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Type | Label | Description |
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Statement | ||
Theorem | hlln 26401 | The half-line relation implies colinearity, part of Theorem 6.4 of [Schwabhauser] p. 44. (Contributed by Thierry Arnoux, 22-Feb-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐴(𝐾‘𝐶)𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐿𝐶)) | ||
Theorem | hleqnid 26402 | The endpoint does not belong to the half-line. (Contributed by Thierry Arnoux, 3-Mar-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) ⇒ ⊢ (𝜑 → ¬ 𝐴(𝐾‘𝐴)𝐵) | ||
Theorem | hlid 26403 | The half-line relation is reflexive. Theorem 6.5 of [Schwabhauser] p. 44. (Contributed by Thierry Arnoux, 21-Feb-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ≠ 𝐶) ⇒ ⊢ (𝜑 → 𝐴(𝐾‘𝐶)𝐴) | ||
Theorem | hltr 26404 | The half-line relation is transitive. Theorem 6.7 of [Schwabhauser] p. 44. (Contributed by Thierry Arnoux, 23-Feb-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐴(𝐾‘𝐷)𝐵) & ⊢ (𝜑 → 𝐵(𝐾‘𝐷)𝐶) ⇒ ⊢ (𝜑 → 𝐴(𝐾‘𝐷)𝐶) | ||
Theorem | hlbtwn 26405 | Betweenness is a sufficient condition to swap half-lines. (Contributed by Thierry Arnoux, 21-Feb-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ (𝐶𝐼𝐵)) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) & ⊢ (𝜑 → 𝐷 ≠ 𝐶) ⇒ ⊢ (𝜑 → (𝐴(𝐾‘𝐶)𝐵 ↔ 𝐴(𝐾‘𝐶)𝐷)) | ||
Theorem | btwnhl1 26406 | Deduce half-line from betweenness. (Contributed by Thierry Arnoux, 4-Mar-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐵)) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐶 ≠ 𝐴) ⇒ ⊢ (𝜑 → 𝐶(𝐾‘𝐴)𝐵) | ||
Theorem | btwnhl2 26407 | Deduce half-line from betweenness. (Contributed by Thierry Arnoux, 4-Mar-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐵)) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐶 ≠ 𝐵) ⇒ ⊢ (𝜑 → 𝐶(𝐾‘𝐵)𝐴) | ||
Theorem | btwnhl 26408 | Swap betweenness for a half-line. (Contributed by Thierry Arnoux, 2-Mar-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐴(𝐾‘𝐷)𝐵) & ⊢ (𝜑 → 𝐷 ∈ (𝐴𝐼𝐶)) ⇒ ⊢ (𝜑 → 𝐷 ∈ (𝐵𝐼𝐶)) | ||
Theorem | lnhl 26409 | Either a point 𝐶 on the line AB is on the same side as 𝐴 or on the opposite side. (Contributed by Thierry Arnoux, 21-Sep-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐿𝐵)) ⇒ ⊢ (𝜑 → (𝐶(𝐾‘𝐵)𝐴 ∨ 𝐵 ∈ (𝐴𝐼𝐶))) | ||
Theorem | hlcgrex 26410* | Construct a point on a half-line, at a given distance of its origin. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝐷 ≠ 𝐴) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥(𝐾‘𝐴)𝐷 ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶))) | ||
Theorem | hlcgreulem 26411 | Lemma for hlcgreu 26412. (Contributed by Thierry Arnoux, 9-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝐷 ≠ 𝐴) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑋(𝐾‘𝐴)𝐷) & ⊢ (𝜑 → 𝑌(𝐾‘𝐴)𝐷) & ⊢ (𝜑 → (𝐴 − 𝑋) = (𝐵 − 𝐶)) & ⊢ (𝜑 → (𝐴 − 𝑌) = (𝐵 − 𝐶)) ⇒ ⊢ (𝜑 → 𝑋 = 𝑌) | ||
Theorem | hlcgreu 26412* | The point constructed in hlcgrex 26410 is unique. Theorem 6.11 of [Schwabhauser] p. 44. (Contributed by Thierry Arnoux, 9-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝐷 ≠ 𝐴) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) ⇒ ⊢ (𝜑 → ∃!𝑥 ∈ 𝑃 (𝑥(𝐾‘𝐴)𝐷 ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶))) | ||
Theorem | btwnlng1 26413 | Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) & ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐼𝑌)) ⇒ ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌)) | ||
Theorem | btwnlng2 26414 | Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) & ⊢ (𝜑 → 𝑋 ∈ (𝑍𝐼𝑌)) ⇒ ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌)) | ||
Theorem | btwnlng3 26415 | Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) & ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼𝑍)) ⇒ ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌)) | ||
