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Type | Label | Description |
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Statement | ||
Theorem | tgcgrcoml 28501 | Congruence commutes on the LHS. Variant of Theorem 2.5 of [Schwabhauser] p. 27, but in a convenient form for a common case. (Contributed by David A. Wheeler, 29-Jun-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷)) ⇒ ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐶 − 𝐷)) | ||
Theorem | tgcgrcomlr 28502 | Congruence commutes on both sides. (Contributed by Thierry Arnoux, 23-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷)) ⇒ ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐷 − 𝐶)) | ||
Theorem | tgcgreqb 28503 | Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷)) ⇒ ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷)) | ||
Theorem | tgcgreq 28504 | Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷)) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 𝐶 = 𝐷) | ||
Theorem | tgcgrneq 28505 | Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷)) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → 𝐶 ≠ 𝐷) | ||
Theorem | tgcgrtriv 28506 | Degenerate segments are congruent. Theorem 2.8 of [Schwabhauser] p. 28. (Contributed by Thierry Arnoux, 23-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝐴 − 𝐴) = (𝐵 − 𝐵)) | ||
Theorem | tgcgrextend 28507 | Link congruence over a pair of line segments. Theorem 2.11 of [Schwabhauser] p. 29. (Contributed by Thierry Arnoux, 23-Mar-2019.) (Shortened by David A. Wheeler and Thierry Arnoux, 22-Apr-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) & ⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝐹)) & ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸)) & ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹)) ⇒ ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹)) | ||
Theorem | tgsegconeq 28508 | Two points that satisfy the conclusion of axtgsegcon 28486 are identical. Uniqueness portion of Theorem 2.12 of [Schwabhauser] p. 29. (Contributed by Thierry Arnoux, 23-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ≠ 𝐴) & ⊢ (𝜑 → 𝐴 ∈ (𝐷𝐼𝐸)) & ⊢ (𝜑 → 𝐴 ∈ (𝐷𝐼𝐹)) & ⊢ (𝜑 → (𝐴 − 𝐸) = (𝐵 − 𝐶)) & ⊢ (𝜑 → (𝐴 − 𝐹) = (𝐵 − 𝐶)) ⇒ ⊢ (𝜑 → 𝐸 = 𝐹) | ||
Theorem | tgbtwntriv2 28509 | Betweenness always holds for the second endpoint. Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) ⇒ ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐵)) | ||
Theorem | tgbtwncom 28510 | Betweenness commutes. Theorem 3.2 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) ⇒ ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴)) | ||
Theorem | tgbtwncomb 28511 | Betweenness commutes, biconditional version. (Contributed by Thierry Arnoux, 3-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝐵 ∈ (𝐴𝐼𝐶) ↔ 𝐵 ∈ (𝐶𝐼𝐴))) | ||
Theorem | tgbtwnne 28512 | Betweenness and inequality. (Contributed by Thierry Arnoux, 1-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) & ⊢ (𝜑 → 𝐵 ≠ 𝐴) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐶) | ||
Theorem | tgbtwntriv1 28513 | Betweenness always holds for the first endpoint. Theorem 3.3 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) ⇒ ⊢ (𝜑 → 𝐴 ∈ (𝐴𝐼𝐵)) | ||
Theorem | tgbtwnswapid 28514 | If you can swap the first two arguments of a betweenness statement, then those arguments are identical. Theorem 3.4 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 16-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼𝐶)) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | tgbtwnintr 28515 | Inner transitivity law for betweenness. Left-hand side of Theorem 3.5 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 18-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼𝐷)) & ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐷)) ⇒ ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) | ||
Theorem | tgbtwnexch3 28516 | Exchange the first endpoint in betweenness. Left-hand side of Theorem 3.6 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 18-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) & ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷)) ⇒ ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷)) | ||
Theorem | tgbtwnouttr2 28517 | Outer transitivity law for betweenness. Left-hand side of Theorem 3.7 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 18-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) & ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷)) ⇒ ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷)) | ||
Theorem | tgbtwnexch2 28518 | Exchange the outer point of two betweenness statements. Right-hand side of Theorem 3.5 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 23-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) & ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷)) ⇒ ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷)) | ||
