵”〉 ∈ (∟G‘𝐺)) ⇒ ⊢ (𝜑 → 𝐵 = 𝐶) | ||
Theorem | ragtriva 26024 | Trivial right angle. Theorem 8.8 of [Schwabhauser] p. 58. (Contributed by Thierry Arnoux, 3-Sep-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐴”〉 ∈ (∟G‘𝐺)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | ragflat3 26025 | Right angle and colinearity. Theorem 8.9 of [Schwabhauser] p. 58. (Contributed by Thierry Arnoux, 4-Sep-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) & ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)) ⇒ ⊢ (𝜑 → (𝐴 = 𝐵 ∨ 𝐶 = 𝐵)) | ||
Theorem | ragcgr 26026 | Right angle and colinearity. Theorem 8.10 of [Schwabhauser] p. 58. (Contributed by Thierry Arnoux, 4-Sep-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ ∼ = (cgrG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝐸𝐹”〉) ⇒ ⊢ (𝜑 → 〈“𝐷𝐸𝐹”〉 ∈ (∟G‘𝐺)) | ||
Theorem | motrag 26027 | Right angles are preserved by motions. (Contributed by Thierry Arnoux, 16-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ (𝐺Ismt𝐺)) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) ⇒ ⊢ (𝜑 → 〈“(𝐹‘𝐴)(𝐹‘𝐵)(𝐹‘𝐶)”〉 ∈ (∟G‘𝐺)) | ||
Theorem | ragncol 26028 | Right angle implies non-colinearity. A consequence of theorem 8.9 of [Schwabhauser] p. 58. (Contributed by Thierry Arnoux, 1-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐶 ≠ 𝐵) ⇒ ⊢ (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)) | ||
Theorem | perpln1 26029 | Derive a line from perpendicularity. (Contributed by Thierry Arnoux, 27-Nov-2019.) |
⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) | ||
Theorem | perpln2 26030 | Derive a line from perpendicularity. (Contributed by Thierry Arnoux, 27-Nov-2019.) |
⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵) ⇒ ⊢ (𝜑 → 𝐵 ∈ ran 𝐿) | ||
Theorem | isperp 26031* | Property for 2 lines A, B to be perpendicular. Item (ii) of definition 8.11 of [Schwabhauser] p. 59. (Contributed by Thierry Arnoux, 16-Oct-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐵 ∈ ran 𝐿) ⇒ ⊢ (𝜑 → (𝐴(⟂G‘𝐺)𝐵 ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺))) | ||
Theorem | perpcom 26032 | The "perpendicular" relation commutes. Theorem 8.12 of [Schwabhauser] p. 59. (Contributed by Thierry Arnoux, 16-Oct-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐵 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵) ⇒ ⊢ (𝜑 → 𝐵(⟂G‘𝐺)𝐴) | ||
Theorem | perpneq 26033 | Two perpendicular lines are different. Theorem 8.14 of [Schwabhauser] p. 59. (Contributed by Thierry Arnoux, 18-Oct-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐵 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐵) | ||
Theorem | isperp2 26034* | Property for 2 lines A, B, intersecting at a point X to be perpendicular. Item (i) of definition 8.13 of [Schwabhauser] p. 59. (Contributed by Thierry Arnoux, 16-Oct-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐵 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) ⇒ ⊢ (𝜑 → (𝐴(⟂G‘𝐺)𝐵 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺))) | ||