Theorem | lncom 26416 | Swapping the points defining a line keeps it unchanged. Part of Theorem 4.11 of [Schwabhauser] p. 34. (Contributed by Thierry Arnoux, 3-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) & ⊢ (𝜑 → 𝑍 ∈ (𝑌𝐿𝑋)) ⇒ ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌)) | ||
Theorem | lnrot1 26417 | Rotating the points defining a line. Part of Theorem 4.11 of [Schwabhauser] p. 34. (Contributed by Thierry Arnoux, 3-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) & ⊢ (𝜑 → 𝑌 ∈ (𝑍𝐿𝑋)) & ⊢ (𝜑 → 𝑍 ≠ 𝑋) ⇒ ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌)) | ||
Theorem | lnrot2 26418 | Rotating the points defining a line. Part of Theorem 4.11 of [Schwabhauser] p. 34. (Contributed by Thierry Arnoux, 3-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) & ⊢ (𝜑 → 𝑋 ∈ (𝑌𝐿𝑍)) & ⊢ (𝜑 → 𝑌 ≠ 𝑍) ⇒ ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌)) | ||
Theorem | ncolne1 26419 | Non-colinear points are different. (Contributed by Thierry Arnoux, 8-Aug-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ⇒ ⊢ (𝜑 → 𝑋 ≠ 𝑌) | ||
Theorem | ncolne2 26420 | Non-colinear points are different. (Contributed by Thierry Arnoux, 8-Aug-2019.) TODO (NM): maybe ncolne2 26420 could be simplified out and deleted, replaced by ncolcom 26355. |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ⇒ ⊢ (𝜑 → 𝑋 ≠ 𝑍) | ||
Theorem | tgisline 26421* | The property of being a proper line, generated by two distinct points. (Contributed by Thierry Arnoux, 25-May-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) | ||
Theorem | tglnne 26422 | It takes two different points to form a line. (Contributed by Thierry Arnoux, 27-Nov-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿) ⇒ ⊢ (𝜑 → 𝑋 ≠ 𝑌) | ||
Theorem | tglndim0 26423 | There are no lines in dimension 0. (Contributed by Thierry Arnoux, 18-Oct-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → (♯‘𝐵) = 1) ⇒ ⊢ (𝜑 → ¬ 𝐴 ∈ ran 𝐿) | ||
Theorem | tgelrnln 26424 | The property of being a proper line, generated by two distinct points. (Contributed by Thierry Arnoux, 25-May-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) ⇒ ⊢ (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿) | ||
Theorem | tglineeltr 26425 | Transitivity law for lines, one half of tglineelsb2 26426. (Contributed by Thierry Arnoux, 25-May-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → 𝑄 ∈ 𝐵) & ⊢ (𝜑 → 𝑃 ≠ 𝑄) & ⊢ (𝜑 → 𝑆 ∈ 𝐵) & ⊢ (𝜑 → 𝑆 ≠ 𝑃) & ⊢ (𝜑 → 𝑆 ∈ (𝑃𝐿𝑄)) & ⊢ (𝜑 → 𝑅 ∈ 𝐵) & ⊢ (𝜑 → 𝑅 ∈ (𝑃𝐿𝑆)) ⇒ ⊢ (𝜑 → 𝑅 ∈ (𝑃𝐿𝑄)) | ||
Theorem | tglineelsb2 26426 | If 𝑆 lies on PQ , then PQ = PS . Theorem 6.16 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 17-May-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → 𝑄 ∈ 𝐵) & ⊢ (𝜑 → 𝑃 ≠ 𝑄) & ⊢ (𝜑 → 𝑆 ∈ 𝐵) & ⊢ (𝜑 → 𝑆 ≠ 𝑃) & ⊢ (𝜑 → 𝑆 ∈ (𝑃𝐿𝑄)) ⇒ ⊢ (𝜑 → (𝑃𝐿𝑄) = (𝑃𝐿𝑆)) | ||
Theorem | tglinerflx1 26427 | Reflexivity law for line membership. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 17-May-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → 𝑄 ∈ 𝐵) & ⊢ (𝜑 → 𝑃 ≠ 𝑄) ⇒ ⊢ (𝜑 → 𝑃 ∈ (𝑃𝐿𝑄)) | ||
Theorem | tglinerflx2 26428 | Reflexivity law for line membership. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 17-May-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → 𝑄 ∈ 𝐵) & ⊢ (𝜑 → 𝑃 ≠ 𝑄) ⇒ ⊢ (𝜑 → 𝑄 ∈ (𝑃𝐿𝑄)) | ||
Theorem | tglinecom 26429 | Commutativity law for lines. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 17-May-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → 𝑄 ∈ 𝐵) & ⊢ (𝜑 → 𝑃 ≠ 𝑄) ⇒ ⊢ (𝜑 → (𝑃𝐿𝑄) = (𝑄𝐿𝑃)) | ||
Theorem | tglinethru 26430 | If 𝐴 is a line containing two distinct points 𝑃 and 𝑄, then 𝐴 is the line through 𝑃 and 𝑄. Theorem 6.18 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 25-May-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → 𝑄 ∈ 𝐵) & ⊢ (𝜑 → 𝑃 ≠ 𝑄) & ⊢ (𝜑 → 𝑃 ≠ 𝑄) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑃 ∈ 𝐴) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝐴 = (𝑃𝐿𝑄)) | ||