Theorem | tgbtwnouttr 28519 | Outer transitivity law for betweenness. Right-hand side of Theorem 3.7 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 23-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) & ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷)) ⇒ ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) | ||
Theorem | tgbtwnexch 28520 | Outer transitivity law for betweenness. Right-hand side of Theorem 3.6 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 23-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) & ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷)) ⇒ ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) | ||
Theorem | tgtrisegint 28521* | A line segment between two sides of a triange intersects a segment crossing from the remaining side to the opposite vertex. Theorem 3.17 of [Schwabhauser] p. 33. (Contributed by Thierry Arnoux, 23-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) & ⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝐶)) & ⊢ (𝜑 → 𝐹 ∈ (𝐴𝐼𝐷)) ⇒ ⊢ (𝜑 → ∃𝑞 ∈ 𝑃 (𝑞 ∈ (𝐹𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸))) | ||
Theorem | tglowdim1 28522* | Lower dimension axiom for one dimension. In dimension at least 1, there are at least two distinct points. The condition "the space is of dimension 1 or more" is written here as 2 ≤ (♯‘𝑃) to avoid a new definition, but a different convention could be chosen. (Contributed by Thierry Arnoux, 23-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 2 ≤ (♯‘𝑃)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 𝑥 ≠ 𝑦) | ||
Theorem | tglowdim1i 28523* | Lower dimension axiom for one dimension. (Contributed by Thierry Arnoux, 28-May-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 2 ≤ (♯‘𝑃)) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 𝑋 ≠ 𝑦) | ||
Theorem | tgldimor 28524 | Excluded-middle like statement allowing to treat dimension zero as a special case. (Contributed by Thierry Arnoux, 11-Apr-2019.) |
⊢ 𝑃 = (𝐸‘𝐹) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) ⇒ ⊢ (𝜑 → ((♯‘𝑃) = 1 ∨ 2 ≤ (♯‘𝑃))) | ||
Theorem | tgldim0eq 28525 | In dimension zero, any two points are equal. (Contributed by Thierry Arnoux, 11-Apr-2019.) |
⊢ 𝑃 = (𝐸‘𝐹) & ⊢ (𝜑 → (♯‘𝑃) = 1) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | tgldim0itv 28526 | In dimension zero, any two points are equal. (Contributed by Thierry Arnoux, 12-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → (♯‘𝑃) = 1) ⇒ ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼𝐶)) | ||
Theorem | tgldim0cgr 28527 | In dimension zero, any two pairs of points are congruent. (Contributed by Thierry Arnoux, 12-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → (♯‘𝑃) = 1) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷)) | ||
Theorem | tgbtwndiff 28528* | There is always a 𝑐 distinct from 𝐵 such that 𝐵 lies between 𝐴 and 𝑐. Theorem 3.14 of [Schwabhauser] p. 32. The condition "the space is of dimension 1 or more" is written here as 2 ≤ (♯‘𝑃) for simplicity. (Contributed by Thierry Arnoux, 23-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 2 ≤ (♯‘𝑃)) ⇒ ⊢ (𝜑 → ∃𝑐 ∈ 𝑃 (𝐵 ∈ (𝐴𝐼𝑐) ∧ 𝐵 ≠ 𝑐)) | ||
Theorem | tgdim01 28529 | In geometries of dimension less than 2, all points are colinear. (Contributed by Thierry Arnoux, 27-Aug-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝐺DimTarskiG≥2) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))) | ||
Theorem | tgifscgr 28530 | Inner five segment congruence. Take two triangles, 𝐴𝐷𝐶 and 𝐸𝐻𝐾, with 𝐵 between 𝐴 and 𝐶 and 𝐹 between 𝐸 and 𝐾. If the other components of the triangles are congruent, then so are 𝐵𝐷 and 𝐹𝐻. Theorem 4.2 of [Schwabhauser] p. 34. (Contributed by Thierry Arnoux, 24-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 𝐾 ∈ 𝑃) & ⊢ (𝜑 → 𝐻 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) & ⊢ (𝜑 → 𝐹 ∈ (𝐸𝐼𝐾)) & ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐸 − 𝐾)) & ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐹 − 𝐾)) & ⊢ (𝜑 → (𝐴 − 𝐷) = (𝐸 − 𝐻)) & ⊢ (𝜑 → (𝐶 − 𝐷) = (𝐾 − 𝐻)) ⇒ ⊢ (𝜑 → (𝐵 − 𝐷) = (𝐹 − 𝐻)) | ||
Theorem | tgcgrsub 28531 | Removing identical parts from the end of a line segment preserves congruence. Theorem 4.3 of [Schwabhauser] p. 35. (Contributed by Thierry Arnoux, 3-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) & ⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝐹)) & ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹)) & ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹)) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸)) | ||
Syntax | ccgrg 28532 | Declare the constant for the congruence between shapes relation. |
class cgrG | ||
Definition | df-cgrg 28533* |
Define the relation of congruence between shapes. Definition 4.4 of
[Schwabhauser] p. 35. A
"shape" is a finite sequence of points, and a
triangle can be represented as a shape with three points. Two shapes
are congruent if all corresponding segments between all corresponding
points are congruent.