Theorem | isperp2d 26035 | One direction of isperp2 26034. (Contributed by Thierry Arnoux, 10-Nov-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐵 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) & ⊢ (𝜑 → 𝑈 ∈ 𝐴) & ⊢ (𝜑 → 𝑉 ∈ 𝐵) & ⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵) ⇒ ⊢ (𝜑 → 〈“𝑈𝑋𝑉”〉 ∈ (∟G‘𝐺)) | ||
Theorem | ragperp 26036 | Deduce that two lines are perpendicular from a right angle statement. One direction of theorem 8.13 of [Schwabhauser] p. 59. (Contributed by Thierry Arnoux, 20-Oct-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐵 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) & ⊢ (𝜑 → 𝑈 ∈ 𝐴) & ⊢ (𝜑 → 𝑉 ∈ 𝐵) & ⊢ (𝜑 → 𝑈 ≠ 𝑋) & ⊢ (𝜑 → 𝑉 ≠ 𝑋) & ⊢ (𝜑 → 〈“𝑈𝑋𝑉”〉 ∈ (∟G‘𝐺)) ⇒ ⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵) | ||
Theorem | footex 26037* | Lemma for foot 26038: existence part. (Contributed by Thierry Arnoux, 19-Oct-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴) | ||
Theorem | foot 26038* | From a point 𝐶 outside of a line 𝐴, there exists a unique point 𝑥 on 𝐴 such that (𝐶𝐿𝑥) is perpendicular to 𝐴. That point is called the foot from 𝐶 on 𝐴. Theorem 8.18 of [Schwabhauser] p. 60. (Contributed by Thierry Arnoux, 19-Oct-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴) ⇒ ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴) | ||
Theorem | footne 26039 | Uniqueness of the foot point. (Contributed by Thierry Arnoux, 28-Feb-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → (𝑋𝐿𝑌)(⟂G‘𝐺)𝐴) ⇒ ⊢ (𝜑 → ¬ 𝑌 ∈ 𝐴) | ||
Theorem | footeq 26040 | Uniqueness of the foot point. (Contributed by Thierry Arnoux, 1-Mar-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → (𝑋𝐿𝑍)(⟂G‘𝐺)𝐴) & ⊢ (𝜑 → (𝑌𝐿𝑍)(⟂G‘𝐺)𝐴) ⇒ ⊢ (𝜑 → 𝑋 = 𝑌) | ||
Theorem | hlperpnel 26041 | A point on a half-line which is perpendicular to a line cannot be on that line. (Contributed by Thierry Arnoux, 1-Mar-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝑈 ∈ 𝐴) & ⊢ (𝜑 → 𝑉 ∈ 𝑃) & ⊢ (𝜑 → 𝑊 ∈ 𝑃) & ⊢ (𝜑 → 𝐴(⟂G‘𝐺)(𝑈𝐿𝑉)) & ⊢ (𝜑 → 𝑉(𝐾‘𝑈)𝑊) ⇒ ⊢ (𝜑 → ¬ 𝑊 ∈ 𝐴) | ||
Theorem | perprag 26042 | Deduce a right angle from perpendicular lines. (Contributed by Thierry Arnoux, 10-Nov-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐿𝐵)) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐶𝐿𝐷)) ⇒ ⊢ (𝜑 → 〈“𝐴𝐶𝐷”〉 ∈ (∟G‘𝐺)) | ||
Theorem | perpdragALT 26043 | Deduce a right angle from perpendicular lines. (Contributed by Thierry Arnoux, 12-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ (𝜑 → 𝐵 ∈ 𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐵𝐿𝐶)) ⇒ ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) | ||
Theorem | perpdrag 26044 | Deduce a right angle from perpendicular lines. (Contributed by Thierry Arnoux, 12-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ (𝜑 → 𝐵 ∈ 𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐵𝐿𝐶)) ⇒ ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) | ||