Theorem | tghilberti1 26431* | There is a line through any two distinct points. Hilbert's axiom I.1 for geometry. (Contributed by Thierry Arnoux, 25-May-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → 𝑄 ∈ 𝐵) & ⊢ (𝜑 → 𝑃 ≠ 𝑄) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ran 𝐿(𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) | ||
Theorem | tghilberti2 26432* | There is at most one line through any two distinct points. Hilbert's axiom I.2 for geometry. (Contributed by Thierry Arnoux, 25-May-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → 𝑄 ∈ 𝐵) & ⊢ (𝜑 → 𝑃 ≠ 𝑄) ⇒ ⊢ (𝜑 → ∃*𝑥 ∈ ran 𝐿(𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) | ||
Theorem | tglinethrueu 26433* | There is a unique line going through any two distinct points. Theorem 6.19 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 25-May-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → 𝑄 ∈ 𝐵) & ⊢ (𝜑 → 𝑃 ≠ 𝑄) ⇒ ⊢ (𝜑 → ∃!𝑥 ∈ ran 𝐿(𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) | ||
Theorem | tglnne0 26434 | A line 𝐴 has at least one point. (Contributed by Thierry Arnoux, 4-Mar-2020.) |
⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) ⇒ ⊢ (𝜑 → 𝐴 ≠ ∅) | ||
Theorem | tglnpt2 26435* | Find a second point on a line. (Contributed by Thierry Arnoux, 18-Oct-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝑋 ≠ 𝑦) | ||
Theorem | tglineintmo 26436* | Two distinct lines intersect in at most one point. Theorem 6.21 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 25-May-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐵 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | ||
Theorem | tglineineq 26437 | Two distinct lines intersect in at most one point, variation. Theorem 6.21 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 6-Aug-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐵 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) & ⊢ (𝜑 → 𝑌 ∈ (𝐴 ∩ 𝐵)) ⇒ ⊢ (𝜑 → 𝑋 = 𝑌) | ||
Theorem | tglineneq 26438 | Given three non-colinear points, build two different lines. (Contributed by Thierry Arnoux, 6-Aug-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶)) ⇒ ⊢ (𝜑 → (𝐴𝐿𝐵) ≠ (𝐶𝐿𝐷)) | ||
Theorem | tglineinteq 26439 | Two distinct lines intersect in at most one point. Theorem 6.21 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 6-Aug-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶)) & ⊢ (𝜑 → 𝑋 ∈ (𝐴𝐿𝐵)) & ⊢ (𝜑 → 𝑌 ∈ (𝐴𝐿𝐵)) & ⊢ (𝜑 → 𝑋 ∈ (𝐶𝐿𝐷)) & ⊢ (𝜑 → 𝑌 ∈ (𝐶𝐿𝐷)) ⇒ ⊢ (𝜑 → 𝑋 = 𝑌) | ||
Theorem | ncolncol 26440 | Deduce non-colinearity from non-colinearity and colinearity. (Contributed by Thierry Arnoux, 27-Aug-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶)) & ⊢ (𝜑 → 𝐷 ∈ (𝐴𝐿𝐵)) & ⊢ (𝜑 → 𝐷 ≠ 𝐵) ⇒ ⊢ (𝜑 → ¬ (𝐷 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶)) | ||
Theorem | coltr 26441 | A transitivity law for colinearity. (Contributed by Thierry Arnoux, 27-Nov-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐿𝐶)) & ⊢ (𝜑 → (𝐵 ∈ (𝐶𝐿𝐷) ∨ 𝐶 = 𝐷)) ⇒ ⊢ (𝜑 → (𝐴 ∈ (𝐶𝐿𝐷) ∨ 𝐶 = 𝐷)) | ||
Theorem | coltr3 26442 | A transitivity law for colinearity. (Contributed by Thierry Arnoux, 27-Nov-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐿𝐶)) & ⊢ (𝜑 → 𝐷 ∈ (𝐴𝐼𝐶)) ⇒ ⊢ (𝜑 → 𝐷 ∈ (𝐵𝐿𝐶)) | ||
Theorem | colline 26443* | Three points are colinear iff there is a line through all three of them. Theorem 6.23 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 28-May-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 2 ≤ (♯‘𝑃)) ⇒ ⊢ (𝜑 → ((𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍) ↔ ∃𝑎 ∈ ran 𝐿(𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎))) | ||