Many systems of geometry define triangle congruence as requiring both segment congruence and angle congruence. Such systems, such as Hilbert's axiomatic system, typically have a primitive notion of angle congruence in addition to segment congruence. Here, angle congruence is instead a derived notion, defined later in df-cgra 28830 and expanded in iscgra 28831. This does not mean our system is weaker; dfcgrg2 28885 proves that these two definitions are equivalent, and using the Tarski definition instead (given in [Schwabhauser] p. 35) is simpler. Once two triangles are proven congruent as defined here, you can use various theorems to prove that corresponding parts of congruent triangles are congruent (CPCTC). For example, see cgr3simp1 28542, cgr3simp2 28543, cgr3simp3 28544, cgrcgra 28843, and permutation laws such as cgr3swap12 28545 and dfcgrg2 28885. Ideally, we would define this for functions of any set, but we will use words (see df-word 14549) in most cases. (Contributed by Thierry Arnoux, 3-Apr-2019.) |
⊢ cgrG = (𝑔 ∈ V ↦ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))))}) | ||
Theorem | iscgrg 28534* | The congruence property for sequences of points. (Contributed by Thierry Arnoux, 3-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ ∼ = (cgrG‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑉 → (𝐴 ∼ 𝐵 ↔ ((𝐴 ∈ (𝑃 ↑pm ℝ) ∧ 𝐵 ∈ (𝑃 ↑pm ℝ)) ∧ (dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))))) | ||
Theorem | iscgrgd 28535* | The property for two sequences 𝐴 and 𝐵 of points to be congruent. (Contributed by Thierry Arnoux, 3-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ ∼ = (cgrG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ⊆ ℝ) & ⊢ (𝜑 → 𝐴:𝐷⟶𝑃) & ⊢ (𝜑 → 𝐵:𝐷⟶𝑃) ⇒ ⊢ (𝜑 → (𝐴 ∼ 𝐵 ↔ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))) | ||
Theorem | iscgrglt 28536* | The property for two sequences 𝐴 and 𝐵 of points to be congruent, where the congruence is only required for indices verifying a less-than relation. (Contributed by Thierry Arnoux, 7-Oct-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ ∼ = (cgrG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐷 ⊆ ℝ) & ⊢ (𝜑 → 𝐴:𝐷⟶𝑃) & ⊢ (𝜑 → 𝐵:𝐷⟶𝑃) ⇒ ⊢ (𝜑 → (𝐴 ∼ 𝐵 ↔ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴(𝑖 < 𝑗 → ((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗))))) | ||
Theorem | trgcgrg 28537 | The property for two triangles to be congruent to each other. (Contributed by Thierry Arnoux, 3-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ ∼ = (cgrG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) ⇒ ⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝐸𝐹”〉 ↔ ((𝐴 − 𝐵) = (𝐷 − 𝐸) ∧ (𝐵 − 𝐶) = (𝐸 − 𝐹) ∧ (𝐶 − 𝐴) = (𝐹 − 𝐷)))) | ||
Theorem | trgcgr 28538 | Triangle congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ ∼ = (cgrG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸)) & ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹)) & ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐹 − 𝐷)) ⇒ ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝐸𝐹”〉) | ||
Theorem | ercgrg 28539 | The shape congruence relation is an equivalence relation. Statement 4.4 of [Schwabhauser] p. 35. (Contributed by Thierry Arnoux, 9-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) ⇒ ⊢ (𝐺 ∈ TarskiG → (cgrG‘𝐺) Er (𝑃 ↑pm ℝ)) | ||