Theorem | colperp 26045 | Deduce a perpendicularity from perpendicularity and colinearity. (Contributed by Thierry Arnoux, 8-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)𝐷) & ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)) & ⊢ (𝜑 → 𝐴 ≠ 𝐶) ⇒ ⊢ (𝜑 → (𝐴𝐿𝐶)(⟂G‘𝐺)𝐷) | ||
Theorem | colperpexlem1 26046 | Lemma for colperp 26045. First part of lemma 8.20 of [Schwabhauser] p. 62. (Contributed by Thierry Arnoux, 27-Oct-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ 𝑀 = (𝑆‘𝐴) & ⊢ 𝑁 = (𝑆‘𝐵) & ⊢ 𝐾 = (𝑆‘𝑄) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) & ⊢ (𝜑 → (𝐾‘(𝑀‘𝐶)) = (𝑁‘𝐶)) ⇒ ⊢ (𝜑 → 〈“𝐵𝐴𝑄”〉 ∈ (∟G‘𝐺)) | ||
Theorem | colperpexlem2 26047 | Lemma for colperpex 26049. Second part of lemma 8.20 of [Schwabhauser] p. 62. (Contributed by Thierry Arnoux, 10-Nov-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ 𝑀 = (𝑆‘𝐴) & ⊢ 𝑁 = (𝑆‘𝐵) & ⊢ 𝐾 = (𝑆‘𝑄) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) & ⊢ (𝜑 → (𝐾‘(𝑀‘𝐶)) = (𝑁‘𝐶)) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝑄) | ||
Theorem | colperpexlem3 26048* | Lemma for colperpex 26049. Case 1 of theorem 8.21 of [Schwabhauser] p. 63. (Contributed by Thierry Arnoux, 20-Nov-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴𝐿𝐵)) ⇒ ⊢ (𝜑 → ∃𝑝 ∈ 𝑃 ((𝐴𝐿𝑝)(⟂G‘𝐺)(𝐴𝐿𝐵) ∧ ∃𝑡 ∈ 𝑃 ((𝑡 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑡 ∈ (𝐶𝐼𝑝)))) | ||
Theorem | colperpex 26049* | In dimension 2 and above, on a line (𝐴𝐿𝐵) there is always a perpendicular 𝑃 from 𝐴 on a given plane (here given by 𝐶, in case 𝐶 does not lie on the line). Theorem 8.21 of [Schwabhauser] p. 63. (Contributed by Thierry Arnoux, 20-Nov-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) ⇒ ⊢ (𝜑 → ∃𝑝 ∈ 𝑃 ((𝐴𝐿𝑝)(⟂G‘𝐺)(𝐴𝐿𝐵) ∧ ∃𝑡 ∈ 𝑃 ((𝑡 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑡 ∈ (𝐶𝐼𝑝)))) | ||
Theorem | mideulem2 26050 | Lemma for opphllem 26051, which is itself used for mideu 26054. (Contributed by Thierry Arnoux, 19-Feb-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) & ⊢ (𝜑 → 𝑂 ∈ 𝑃) & ⊢ (𝜑 → 𝑇 ∈ 𝑃) & ⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵)) & ⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂)) & ⊢ (𝜑 → 𝑇 ∈ (𝐴𝐿𝐵)) & ⊢ (𝜑 → 𝑇 ∈ (𝑄𝐼𝑂)) & ⊢ (𝜑 → 𝑅 ∈ 𝑃) & ⊢ (𝜑 → 𝑅 ∈ (𝐵𝐼𝑄)) & ⊢ (𝜑 → (𝐴 − 𝑂) = (𝐵 − 𝑅)) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ (𝑇𝐼𝐵)) & ⊢ (𝜑 → 𝑋 ∈ (𝑅𝐼𝑂)) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ (((𝑆‘𝐴)‘𝑂)𝐼𝑍)) & ⊢ (𝜑 → (𝑋 − 𝑍) = (𝑋 − 𝑅)) & ⊢ (𝜑 → 𝑀 ∈ 𝑃) & ⊢ (𝜑 → 𝑅 = ((𝑆‘𝑀)‘𝑍)) ⇒ ⊢ (𝜑 → 𝐵 = 𝑀) | ||
Theorem | opphllem 26051* | Lemma 8.24 of [Schwabhauser] p. 66. This is used later for mideulem 26052 and later for opphl 26070. (Contributed by Thierry Arnoux, 21-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) & ⊢ (𝜑 → 𝑂 ∈ 𝑃) & ⊢ (𝜑 → 𝑇 ∈ 𝑃) & ⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵)) & ⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂)) & ⊢ (𝜑 → 𝑇 ∈ (𝐴𝐿𝐵)) & ⊢ (𝜑 → 𝑇 ∈ (𝑄𝐼𝑂)) & ⊢ (𝜑 → 𝑅 ∈ 𝑃) & ⊢ (𝜑 → 𝑅 ∈ (𝐵𝐼𝑄)) & ⊢ (𝜑 → (𝐴 − 𝑂) = (𝐵 − 𝑅)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝑂 = ((𝑆‘𝑥)‘𝑅))) | ||
