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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | oppne2 26701* | 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 26702* | 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 26703* | 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 26704* | 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 26705* | 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 26706* | Lemma for opphl 26713. (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 26707* | Lemma for opphl 26713. 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 26708* | Lemma for opphl 26713: 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 26709* | Lemma for opphl 26713. (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 26710* | 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 26711* | 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 26712* | Restating colperpex 26692 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 26713* | If two points 𝐴 and 𝐶 lie on opposite sides of a line 𝐷, then any point of the half line (𝑅𝐴) 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 26714* | 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 26715* | 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 26716 | "Belong to the same open half-plane" relation for points in a geometry. |
class hpG | ||
Definition | df-hpg 26717* | Define the open half plane relation for a geometry 𝐺. Definition 9.7 of [Schwabhauser] p. 71. See hpgbr 26719 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 26718* | 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 26719* | 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 26720* | 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 26721* | 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 26722* | 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 26723* | 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 26724* | 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 26725* | 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 26726* | 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 26727* | 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 26728* | 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 26729* | 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 26730* | 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 26731 | Declare the constant for the midpoint operation. |
class midG | ||
Syntax | clmi 26732 | Declare the constant for the line mirroring function. |
class lInvG | ||
Definition | df-mid 26733* | Define the midpoint operation. Definition 10.1 of [Schwabhauser] p. 88. See ismidb 26737, midbtwn 26738, and midcgr 26739. (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 26734* | Define the line mirroring function. Definition 10.3 of [Schwabhauser] p. 89. See islmib 26746. (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 26735 | 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 26736 | 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 26737 | 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 26738 | 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 26739 | 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 26740 | 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 26741 | 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 26742 | 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 26743* | 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))) | ||
Theorem | lmif 26744 | Line mirror as a function. (Contributed by Thierry Arnoux, 11-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) & ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) ⇒ ⊢ (𝜑 → 𝑀:𝑃⟶𝑃) | ||
Theorem | lmicl 26745 | Closure of the line mirror. (Contributed by Thierry Arnoux, 11-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) & ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃) | ||
Theorem | islmib 26746 | Property of the line mirror. (Contributed by Thierry Arnoux, 11-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) & ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝐵 = (𝑀‘𝐴) ↔ ((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)))) | ||