Theorem | tgcgrxfr 28540* | A line segment can be divided at the same place as a congruent line segment is divided. Theorem 4.5 of [Schwabhauser] p. 35. (Contributed by Thierry Arnoux, 9-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ ∼ = (cgrG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) & ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹)) ⇒ ⊢ (𝜑 → ∃𝑒 ∈ 𝑃 (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) | ||
Theorem | cgr3id 28541 | Reflexivity law for three-place congruence. (Contributed by Thierry Arnoux, 28-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ ∼ = (cgrG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) ⇒ ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐴𝐵𝐶”〉) | ||
Theorem | cgr3simp1 28542 | Deduce segment congruence from a triangle congruence. This is a portion of the theorem that corresponding parts of congruent triangles are congruent (CPCTC), focusing on a specific segment. (Contributed by Thierry Arnoux, 27-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ ∼ = (cgrG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝐸𝐹”〉) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸)) | ||
Theorem | cgr3simp2 28543 | Deduce segment congruence from a triangle congruence. This is a portion of CPCTC, focusing on a specific segment. (Contributed by Thierry Arnoux, 27-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ ∼ = (cgrG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝐸𝐹”〉) ⇒ ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹)) | ||
Theorem | cgr3simp3 28544 | Deduce segment congruence from a triangle congruence. This is a portion of CPCTC, focusing on a specific segment. (Contributed by Thierry Arnoux, 27-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ ∼ = (cgrG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝐸𝐹”〉) ⇒ ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐹 − 𝐷)) | ||
Theorem | cgr3swap12 28545 | Permutation law for three-place congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ ∼ = (cgrG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝐸𝐹”〉) ⇒ ⊢ (𝜑 → 〈“𝐵𝐴𝐶”〉 ∼ 〈“𝐸𝐷𝐹”〉) | ||
Theorem | cgr3swap23 28546 | Permutation law for three-place congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ ∼ = (cgrG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝐸𝐹”〉) ⇒ ⊢ (𝜑 → 〈“𝐴𝐶𝐵”〉 ∼ 〈“𝐷𝐹𝐸”〉) | ||
Theorem | cgr3swap13 28547 | Permutation law for three-place congruence. (Contributed by Thierry Arnoux, 3-Oct-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ ∼ = (cgrG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝐸𝐹”〉) ⇒ ⊢ (𝜑 → 〈“𝐶𝐵𝐴”〉 ∼ 〈“𝐹𝐸𝐷”〉) | ||
Theorem | cgr3rotr 28548 | Permutation law for three-place congruence. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ ∼ = (cgrG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝐸𝐹”〉) ⇒ ⊢ (𝜑 → 〈“𝐶𝐴𝐵”〉 ∼ 〈“𝐹𝐷𝐸”〉) | ||
Theorem | cgr3rotl 28549 | Permutation law for three-place congruence. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ ∼ = (cgrG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝐸𝐹”〉) ⇒ ⊢ (𝜑 → 〈“𝐵𝐶𝐴”〉 ∼ 〈“𝐸𝐹𝐷”〉) | ||
Theorem | trgcgrcom 28550 | Commutative law for three-place congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ ∼ = (cgrG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝐸𝐹”〉) ⇒ ⊢ (𝜑 → 〈“𝐷𝐸𝐹”〉 ∼ 〈“𝐴𝐵𝐶”〉) | ||
Theorem | cgr3tr 28551 | Transitivity law for three-place congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ ∼ = (cgrG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝐸𝐹”〉) & ⊢ (𝜑 → 𝐽 ∈ 𝑃) & ⊢ (𝜑 → 𝐾 ∈ 𝑃) & ⊢ (𝜑 → 𝐿 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐷𝐸𝐹”〉 ∼ 〈“𝐽𝐾𝐿”〉) ⇒ ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐽𝐾𝐿”〉) | ||