Theorem | mideulem 26052* | Lemma for mideu 26054. We can assume mideulem.9 "without loss of generality" (Contributed by Thierry Arnoux, 25-Nov-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) & ⊢ (𝜑 → 𝑂 ∈ 𝑃) & ⊢ (𝜑 → 𝑇 ∈ 𝑃) & ⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵)) & ⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂)) & ⊢ (𝜑 → 𝑇 ∈ (𝐴𝐿𝐵)) & ⊢ (𝜑 → 𝑇 ∈ (𝑄𝐼𝑂)) & ⊢ (𝜑 → (𝐴 − 𝑂)(≤G‘𝐺)(𝐵 − 𝑄)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴)) | ||
Theorem | midex 26053* | Existence of the midpoint, part Theorem 8.22 of [Schwabhauser] p. 64. Note that this proof requires a construction in 2 dimensions or more, i.e. it does not prove the existence of a midpoint in dimension 1, for a geometry restricted to a line. (Contributed by Thierry Arnoux, 25-Nov-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴)) | ||
Theorem | mideu 26054* | Existence and uniqueness of the midpoint, Theorem 8.22 of [Schwabhauser] p. 64. (Contributed by Thierry Arnoux, 25-Nov-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) ⇒ ⊢ (𝜑 → ∃!𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴)) | ||
Theorem | islnopp 26055* | The property for two points 𝐴 and 𝐵 to lie on the opposite sides of a set 𝐷 Definition 9.1 of [Schwabhauser] p. 67. (Contributed by Thierry Arnoux, 19-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵)))) | ||
Theorem | islnoppd 26056* | Deduce that 𝐴 and 𝐵 lie on opposite sides of line 𝐿. (Contributed by Thierry Arnoux, 16-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝐷) & ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷) & ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐷) & ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐵)) ⇒ ⊢ (𝜑 → 𝐴𝑂𝐵) | ||
Theorem | oppne1 26057* | Points lying on opposite sides of a line cannot be on the line. (Contributed by Thierry Arnoux, 3-Mar-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐴𝑂𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷) | ||
Theorem | oppne2 26058* | Points lying on opposite sides of a line cannot be on the line. (Contributed by Thierry Arnoux, 3-Mar-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐴𝑂𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐷) | ||
Theorem | oppne3 26059* | Points lying on opposite sides of a line cannot be equal. (Contributed by Thierry Arnoux, 3-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐴𝑂𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐵) | ||
Theorem | oppcom 26060* | Commutativity rule for "opposite" Theorem 9.2 of [Schwabhauser] p. 67. (Contributed by Thierry Arnoux, 19-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐴𝑂𝐵) ⇒ ⊢ (𝜑 → 𝐵𝑂𝐴) | ||