Theorem | tglowdim2l 26444* | Reformulation of the lower dimension axiom for dimension two. There exist three non-colinear points. Theorem 6.24 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 30-May-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) ⇒ ⊢ (𝜑 → ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ¬ (𝑐 ∈ (𝑎𝐿𝑏) ∨ 𝑎 = 𝑏)) | ||
Theorem | tglowdim2ln 26445* | There is always one point outside of any line. Theorem 6.25 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 16-Nov-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → ∃𝑐 ∈ 𝑃 ¬ 𝑐 ∈ (𝐴𝐿𝐵)) | ||
Syntax | cmir 26446 | Declare the constant for the point inversion function. |
class pInvG | ||
Definition | df-mir 26447* | Define the point inversion ("mirror") function. Definition 7.5 of [Schwabhauser] p. 49. See mirval 26449 and ismir 26453. (Contributed by Thierry Arnoux, 30-May-2019.) |
⊢ pInvG = (𝑔 ∈ V ↦ (𝑚 ∈ (Base‘𝑔) ↦ (𝑎 ∈ (Base‘𝑔) ↦ (℩𝑏 ∈ (Base‘𝑔)((𝑚(dist‘𝑔)𝑏) = (𝑚(dist‘𝑔)𝑎) ∧ 𝑚 ∈ (𝑏(Itv‘𝑔)𝑎)))))) | ||
Theorem | mirreu3 26448* | Existential uniqueness of the mirror point. Theorem 7.8 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 30-May-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝑀 ∈ 𝑃) ⇒ ⊢ (𝜑 → ∃!𝑏 ∈ 𝑃 ((𝑀 − 𝑏) = (𝑀 − 𝐴) ∧ 𝑀 ∈ (𝑏𝐼𝐴))) | ||
Theorem | mirval 26449* | Value of the point inversion function 𝑆. Definition 7.5 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 30-May-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝑆‘𝐴) = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))))) | ||
Theorem | mirfv 26450* | Value of the point inversion function 𝑀. Definition 7.5 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 30-May-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ 𝑀 = (𝑆‘𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝑀‘𝐵) = (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)))) | ||
Theorem | mircgr 26451 | Property of the image by the point inversion function. Definition 7.5 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 3-Jun-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ 𝑀 = (𝑆‘𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝐴 − (𝑀‘𝐵)) = (𝐴 − 𝐵)) | ||
Theorem | mirbtwn 26452 | Property of the image by the point inversion function. Definition 7.5 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 3-Jun-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ 𝑀 = (𝑆‘𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) ⇒ ⊢ (𝜑 → 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵)) | ||
Theorem | ismir 26453 | Property of the image by the point inversion function. Definition 7.5 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 3-Jun-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ 𝑀 = (𝑆‘𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐴 − 𝐵)) & ⊢ (𝜑 → 𝐴 ∈ (𝐶𝐼𝐵)) ⇒ ⊢ (𝜑 → 𝐶 = (𝑀‘𝐵)) | ||
Theorem | mirf 26454 | Point inversion as function. (Contributed by Thierry Arnoux, 30-May-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ 𝑀 = (𝑆‘𝐴) ⇒ ⊢ (𝜑 → 𝑀:𝑃⟶𝑃) | ||
Theorem | mircl 26455 | Closure of the point inversion function. (Contributed by Thierry Arnoux, 20-Oct-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ 𝑀 = (𝑆‘𝐴) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝑀‘𝑋) ∈ 𝑃) | ||
Theorem | mirmir 26456 | The point inversion function is an involution. Theorem 7.7 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 3-Jun-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ 𝑀 = (𝑆‘𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝑀‘(𝑀‘𝐵)) = 𝐵) | ||
Theorem | mircom 26457 | Variation on mirmir 26456. (Contributed by Thierry Arnoux, 10-Nov-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ 𝑀 = (𝑆‘𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → (𝑀‘𝐵) = 𝐶) ⇒ ⊢ (𝜑 → (𝑀‘𝐶) = 𝐵) | ||