Theorem | tgbtwnxfr 28552 | A condition for extending betweenness to a new set of points based on congruence with another set of points. Theorem 4.6 of [Schwabhauser] p. 36. (Contributed by Thierry Arnoux, 27-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ ∼ = (cgrG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝐸𝐹”〉) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) ⇒ ⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝐹)) | ||
Theorem | tgcgr4 28553 | Two quadrilaterals to be congruent to each other if one triangle formed by their vertices is, and the additional points are equidistant too. (Contributed by Thierry Arnoux, 8-Oct-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ ∼ = (cgrG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝑊 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) ⇒ ⊢ (𝜑 → (〈“𝐴𝐵𝐶𝐷”〉 ∼ 〈“𝑊𝑋𝑌𝑍”〉 ↔ (〈“𝐴𝐵𝐶”〉 ∼ 〈“𝑊𝑋𝑌”〉 ∧ ((𝐴 − 𝐷) = (𝑊 − 𝑍) ∧ (𝐵 − 𝐷) = (𝑋 − 𝑍) ∧ (𝐶 − 𝐷) = (𝑌 − 𝑍))))) | ||
Syntax | cismt 28554 | Declare the constant for the isometry builder. |
class Ismt | ||
Definition | df-ismt 28555* | Define the set of isometries between two structures. Definition 4.8 of [Schwabhauser] p. 36. See isismt 28556. (Contributed by Thierry Arnoux, 13-Dec-2019.) |
⊢ Ismt = (𝑔 ∈ V, ℎ ∈ V ↦ {𝑓 ∣ (𝑓:(Base‘𝑔)–1-1-onto→(Base‘ℎ) ∧ ∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎(dist‘𝑔)𝑏))}) | ||
Theorem | isismt 28556* | Property of being an isometry. Compare with isismty 37787. (Contributed by Thierry Arnoux, 13-Dec-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑃 = (Base‘𝐻) & ⊢ 𝐷 = (dist‘𝐺) & ⊢ − = (dist‘𝐻) ⇒ ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐻 ∈ 𝑊) → (𝐹 ∈ (𝐺Ismt𝐻) ↔ (𝐹:𝐵–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎𝐷𝑏)))) | ||
Theorem | ismot 28557* | Property of being an isometry mapping to the same space. In geometry, this is also called a motion. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑉 → (𝐹 ∈ (𝐺Ismt𝐺) ↔ (𝐹:𝑃–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎 − 𝑏)))) | ||
Theorem | motcgr 28558 | Property of a motion: distances are preserved. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ (𝐺Ismt𝐺)) ⇒ ⊢ (𝜑 → ((𝐹‘𝐴) − (𝐹‘𝐵)) = (𝐴 − 𝐵)) | ||
Theorem | idmot 28559 | The identity is a motion. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) ⇒ ⊢ (𝜑 → ( I ↾ 𝑃) ∈ (𝐺Ismt𝐺)) | ||
Theorem | motf1o 28560 | Motions are bijections. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ (𝐺Ismt𝐺)) ⇒ ⊢ (𝜑 → 𝐹:𝑃–1-1-onto→𝑃) | ||
Theorem | motcl 28561 | Closure of motions. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ (𝐺Ismt𝐺)) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝑃) | ||
Theorem | motco 28562 | The composition of two motions is a motion. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ (𝐺Ismt𝐺)) & ⊢ (𝜑 → 𝐻 ∈ (𝐺Ismt𝐺)) ⇒ ⊢ (𝜑 → (𝐹 ∘ 𝐻) ∈ (𝐺Ismt𝐺)) | ||
Theorem | cnvmot 28563 | The converse of a motion is a motion. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ (𝐺Ismt𝐺)) ⇒ ⊢ (𝜑 → ◡𝐹 ∈ (𝐺Ismt𝐺)) | ||
Theorem | motplusg 28564* | The operation for motions is their composition. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ 𝐼 = {〈(Base‘ndx), (𝐺Ismt𝐺)〉, 〈(+g‘ndx), (𝑓 ∈ (𝐺Ismt𝐺), 𝑔 ∈ (𝐺Ismt𝐺) ↦ (𝑓 ∘ 𝑔))〉} & ⊢ (𝜑 → 𝐹 ∈ (𝐺Ismt𝐺)) & ⊢ (𝜑 → 𝐻 ∈ (𝐺Ismt𝐺)) ⇒ ⊢ (𝜑 → (𝐹(+g‘𝐼)𝐻) = (𝐹 ∘ 𝐻)) | ||
Theorem | motgrp 28565* | The motions of a geometry form a group with respect to function composition, called the Isometry group. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ 𝐼 = {〈(Base‘ndx), (𝐺Ismt𝐺)〉, 〈(+g‘ndx), (𝑓 ∈ (𝐺Ismt𝐺), 𝑔 ∈ (𝐺Ismt𝐺) ↦ (𝑓 ∘ 𝑔))〉} ⇒ ⊢ (𝜑 → 𝐼 ∈ Grp) | ||