Theorem | opptgdim2 26061* | If two points opposite to a line exist, dimension must be 2 or more. (Contributed by Thierry Arnoux, 3-Mar-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐴𝑂𝐵) ⇒ ⊢ (𝜑 → 𝐺DimTarskiG≥2) | ||
Theorem | oppnid 26062* | The "opposite to a line" relation is irreflexive. (Contributed by Thierry Arnoux, 4-Mar-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) ⇒ ⊢ (𝜑 → ¬ 𝐴𝑂𝐴) | ||
Theorem | opphllem1 26063* | Lemma for opphl 26070. (Contributed by Thierry Arnoux, 20-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝑆 = ((pInvG‘𝐺)‘𝑀) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝑅 ∈ 𝐷) & ⊢ (𝜑 → 𝐴𝑂𝐶) & ⊢ (𝜑 → 𝑀 ∈ 𝐷) & ⊢ (𝜑 → 𝐴 = (𝑆‘𝐶)) & ⊢ (𝜑 → 𝐴 ≠ 𝑅) & ⊢ (𝜑 → 𝐵 ≠ 𝑅) & ⊢ (𝜑 → 𝐵 ∈ (𝑅𝐼𝐴)) ⇒ ⊢ (𝜑 → 𝐵𝑂𝐶) | ||
Theorem | opphllem2 26064* | Lemma for opphl 26070. Lemma 9.3 of [Schwabhauser] p. 68. (Contributed by Thierry Arnoux, 21-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝑆 = ((pInvG‘𝐺)‘𝑀) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝑅 ∈ 𝐷) & ⊢ (𝜑 → 𝐴𝑂𝐶) & ⊢ (𝜑 → 𝑀 ∈ 𝐷) & ⊢ (𝜑 → 𝐴 = (𝑆‘𝐶)) & ⊢ (𝜑 → 𝐴 ≠ 𝑅) & ⊢ (𝜑 → 𝐵 ≠ 𝑅) & ⊢ (𝜑 → (𝐴 ∈ (𝑅𝐼𝐵) ∨ 𝐵 ∈ (𝑅𝐼𝐴))) ⇒ ⊢ (𝜑 → 𝐵𝑂𝐶) | ||
Theorem | opphllem3 26065* | Lemma for opphl 26070: We assume opphllem3.l "without loss of generality". (Contributed by Thierry Arnoux, 21-Feb-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ 𝑁 = ((pInvG‘𝐺)‘𝑀) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝑅 ∈ 𝐷) & ⊢ (𝜑 → 𝑆 ∈ 𝐷) & ⊢ (𝜑 → 𝑀 ∈ 𝑃) & ⊢ (𝜑 → 𝐴𝑂𝐶) & ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅)) & ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆)) & ⊢ (𝜑 → 𝑅 ≠ 𝑆) & ⊢ (𝜑 → (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) & ⊢ (𝜑 → 𝑈 ∈ 𝑃) & ⊢ (𝜑 → (𝑁‘𝑅) = 𝑆) ⇒ ⊢ (𝜑 → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶)) | ||
Theorem | opphllem4 26066* | Lemma for opphl 26070. (Contributed by Thierry Arnoux, 22-Feb-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ 𝑁 = ((pInvG‘𝐺)‘𝑀) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝑅 ∈ 𝐷) & ⊢ (𝜑 → 𝑆 ∈ 𝐷) & ⊢ (𝜑 → 𝑀 ∈ 𝑃) & ⊢ (𝜑 → 𝐴𝑂𝐶) & ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅)) & ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆)) & ⊢ (𝜑 → 𝑅 ≠ 𝑆) & ⊢ (𝜑 → (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) & ⊢ (𝜑 → 𝑈 ∈ 𝑃) & ⊢ (𝜑 → (𝑁‘𝑅) = 𝑆) & ⊢ (𝜑 → 𝑉 ∈ 𝑃) & ⊢ (𝜑 → 𝑈(𝐾‘𝑅)𝐴) & ⊢ (𝜑 → 𝑉(𝐾‘𝑆)𝐶) ⇒ ⊢ (𝜑 → 𝑈𝑂𝑉) | ||
Theorem | opphllem5 26067* | Second part of Lemma 9.4 of [Schwabhauser] p. 68. (Contributed by Thierry Arnoux, 2-Mar-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ 𝑁 = ((pInvG‘𝐺)‘𝑀) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝑅 ∈ 𝐷) & ⊢ (𝜑 → 𝑆 ∈ 𝐷) & ⊢ (𝜑 → 𝑀 ∈ 𝑃) & ⊢ (𝜑 → 𝐴𝑂𝐶) & ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅)) & ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆)) & ⊢ (𝜑 → 𝑈 ∈ 𝑃) & ⊢ (𝜑 → 𝑉 ∈ 𝑃) & ⊢ (𝜑 → 𝑈(𝐾‘𝑅)𝐴) & ⊢ (𝜑 → 𝑉(𝐾‘𝑆)𝐶) ⇒ ⊢ (𝜑 → 𝑈𝑂𝑉) | ||