Theorem | mirreu 26458* | Any point has a unique antecedent through point inversion. Theorem 7.8 of [Schwabhauser] p. 50. (Contributed by Thierry Arnoux, 3-Jun-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ 𝑀 = (𝑆‘𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) ⇒ ⊢ (𝜑 → ∃!𝑎 ∈ 𝑃 (𝑀‘𝑎) = 𝐵) | ||
Theorem | mireq 26459 | Equality deduction for point inversion. Theorem 7.9 of [Schwabhauser] p. 50. (Contributed by Thierry Arnoux, 30-May-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ 𝑀 = (𝑆‘𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → (𝑀‘𝐵) = (𝑀‘𝐶)) ⇒ ⊢ (𝜑 → 𝐵 = 𝐶) | ||
Theorem | mirinv 26460 | The only invariant point of a point inversion Theorem 7.3 of [Schwabhauser] p. 49, Theorem 7.10 of [Schwabhauser] p. 50. (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) & ⊢ 𝑀 = (𝑆‘𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) ⇒ ⊢ (𝜑 → ((𝑀‘𝐵) = 𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | mirne 26461 | Mirror of non-center point cannot be the center point. (Contributed by Thierry Arnoux, 27-Sep-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ 𝑀 = (𝑆‘𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ≠ 𝐴) ⇒ ⊢ (𝜑 → (𝑀‘𝐵) ≠ 𝐴) | ||
Theorem | mircinv 26462 | The center point is invariant of a point inversion. (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) & ⊢ 𝑀 = (𝑆‘𝐴) ⇒ ⊢ (𝜑 → (𝑀‘𝐴) = 𝐴) | ||
Theorem | mirf1o 26463 | The point inversion function 𝑀 is a bijection. Theorem 7.11 of [Schwabhauser] p. 50. (Contributed by Thierry Arnoux, 6-Jun-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ 𝑀 = (𝑆‘𝐴) ⇒ ⊢ (𝜑 → 𝑀:𝑃–1-1-onto→𝑃) | ||
Theorem | miriso 26464 | The point inversion function is an isometry, i.e. it is conserves congruence. Because it is also a bijection, it is also a motion. Theorem 7.13 of [Schwabhauser] p. 50. (Contributed by Thierry Arnoux, 6-Jun-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ 𝑀 = (𝑆‘𝐴) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) ⇒ ⊢ (𝜑 → ((𝑀‘𝑋) − (𝑀‘𝑌)) = (𝑋 − 𝑌)) | ||
Theorem | mirbtwni 26465 | Point inversion preserves betweenness, first half of Theorem 7.15 of [Schwabhauser] p. 51. (Contributed by Thierry Arnoux, 9-Jun-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ 𝑀 = (𝑆‘𝐴) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼𝑍)) ⇒ ⊢ (𝜑 → (𝑀‘𝑌) ∈ ((𝑀‘𝑋)𝐼(𝑀‘𝑍))) | ||
Theorem | mirbtwnb 26466 | Point inversion preserves betweenness. Theorem 7.15 of [Schwabhauser] p. 51. (Contributed by Thierry Arnoux, 9-Jun-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ 𝑀 = (𝑆‘𝐴) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐼𝑍) ↔ (𝑀‘𝑌) ∈ ((𝑀‘𝑋)𝐼(𝑀‘𝑍)))) | ||
Theorem | mircgrs 26467 | Point inversion preserves congruence. Theorem 7.16 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) & ⊢ 𝑀 = (𝑆‘𝐴) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑇 ∈ 𝑃) & ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑍 − 𝑇)) ⇒ ⊢ (𝜑 → ((𝑀‘𝑋) − (𝑀‘𝑌)) = ((𝑀‘𝑍) − (𝑀‘𝑇))) | ||
Theorem | mirmir2 26468 | Point inversion of a point inversion through another point. (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) & ⊢ 𝑀 = (𝑆‘𝐴) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝑀‘((𝑆‘𝑌)‘𝑋)) = ((𝑆‘(𝑀‘𝑌))‘(𝑀‘𝑋))) | ||
Theorem | mirmot 26469 | Point investion is a motion of the geometric space. Theorem 7.14 of [Schwabhauser] p. 51. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝑀 = (𝑆‘𝐴) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) ⇒ ⊢ (𝜑 → 𝑀 ∈ (𝐺Ismt𝐺)) | ||
Theorem | mirln 26470 | If two points are on the same line, so is the mirror point of one through the other. (Contributed by Thierry Arnoux, 21-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝑀 = (𝑆‘𝐴) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ (𝜑 → 𝐵 ∈ 𝐷) ⇒ ⊢ (𝜑 → (𝑀‘𝐵) ∈ 𝐷) | ||