Theorem | motcgrg 28566* | Property of a motion: distances are preserved. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ 𝐼 = {〈(Base‘ndx), (𝐺Ismt𝐺)〉, 〈(+g‘ndx), (𝑓 ∈ (𝐺Ismt𝐺), 𝑔 ∈ (𝐺Ismt𝐺) ↦ (𝑓 ∘ 𝑔))〉} & ⊢ ∼ = (cgrG‘𝐺) & ⊢ (𝜑 → 𝑇 ∈ Word 𝑃) & ⊢ (𝜑 → 𝐹 ∈ (𝐺Ismt𝐺)) ⇒ ⊢ (𝜑 → (𝐹 ∘ 𝑇) ∼ 𝑇) | ||
Theorem | motcgr3 28567 | Property of a motion: distances are preserved, special case of triangles. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ ∼ = (cgrG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 = (𝐻‘𝐴)) & ⊢ (𝜑 → 𝐸 = (𝐻‘𝐵)) & ⊢ (𝜑 → 𝐹 = (𝐻‘𝐶)) & ⊢ (𝜑 → 𝐻 ∈ (𝐺Ismt𝐺)) ⇒ ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝐸𝐹”〉) | ||
Theorem | tglng 28568* | Lines of a Tarski Geometry. This relates to both Definition 4.10 of [Schwabhauser] p. 36. and Definition 6.14 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 28-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) ⇒ ⊢ (𝐺 ∈ TarskiG → 𝐿 = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})) | ||
Theorem | tglnfn 28569 | Lines as functions. (Contributed by Thierry Arnoux, 25-May-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) ⇒ ⊢ (𝐺 ∈ TarskiG → 𝐿 Fn ((𝑃 × 𝑃) ∖ I )) | ||
Theorem | tglnunirn 28570 | Lines are sets of points. (Contributed by Thierry Arnoux, 25-May-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) ⇒ ⊢ (𝐺 ∈ TarskiG → ∪ ran 𝐿 ⊆ 𝑃) | ||
Theorem | tglnpt 28571 | Lines are sets of points. (Contributed by Thierry Arnoux, 17-Oct-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝑋 ∈ 𝑃) | ||
Theorem | tglngne 28572 | It takes two different points to form a line. (Contributed by Thierry Arnoux, 6-Aug-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌)) ⇒ ⊢ (𝜑 → 𝑋 ≠ 𝑌) | ||
Theorem | tglngval 28573* | The line going through points 𝑋 and 𝑌. (Contributed by Thierry Arnoux, 28-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) ⇒ ⊢ (𝜑 → (𝑋𝐿𝑌) = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))}) | ||
Theorem | tglnssp 28574 | Lines are subset of the geometry base set. That is, lines are sets of points. (Contributed by Thierry Arnoux, 17-May-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) ⇒ ⊢ (𝜑 → (𝑋𝐿𝑌) ⊆ 𝑃) | ||
Theorem | tgellng 28575 | Property of lying on the line going through points 𝑋 and 𝑌. Definition 4.10 of [Schwabhauser] p. 36. We choose the notation 𝑍 ∈ (𝑋(LineG‘𝐺)𝑌) instead of "colinear" because LineG is a common structure slot for other axiomatizations of geometry. (Contributed by Thierry Arnoux, 28-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))) | ||
Theorem | tgcolg 28576 | We choose the notation (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) instead of "colinear" in order to avoid defining an additional symbol for colinearity because LineG is a common structure slot for other axiomatizations of geometry. (Contributed by Thierry Arnoux, 25-May-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) ⇒ ⊢ (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))) | ||
Theorem | btwncolg1 28577 | Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐼𝑌)) ⇒ ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) | ||
Theorem | btwncolg2 28578 | Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ (𝑍𝐼𝑌)) ⇒ ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) | ||
Theorem | btwncolg3 28579 | Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼𝑍)) ⇒ ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) | ||