Theorem | opphllem6 26068* | First part of Lemma 9.4 of [Schwabhauser] p. 68. (Contributed by Thierry Arnoux, 3-Mar-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ 𝑁 = ((pInvG‘𝐺)‘𝑀) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝑅 ∈ 𝐷) & ⊢ (𝜑 → 𝑆 ∈ 𝐷) & ⊢ (𝜑 → 𝑀 ∈ 𝑃) & ⊢ (𝜑 → 𝐴𝑂𝐶) & ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅)) & ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆)) & ⊢ (𝜑 → 𝑈 ∈ 𝑃) & ⊢ (𝜑 → (𝑁‘𝑅) = 𝑆) ⇒ ⊢ (𝜑 → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶)) | ||
Theorem | oppperpex 26069* | Restating colperpex 26049 using the "opposite side of a line" relation. (Contributed by Thierry Arnoux, 2-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐷) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) ⇒ ⊢ (𝜑 → ∃𝑝 ∈ 𝑃 ((𝐴𝐿𝑝)(⟂G‘𝐺)𝐷 ∧ 𝐶𝑂𝑝)) | ||
Theorem | opphl 26070* | If two points 𝐴 and 𝐶 lie on the opposite side of a line 𝐷 then any point of the half line (𝑅 𝐴) also lies opposite to 𝐶. Theorem 9.5 of [Schwabhauser] p. 69. (Contributed by Thierry Arnoux, 3-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐴𝑂𝐶) & ⊢ (𝜑 → 𝑅 ∈ 𝐷) & ⊢ (𝜑 → 𝐴(𝐾‘𝑅)𝐵) ⇒ ⊢ (𝜑 → 𝐵𝑂𝐶) | ||
Theorem | outpasch 26071* | Axiom of Pasch, outer form. This was proven by Gupta from other axioms and is therefore presented as Theorem 9.6 in [Schwabhauser] p. 70. (Contributed by Thierry Arnoux, 16-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝑅 ∈ 𝑃) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝑅)) & ⊢ (𝜑 → 𝑄 ∈ (𝐵𝐼𝐶)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥))) | ||
Theorem | hlpasch 26072* | An application of the axiom of Pasch for half-lines. (Contributed by Thierry Arnoux, 15-Sep-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐶(𝐾‘𝐵)𝐷) & ⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼𝐶)) ⇒ ⊢ (𝜑 → ∃𝑒 ∈ 𝑃 (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷))) | ||
Syntax | chpg 26073 | "Belong to the same open half-plane" relation for points in a geometry. |
class hpG | ||
Definition | df-hpg 26074* | Define the open half plane relation for a geometry 𝐺. Definition 9.7 of [Schwabhauser] p. 71. See hpgbr 26076 to find the same formulation. (Contributed by Thierry Arnoux, 4-Mar-2020.) |
⊢ hpG = (𝑔 ∈ V ↦ (𝑑 ∈ ran (LineG‘𝑔) ↦ {〈𝑎, 𝑏〉 ∣ [(Base‘𝑔) / 𝑝][(Itv‘𝑔) / 𝑖]∃𝑐 ∈ 𝑝 (((𝑎 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝑖𝑐)) ∧ ((𝑏 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝑖𝑐)))})) | ||
Theorem | ishpg 26075* | Value of the half-plane relation for a given line 𝐷. (Contributed by Thierry Arnoux, 4-Mar-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) ⇒ ⊢ (𝜑 → ((hpG‘𝐺)‘𝐷) = {〈𝑎, 𝑏〉 ∣ ∃𝑐 ∈ 𝑃 (𝑎𝑂𝑐 ∧ 𝑏𝑂𝑐)}) | ||
Theorem | hpgbr 26076* | Half-planes : property for points 𝐴 and 𝐵 to belong to the same open half plane delimited by line 𝐷. Definition 9.7 of [Schwabhauser] p. 71. (Contributed by Thierry Arnoux, 4-Mar-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐))) | ||