Theorem | mirln2 26471 | If a point and its mirror point are both on the same line, so is the center of the point inversion. (Contributed by Thierry Arnoux, 3-Mar-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝑀 = (𝑆‘𝐴) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘𝐵) ∈ 𝐷) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐷) | ||
Theorem | mirconn 26472 | Point inversion of connectedness. (Contributed by Thierry Arnoux, 2-Mar-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝑀 = (𝑆‘𝐴) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → (𝑋 ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼𝑋))) ⇒ ⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑌))) | ||
Theorem | mirhl 26473 | If two points 𝑋 and 𝑌 are on the same half-line from 𝑍, the same applies to the mirror points. (Contributed by Thierry Arnoux, 21-Feb-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝑀 = (𝑆‘𝐴) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑋(𝐾‘𝑍)𝑌) ⇒ ⊢ (𝜑 → (𝑀‘𝑋)(𝐾‘(𝑀‘𝑍))(𝑀‘𝑌)) | ||
Theorem | mirbtwnhl 26474 | If the center of the point inversion 𝐴 is between two points 𝑋 and 𝑌, then the half lines are mirrored. (Contributed by Thierry Arnoux, 3-Mar-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝑀 = (𝑆‘𝐴) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ≠ 𝐴) & ⊢ (𝜑 → 𝑌 ≠ 𝐴) & ⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼𝑌)) ⇒ ⊢ (𝜑 → (𝑍(𝐾‘𝐴)𝑋 ↔ (𝑀‘𝑍)(𝐾‘𝐴)𝑌)) | ||
Theorem | mirhl2 26475 | Deduce half-line relation from mirror point. (Contributed by Thierry Arnoux, 8-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝑀 = (𝑆‘𝐴) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ≠ 𝐴) & ⊢ (𝜑 → 𝑌 ≠ 𝐴) & ⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑌))) ⇒ ⊢ (𝜑 → 𝑋(𝐾‘𝐴)𝑌) | ||
Theorem | mircgrextend 26476 | Link congruence over a pair of mirror points. cf tgcgrextend 26279. (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 26477 | 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 26478 | 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 26479 | 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 26480 | 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 26481 | 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 26482 | 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 26483 | 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 26484 | Lemma for krippen 26485. 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 26485 | 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 26486* | 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 26532 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 26487 | Declare the constant for the class of right angles. |
class ∟G | ||
Definition | df-rag 26488* | Define the class of right angles. Definition 8.1 of [Schwabhauser] p. 57. See israg 26491. (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 26489 | Declare the constant for the perpendicular relation. |
class ⟂G | ||
Definition | df-perpg 26490* | Define the "perpendicular" relation. Definition 8.11 of [Schwabhauser] p. 59. See isperp 26506. (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 26491 | 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 26492 | 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 26493 | 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 26494 | 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 26495 | 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 26496 | 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 26497 | 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 26498 | 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 26499 | 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 26500 | 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)) |
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