Theorem | colcom 28580 | 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) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) ⇒ ⊢ (𝜑 → (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) | ||
Theorem | colrot1 28581 | 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) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) | ||
Theorem | colrot2 28582 | 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) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) ⇒ ⊢ (𝜑 → (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) | ||
Theorem | ncolcom 28583 | Swapping non-colinear points. (Contributed by Thierry Arnoux, 19-Oct-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) ⇒ ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) | ||
Theorem | ncolrot1 28584 | Rotating non-colinear points. (Contributed by Thierry Arnoux, 19-Oct-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) ⇒ ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) | ||
Theorem | ncolrot2 28585 | Rotating non-colinear points. (Contributed by Thierry Arnoux, 19-Oct-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) ⇒ ⊢ (𝜑 → ¬ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) | ||
Theorem | tgdim01ln 28586 | In geometries of dimension less than two, then any three points are colinear. (Contributed by Thierry Arnoux, 27-Aug-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → ¬ 𝐺DimTarskiG≥2) ⇒ ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) | ||
Theorem | ncoltgdim2 28587 | If there are three non-colinear points, then the dimension is at least two. Converse of tglowdim2l 28672. (Contributed by Thierry Arnoux, 23-Feb-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) ⇒ ⊢ (𝜑 → 𝐺DimTarskiG≥2) | ||
Theorem | lnxfr 28588 | Transfer law for colinearity. Theorem 4.13 of [Schwabhauser] p. 37. (Contributed by Thierry Arnoux, 27-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ ∼ = (cgrG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍)) & ⊢ (𝜑 → 〈“𝑋𝑌𝑍”〉 ∼ 〈“𝐴𝐵𝐶”〉) ⇒ ⊢ (𝜑 → (𝐵 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶)) | ||
Theorem | lnext 28589* | Extend a line with a missing point. Theorem 4.14 of [Schwabhauser] p. 37. (Contributed by Thierry Arnoux, 27-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ ∼ = (cgrG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍)) & ⊢ (𝜑 → (𝑋 − 𝑌) = (𝐴 − 𝐵)) ⇒ ⊢ (𝜑 → ∃𝑐 ∈ 𝑃 〈“𝑋𝑌𝑍”〉 ∼ 〈“𝐴𝐵𝑐”〉) | ||
Theorem | tgfscgr 28590 | Congruence law for the general five segment configuration. Theorem 4.16 of [Schwabhauser] p. 37. (Contributed by Thierry Arnoux, 27-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ ∼ = (cgrG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝑇 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍)) & ⊢ (𝜑 → 〈“𝑋𝑌𝑍”〉 ∼ 〈“𝐴𝐵𝐶”〉) & ⊢ (𝜑 → (𝑋 − 𝑇) = (𝐴 − 𝐷)) & ⊢ (𝜑 → (𝑌 − 𝑇) = (𝐵 − 𝐷)) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) ⇒ ⊢ (𝜑 → (𝑍 − 𝑇) = (𝐶 − 𝐷)) | ||
Theorem | lncgr 28591 | Congruence rule for lines. Theorem 4.17 of [Schwabhauser] p. 37. (Contributed by Thierry Arnoux, 28-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ ∼ = (cgrG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) & ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍)) & ⊢ (𝜑 → (𝑋 − 𝐴) = (𝑋 − 𝐵)) & ⊢ (𝜑 → (𝑌 − 𝐴) = (𝑌 − 𝐵)) ⇒ ⊢ (𝜑 → (𝑍 − 𝐴) = (𝑍 − 𝐵)) | ||