Theorem | hpgne1 26077* | Points on the open half plane cannot lie on its border. (Contributed by Thierry Arnoux, 4-Mar-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐴((hpG‘𝐺)‘𝐷)𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷) | ||
Theorem | hpgne2 26078* | Points on the open half plane cannot lie on its border. (Contributed by Thierry Arnoux, 4-Mar-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐴((hpG‘𝐺)‘𝐷)𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐷) | ||
Theorem | lnopp2hpgb 26079* | Theorem 9.8 of [Schwabhauser] p. 71. (Contributed by Thierry Arnoux, 4-Mar-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐴𝑂𝐶) ⇒ ⊢ (𝜑 → (𝐵𝑂𝐶 ↔ 𝐴((hpG‘𝐺)‘𝐷)𝐵)) | ||
Theorem | lnoppnhpg 26080* | If two points lie on the opposite side of a line 𝐷, they are not on the same half-plane. Theorem 9.9 of [Schwabhauser] p. 72. (Contributed by Thierry Arnoux, 4-Mar-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐴𝑂𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐴((hpG‘𝐺)‘𝐷)𝐵) | ||
Theorem | hpgerlem 26081* | Lemma for the proof that the half-plane relation is an equivalence relation. Lemma 9.10 of [Schwabhauser] p. 72. (Contributed by Thierry Arnoux, 4-Mar-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷) ⇒ ⊢ (𝜑 → ∃𝑐 ∈ 𝑃 𝐴𝑂𝑐) | ||
Theorem | hpgid 26082* | The half-plane relation is reflexive. Theorem 9.11 of [Schwabhauser] p. 72. (Contributed by Thierry Arnoux, 4-Mar-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷) ⇒ ⊢ (𝜑 → 𝐴((hpG‘𝐺)‘𝐷)𝐴) | ||
Theorem | hpgcom 26083* | The half-plane relation commutes. Theorem 9.12 of [Schwabhauser] p. 72. (Contributed by Thierry Arnoux, 4-Mar-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐴((hpG‘𝐺)‘𝐷)𝐵) ⇒ ⊢ (𝜑 → 𝐵((hpG‘𝐺)‘𝐷)𝐴) | ||
Theorem | hpgtr 26084* | The half-plane relation is transitive. Theorem 9.13 of [Schwabhauser] p. 72. (Contributed by Thierry Arnoux, 4-Mar-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐴((hpG‘𝐺)‘𝐷)𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐵((hpG‘𝐺)‘𝐷)𝐶) ⇒ ⊢ (𝜑 → 𝐴((hpG‘𝐺)‘𝐷)𝐶) | ||
Theorem | colopp 26085* | Opposite sides of a line for colinear points. Theorem 9.18 of [Schwabhauser] p. 73. (Contributed by Thierry Arnoux, 3-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝐷) & ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)) ⇒ ⊢ (𝜑 → (𝐴𝑂𝐵 ↔ (𝐶 ∈ (𝐴𝐼𝐵) ∧ ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷))) | ||
Theorem | colhp 26086* | Half-plane relation for colinear points. Theorem 9.19 of [Schwabhauser] p. 73. (Contributed by Thierry Arnoux, 3-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝐷) & ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)) & ⊢ 𝐾 = (hlG‘𝐺) ⇒ ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ (𝐴(𝐾‘𝐶)𝐵 ∧ ¬ 𝐴 ∈ 𝐷))) | ||