Theorem | lmicom 26747 | The line mirroring function is an involution. Theorem 10.4 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) & ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → (𝑀‘𝐴) = 𝐵) ⇒ ⊢ (𝜑 → (𝑀‘𝐵) = 𝐴) | ||
Theorem | lmilmi 26748 | Line mirroring is an involution. Theorem 10.5 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) & ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝑀‘(𝑀‘𝐴)) = 𝐴) | ||
Theorem | lmireu 26749* | Any point has a unique antecedent through line mirroring. Theorem 10.6 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) & ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) ⇒ ⊢ (𝜑 → ∃!𝑏 ∈ 𝑃 (𝑀‘𝑏) = 𝐴) | ||
Theorem | lmieq 26750 | Equality deduction for line mirroring. Theorem 10.7 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) & ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → (𝑀‘𝐴) = (𝑀‘𝐵)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | lmiinv 26751 | The invariants of the line mirroring lie on the mirror line. Theorem 10.8 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) & ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) ⇒ ⊢ (𝜑 → ((𝑀‘𝐴) = 𝐴 ↔ 𝐴 ∈ 𝐷)) | ||
Theorem | lmicinv 26752 | The mirroring line is an invariant. (Contributed by Thierry Arnoux, 8-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) & ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ∈ 𝐷) ⇒ ⊢ (𝜑 → (𝑀‘𝐴) = 𝐴) | ||
Theorem | lmimid 26753 | If we have a right angle, then the mirror point is the point inversion. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) & ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ 𝑆 = ((pInvG‘𝐺)‘𝐵) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) & ⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ (𝜑 → 𝐵 ∈ 𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → (𝑀‘𝐶) = (𝑆‘𝐶)) | ||
Theorem | lmif1o 26754 | The line mirroring function 𝑀 is a bijection. Theorem 10.9 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) & ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) ⇒ ⊢ (𝜑 → 𝑀:𝑃–1-1-onto→𝑃) | ||
Theorem | lmiisolem 26755 | Lemma for lmiiso 26756. (Contributed by Thierry Arnoux, 14-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) & ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ 𝑆 = ((pInvG‘𝐺)‘𝑍) & ⊢ 𝑍 = ((𝐴(midG‘𝐺)(𝑀‘𝐴))(midG‘𝐺)(𝐵(midG‘𝐺)(𝑀‘𝐵))) ⇒ ⊢ (𝜑 → ((𝑀‘𝐴) − (𝑀‘𝐵)) = (𝐴 − 𝐵)) | ||
Theorem | lmiiso 26756 | The line mirroring function is an isometry, i.e. it is conserves congruence. Because it is also a bijection, it is also a motion. Theorem 10.10 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) & ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) ⇒ ⊢ (𝜑 → ((𝑀‘𝐴) − (𝑀‘𝐵)) = (𝐴 − 𝐵)) | ||
Theorem | lmimot 26757 | Line mirroring is a motion of the geometric space. Theorem 10.11 of [Schwabhauser] p. 90. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) & ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) ⇒ ⊢ (𝜑 → 𝑀 ∈ (𝐺Ismt𝐺)) | ||
Theorem | hypcgrlem1 26758 | Lemma for hypcgr 26760, case where triangles share a cathetus. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) & ⊢ (𝜑 → 〈“𝐷𝐸𝐹”〉 ∈ (∟G‘𝐺)) & ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸)) & ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹)) & ⊢ (𝜑 → 𝐵 = 𝐸) & ⊢ 𝑆 = ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵)) & ⊢ (𝜑 → 𝐶 = 𝐹) ⇒ ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹)) | ||
Theorem | hypcgrlem2 26759 | Lemma for hypcgr 26760, case where triangles share one vertex 𝐵. (Contributed by Thierry Arnoux, 16-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) & ⊢ (𝜑 → 〈“𝐷𝐸𝐹”〉 ∈ (∟G‘𝐺)) & ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸)) & ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹)) & ⊢ (𝜑 → 𝐵 = 𝐸) & ⊢ 𝑆 = ((lInvG‘𝐺)‘((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵)) ⇒ ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹)) | ||
Theorem | hypcgr 26760 | If the catheti of two right-angled triangles are congruent, so is their hypothenuse. Theorem 10.12 of [Schwabhauser] p. 91. (Contributed by Thierry Arnoux, 16-Dec-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) & ⊢ (𝜑 → 〈“𝐷𝐸𝐹”〉 ∈ (∟G‘𝐺)) & ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸)) & ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹)) ⇒ ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹)) | ||