Theorem | lnid 28592 | Identity law for points on lines. Theorem 4.18 of [Schwabhauser] p. 38. (Contributed by Thierry Arnoux, 28-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ ∼ = (cgrG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) & ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍)) & ⊢ (𝜑 → (𝑋 − 𝑍) = (𝑋 − 𝐴)) & ⊢ (𝜑 → (𝑌 − 𝑍) = (𝑌 − 𝐴)) ⇒ ⊢ (𝜑 → 𝑍 = 𝐴) | ||
Theorem | tgidinside 28593 | Law for finding a point inside a segment. Theorem 4.19 of [Schwabhauser] p. 38. (Contributed by Thierry Arnoux, 28-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ ∼ = (cgrG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐼𝑌)) & ⊢ (𝜑 → (𝑋 − 𝑍) = (𝑋 − 𝐴)) & ⊢ (𝜑 → (𝑌 − 𝑍) = (𝑌 − 𝐴)) ⇒ ⊢ (𝜑 → 𝑍 = 𝐴) | ||
Theorem | tgbtwnconn1lem1 28594 | Lemma for tgbtwnconn1 28597. (Contributed by Thierry Arnoux, 30-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 𝐻 ∈ 𝑃) & ⊢ (𝜑 → 𝐽 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ (𝐴𝐼𝐸)) & ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐹)) & ⊢ (𝜑 → 𝐸 ∈ (𝐴𝐼𝐻)) & ⊢ (𝜑 → 𝐹 ∈ (𝐴𝐼𝐽)) & ⊢ (𝜑 → (𝐸 − 𝐷) = (𝐶 − 𝐷)) & ⊢ (𝜑 → (𝐶 − 𝐹) = (𝐶 − 𝐷)) & ⊢ (𝜑 → (𝐸 − 𝐻) = (𝐵 − 𝐶)) & ⊢ (𝜑 → (𝐹 − 𝐽) = (𝐵 − 𝐷)) ⇒ ⊢ (𝜑 → 𝐻 = 𝐽) | ||
Theorem | tgbtwnconn1lem2 28595 | Lemma for tgbtwnconn1 28597. (Contributed by Thierry Arnoux, 30-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 𝐻 ∈ 𝑃) & ⊢ (𝜑 → 𝐽 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ (𝐴𝐼𝐸)) & ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐹)) & ⊢ (𝜑 → 𝐸 ∈ (𝐴𝐼𝐻)) & ⊢ (𝜑 → 𝐹 ∈ (𝐴𝐼𝐽)) & ⊢ (𝜑 → (𝐸 − 𝐷) = (𝐶 − 𝐷)) & ⊢ (𝜑 → (𝐶 − 𝐹) = (𝐶 − 𝐷)) & ⊢ (𝜑 → (𝐸 − 𝐻) = (𝐵 − 𝐶)) & ⊢ (𝜑 → (𝐹 − 𝐽) = (𝐵 − 𝐷)) ⇒ ⊢ (𝜑 → (𝐸 − 𝐹) = (𝐶 − 𝐷)) | ||
Theorem | tgbtwnconn1lem3 28596 | Lemma for tgbtwnconn1 28597. (Contributed by Thierry Arnoux, 30-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 𝐻 ∈ 𝑃) & ⊢ (𝜑 → 𝐽 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ (𝐴𝐼𝐸)) & ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐹)) & ⊢ (𝜑 → 𝐸 ∈ (𝐴𝐼𝐻)) & ⊢ (𝜑 → 𝐹 ∈ (𝐴𝐼𝐽)) & ⊢ (𝜑 → (𝐸 − 𝐷) = (𝐶 − 𝐷)) & ⊢ (𝜑 → (𝐶 − 𝐹) = (𝐶 − 𝐷)) & ⊢ (𝜑 → (𝐸 − 𝐻) = (𝐵 − 𝐶)) & ⊢ (𝜑 → (𝐹 − 𝐽) = (𝐵 − 𝐷)) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ (𝐶𝐼𝐸)) & ⊢ (𝜑 → 𝑋 ∈ (𝐷𝐼𝐹)) & ⊢ (𝜑 → 𝐶 ≠ 𝐸) ⇒ ⊢ (𝜑 → 𝐷 = 𝐹) | ||
Theorem | tgbtwnconn1 28597 | Connectivity law for betweenness. Theorem 5.1 of [Schwabhauser] p. 39-41. In earlier presentations of Tarski's axioms, this theorem appeared as an additional axiom. It was derived from the other axioms by Gupta, 1965. (Contributed by Thierry Arnoux, 30-Apr-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) ⇒ ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐼𝐷) ∨ 𝐷 ∈ (𝐴𝐼𝐶))) | ||
Theorem | tgbtwnconn2 28598 | Another connectivity law for betweenness. Theorem 5.2 of [Schwabhauser] p. 41. (Contributed by Thierry Arnoux, 17-May-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) ⇒ ⊢ (𝜑 → (𝐶 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐶))) | ||
Theorem | tgbtwnconn3 28599 | Inner connectivity law for betweenness. Theorem 5.3 of [Schwabhauser] p. 41. (Contributed by Thierry Arnoux, 17-May-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) & ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷)) ⇒ ⊢ (𝜑 → (𝐵 ∈ (𝐴𝐼𝐶) ∨ 𝐶 ∈ (𝐴𝐼𝐵))) | ||
Theorem | tgbtwnconnln3 28600 | Derive colinearity from betweenness. (Contributed by Thierry Arnoux, 17-May-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) & ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷)) & ⊢ 𝐿 = (LineG‘𝐺) ⇒ ⊢ (𝜑 → (𝐵 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶)) |
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