Theorem | hphl 26087* | If two points are on the same half-line with endpoint on a line, they are on the same half-plane defined by this line. (Contributed by Thierry Arnoux, 9-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐷) & ⊢ (𝜑 → 𝐵(𝐾‘𝐴)𝐶) ⇒ ⊢ (𝜑 → 𝐵((hpG‘𝐺)‘𝐷)𝐶) | ||
Syntax | cmid 26088 | Declare the constant for the midpoint operation. |
class midG | ||
Syntax | clmi 26089 | Declare the constant for the line mirroring function. |
class lInvG | ||
Definition | df-mid 26090* | Define the midpoint operation. Definition 10.1 of [Schwabhauser] p. 88. See ismidb 26094, midbtwn 26095, and midcgr 26096. (Contributed by Thierry Arnoux, 9-Jun-2019.) |
⊢ midG = (𝑔 ∈ V ↦ (𝑎 ∈ (Base‘𝑔), 𝑏 ∈ (Base‘𝑔) ↦ (℩𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎)))) | ||
Definition | df-lmi 26091* | Define the line mirroring function. Definition 10.3 of [Schwabhauser] p. 89. See islmib 26103. (Contributed by Thierry Arnoux, 1-Dec-2019.) |
⊢ lInvG = (𝑔 ∈ V ↦ (𝑚 ∈ ran (LineG‘𝑔) ↦ (𝑎 ∈ (Base‘𝑔) ↦ (℩𝑏 ∈ (Base‘𝑔)((𝑎(midG‘𝑔)𝑏) ∈ 𝑚 ∧ (𝑚(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ∨ 𝑎 = 𝑏)))))) | ||
Theorem | midf 26092 | Midpoint as a function. (Contributed by Thierry Arnoux, 1-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) ⇒ ⊢ (𝜑 → (midG‘𝐺):(𝑃 × 𝑃)⟶𝑃) | ||
Theorem | midcl 26093 | Closure of the midpoint. (Contributed by Thierry Arnoux, 1-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) ∈ 𝑃) | ||
Theorem | ismidb 26094 | Property of the midpoint. (Contributed by Thierry Arnoux, 1-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝑀 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝐵 = ((𝑆‘𝑀)‘𝐴) ↔ (𝐴(midG‘𝐺)𝐵) = 𝑀)) | ||
Theorem | midbtwn 26095 | Betweenness of midpoint. (Contributed by Thierry Arnoux, 7-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) ∈ (𝐴𝐼𝐵)) | ||
Theorem | midcgr 26096 | Congruence of midpoint. (Contributed by Thierry Arnoux, 7-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) = 𝐶) ⇒ ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐶 − 𝐵)) | ||
Theorem | midid 26097 | Midpoint of a null segment. (Contributed by Thierry Arnoux, 7-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐴) = 𝐴) | ||
Theorem | midcom 26098 | Commutativity rule for the midpoint. (Contributed by Thierry Arnoux, 2-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) = (𝐵(midG‘𝐺)𝐴)) | ||
Theorem | mirmid 26099 | Point inversion preserves midpoints. (Contributed by Thierry Arnoux, 12-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ 𝑆 = ((pInvG‘𝐺)‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ 𝑃) ⇒ ⊢ (𝜑 → ((𝑆‘𝐴)(midG‘𝐺)(𝑆‘𝐵)) = (𝑆‘(𝐴(midG‘𝐺)𝐵))) | ||
Theorem | lmieu 26100* | Uniqueness of the line mirror point. Theorem 10.2 of [Schwabhauser] p. 88. (Contributed by Thierry Arnoux, 1-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) ⇒ ⊢ (𝜑 → ∃!𝑏 ∈ 𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏))) |
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