Theorem | lmiopp 26761* | Line mirroring produces points on the opposite side of the mirroring line. Theorem 10.14 of [Schwabhauser] p. 92. (Contributed by Thierry Arnoux, 2-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷) ⇒ ⊢ (𝜑 → 𝐴𝑂(𝑀‘𝐴)) | ||
Theorem | lnperpex 26762* | Existence of a perpendicular to a line 𝐿 at a given point 𝐴. Theorem 10.15 of [Schwabhauser] p. 92. (Contributed by Thierry Arnoux, 2-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺DimTarskiG≥2) & ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) & ⊢ (𝜑 → ¬ 𝑄 ∈ 𝐷) ⇒ ⊢ (𝜑 → ∃𝑝 ∈ 𝑃 (𝐷(⟂G‘𝐺)(𝑝𝐿𝐴) ∧ 𝑝((hpG‘𝐺)‘𝐷)𝑄)) | ||
Theorem | trgcopy 26763* | Triangle construction: a copy of a given triangle can always be constructed in such a way that one side is lying on a half-line, and the third vertex is on a given half-plane: existence part. First part of Theorem 10.16 of [Schwabhauser] p. 92. (Contributed by Thierry Arnoux, 4-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶)) & ⊢ (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹)) & ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸)) ⇒ ⊢ (𝜑 → ∃𝑓 ∈ 𝑃 (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝐷𝐸𝑓”〉 ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) | ||
Theorem | trgcopyeulem 26764* | Lemma for trgcopyeu 26765. (Contributed by Thierry Arnoux, 8-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶)) & ⊢ (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹)) & ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸)) & ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐷𝐿𝐸))) ∧ ∃𝑡 ∈ (𝐷𝐿𝐸)𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝐷𝐸𝑋”〉) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝐷𝐸𝑌”〉) & ⊢ (𝜑 → 𝑋((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) & ⊢ (𝜑 → 𝑌((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) ⇒ ⊢ (𝜑 → 𝑋 = 𝑌) | ||
Theorem | trgcopyeu 26765* | Triangle construction: a copy of a given triangle can always be constructed in such a way that one side is lying on a half-line, and the third vertex is on a given half-plane: uniqueness part. Second part of Theorem 10.16 of [Schwabhauser] p. 92. (Contributed by Thierry Arnoux, 8-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶)) & ⊢ (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹)) & ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸)) ⇒ ⊢ (𝜑 → ∃!𝑓 ∈ 𝑃 (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝐷𝐸𝑓”〉 ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) | ||
Syntax | ccgra 26766 | Declare the constant for the congruence between angles relation. |
class cgrA | ||
Definition | df-cgra 26767* | Define the congruence relation between angles. As for triangles we use "words of points". See iscgra 26768 for a more human readable version. (Contributed by Thierry Arnoux, 30-Jul-2020.) |
⊢ cgrA = (𝑔 ∈ V ↦ {〈𝑎, 𝑏〉 ∣ [(Base‘𝑔) / 𝑝][(hlG‘𝑔) / 𝑘]((𝑎 ∈ (𝑝 ↑m (0..^3)) ∧ 𝑏 ∈ (𝑝 ↑m (0..^3))) ∧ ∃𝑥 ∈ 𝑝 ∃𝑦 ∈ 𝑝 (𝑎(cgrG‘𝑔)〈“𝑥(𝑏‘1)𝑦”〉 ∧ 𝑥(𝑘‘(𝑏‘1))(𝑏‘0) ∧ 𝑦(𝑘‘(𝑏‘1))(𝑏‘2)))}) | ||
Theorem | iscgra 26768* | Property for two angles ABC and DEF to be congruent. This is a modified version of the definition 11.3 of [Schwabhauser] p. 95. where the number of constructed points has been reduced to two. We chose this version rather than the textbook version because it is shorter and therefore simpler to work with. Theorem dfcgra2 26789 shows that those definitions are indeed equivalent. (Contributed by Thierry Arnoux, 31-Jul-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) ⇒ ⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉 ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹))) | ||
Theorem | iscgra1 26769* | A special version of iscgra 26768 where one distance is known to be equal. In this case, angle congruence can be written with only one quantifier. (Contributed by Thierry Arnoux, 9-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸)) ⇒ ⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉 ↔ ∃𝑥 ∈ 𝑃 (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝐷𝐸𝑥”〉 ∧ 𝑥(𝐾‘𝐸)𝐹))) | ||
Theorem | iscgrad 26770 | Sufficient conditions for angle congruence, deduction version. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑋𝐸𝑌”〉) & ⊢ (𝜑 → 𝑋(𝐾‘𝐸)𝐷) & ⊢ (𝜑 → 𝑌(𝐾‘𝐸)𝐹) ⇒ ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) | ||
Theorem | cgrane1 26771 | Angles imply inequality. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐵) | ||
Theorem | cgrane2 26772 | Angles imply inequality. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) ⇒ ⊢ (𝜑 → 𝐵 ≠ 𝐶) | ||
Theorem | cgrane3 26773 | Angles imply inequality. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) ⇒ ⊢ (𝜑 → 𝐸 ≠ 𝐷) | ||
Theorem | cgrane4 26774 | Angles imply inequality. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) ⇒ ⊢ (𝜑 → 𝐸 ≠ 𝐹) | ||
Theorem | cgrahl1 26775 | Angle congruence is independent of the choice of points on the rays. Proposition 11.10 of [Schwabhauser] p. 95. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑋(𝐾‘𝐸)𝐷) ⇒ ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝑋𝐸𝐹”〉) | ||
Theorem | cgrahl2 26776 | Angle congruence is independent of the choice of points on the rays. Proposition 11.10 of [Schwabhauser] p. 95. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑋(𝐾‘𝐸)𝐹) ⇒ ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑋”〉) | ||
Theorem | cgracgr 26777 | First direction of proposition 11.4 of [Schwabhauser] p. 95. Again, this is "half" of the proposition, i.e. only two additional points are used, while Schwabhauser has four. (Contributed by Thierry Arnoux, 31-Jul-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑋(𝐾‘𝐵)𝐴) & ⊢ (𝜑 → 𝑌(𝐾‘𝐵)𝐶) & ⊢ (𝜑 → (𝐵 − 𝑋) = (𝐸 − 𝐷)) & ⊢ (𝜑 → (𝐵 − 𝑌) = (𝐸 − 𝐹)) ⇒ ⊢ (𝜑 → (𝑋 − 𝑌) = (𝐷 − 𝐹)) | ||
Theorem | cgraid 26778 | Angle congruence is reflexive. Theorem 11.6 of [Schwabhauser] p. 96. (Contributed by Thierry Arnoux, 31-Jul-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) ⇒ ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐴𝐵𝐶”〉) | ||
Theorem | cgraswap 26779 | Swap rays in a congruence relation. Theorem 11.9 of [Schwabhauser] p. 96. (Contributed by Thierry Arnoux, 5-Mar-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) ⇒ ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐶𝐵𝐴”〉) | ||
Theorem | cgrcgra 26780 | Triangle congruence implies angle congruence. This is a portion of CPCTC, focusing on a specific angle. (Contributed by Arnoux, 2-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝐷𝐸𝐹”〉) ⇒ ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) | ||
Theorem | cgracom 26781 | Angle congruence commutes. Theorem 11.7 of [Schwabhauser] p. 97. (Contributed by Thierry Arnoux, 5-Mar-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) ⇒ ⊢ (𝜑 → 〈“𝐷𝐸𝐹”〉(cgrA‘𝐺)〈“𝐴𝐵𝐶”〉) | ||
Theorem | cgratr 26782 | Angle congruence is transitive. Theorem 11.8 of [Schwabhauser] p. 97. (Contributed by Thierry Arnoux, 5-Mar-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) & ⊢ (𝜑 → 𝐻 ∈ 𝑃) & ⊢ (𝜑 → 𝑈 ∈ 𝑃) & ⊢ (𝜑 → 𝐽 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐷𝐸𝐹”〉(cgrA‘𝐺)〈“𝐻𝑈𝐽”〉) ⇒ ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐻𝑈𝐽”〉) | ||
Theorem | flatcgra 26783 | Flat angles are congruent. (Contributed by Thierry Arnoux, 13-Feb-2023.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ − = (dist‘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) & ⊢ (𝜑 → 𝐶 ≠ 𝐵) & ⊢ (𝜑 → 𝐷 ≠ 𝐸) & ⊢ (𝜑 → 𝐹 ≠ 𝐸) ⇒ ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) | ||
Theorem | cgraswaplr 26784 | Swap both side of angle congruence. (Contributed by Thierry Arnoux, 5-Oct-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) ⇒ ⊢ (𝜑 → 〈“𝐶𝐵𝐴”〉(cgrA‘𝐺)〈“𝐹𝐸𝐷”〉) | ||
Theorem | cgrabtwn 26785 | Angle congruence preserves flat angles. Part of Theorem 11.21 of [Schwabhauser] p. 97. (Contributed by Thierry Arnoux, 9-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) ⇒ ⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝐹)) | ||
Theorem | cgrahl 26786 | Angle congruence preserves null angles. Part of Theorem 11.21 of [Schwabhauser] p. 97. (Contributed by Thierry Arnoux, 9-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐴(𝐾‘𝐵)𝐶) ⇒ ⊢ (𝜑 → 𝐷(𝐾‘𝐸)𝐹) | ||
Theorem | cgracol 26787 | Angle congruence preserves colinearity. (Contributed by Thierry Arnoux, 9-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸)) | ||
Theorem | cgrancol 26788 | Angle congruence preserves non-colinearity. (Contributed by Thierry Arnoux, 9-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)) ⇒ ⊢ (𝜑 → ¬ (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸)) | ||
Theorem | dfcgra2 26789* | This is the full statement of definition 11.2 of [Schwabhauser] p. 95. This proof serves to confirm that the definition we have chosen, df-cgra 26767 is indeed equivalent to the textbook's definition. (Contributed by Thierry Arnoux, 2-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) ⇒ ⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉 ↔ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵) ∧ (𝐷 ≠ 𝐸 ∧ 𝐹 ≠ 𝐸) ∧ ∃𝑎 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑓 ∈ 𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 − 𝑎) = (𝐸 − 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 − 𝑐) = (𝐸 − 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 − 𝑑) = (𝐵 − 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝑎 − 𝑐) = (𝑑 − 𝑓))))) | ||
Theorem | sacgr 26790 | Supplementary angles of congruent angles are themselves congruent. Theorem 11.13 of [Schwabhauser] p. 98. (Contributed by Thierry Arnoux, 30-Sep-2020.) (Proof shortened by Igor Ieskov, 16-Feb-2023.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝑋)) & ⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝑌)) & ⊢ (𝜑 → 𝐵 ≠ 𝑋) & ⊢ (𝜑 → 𝐸 ≠ 𝑌) ⇒ ⊢ (𝜑 → 〈“𝑋𝐵𝐶”〉(cgrA‘𝐺)〈“𝑌𝐸𝐹”〉) | ||
Theorem | oacgr 26791 | Vertical angle theorem. Vertical, or opposite angles are the facing pair of angles formed when two lines intersect. Eudemus of Rhodes attributed the proof to Thales of Miletus. The proposition showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles, the vertical angles are equal in measure. We follow the same path. Theorem 11.14 of [Schwabhauser] p. 98. (Contributed by Thierry Arnoux, 27-Sep-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) & ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐹)) & ⊢ (𝜑 → 𝐵 ≠ 𝐴) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) & ⊢ (𝜑 → 𝐵 ≠ 𝐷) & ⊢ (𝜑 → 𝐵 ≠ 𝐹) ⇒ ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐵𝐹”〉) | ||
Theorem | acopy 26792* | Angle construction. Theorem 11.15 of [Schwabhauser] p. 98. This is Hilbert's axiom III.4 for geometry. (Contributed by Thierry Arnoux, 9-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶)) & ⊢ (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹)) ⇒ ⊢ (𝜑 → ∃𝑓 ∈ 𝑃 (〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑓”〉 ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) | ||
Theorem | acopyeu 26793 | Angle construction. Theorem 11.15 of [Schwabhauser] p. 98. This is Hilbert's axiom III.4 for geometry. Akin to a uniqueness theorem, this states that if two points 𝑋 and 𝑌 both fulfill the conditions, then they are on the same half-line. (Contributed by Thierry Arnoux, 9-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶)) & ⊢ (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹)) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑋”〉) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑌”〉) & ⊢ (𝜑 → 𝑋((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) & ⊢ (𝜑 → 𝑌((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) ⇒ ⊢ (𝜑 → 𝑋(𝐾‘𝐸)𝑌) | ||
Syntax | cinag 26794 | Extend class relation with the geometrical "point in angle" relation. |
class inA | ||
Syntax | cleag 26795 | Extend class relation with the "angle less than" relation. |
class ≤∠ | ||
Definition | df-inag 26796* | Definition of the geometrical "in angle" relation. (Contributed by Thierry Arnoux, 15-Aug-2020.) |
⊢ inA = (𝑔 ∈ V ↦ {〈𝑝, 𝑡〉 ∣ ((𝑝 ∈ (Base‘𝑔) ∧ 𝑡 ∈ ((Base‘𝑔) ↑m (0..^3))) ∧ (((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ (Base‘𝑔)(𝑥 ∈ ((𝑡‘0)(Itv‘𝑔)(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥((hlG‘𝑔)‘(𝑡‘1))𝑝))))}) | ||
Theorem | isinag 26797* | Property for point 𝑋 to lie in the angle 〈“𝐴𝐵𝐶”〉. Definition 11.23 of [Schwabhauser] p. 101. (Contributed by Thierry Arnoux, 15-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑋(inA‘𝐺)〈“𝐴𝐵𝐶”〉 ↔ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ 𝑋 ≠ 𝐵) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋))))) | ||
Theorem | isinagd 26798 | Sufficient conditions for in-angle relation, deduction version. (Contributed by Thierry Arnoux, 20-Oct-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐶 ≠ 𝐵) & ⊢ (𝜑 → 𝑋 ≠ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ (𝐴𝐼𝐶)) & ⊢ (𝜑 → (𝑌 = 𝐵 ∨ 𝑌(𝐾‘𝐵)𝑋)) ⇒ ⊢ (𝜑 → 𝑋(inA‘𝐺)〈“𝐴𝐵𝐶”〉) | ||
Theorem | inagflat 26799 | Any point lies in a flat angle. (Contributed by Thierry Arnoux, 13-Feb-2023.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐶 ≠ 𝐵) & ⊢ (𝜑 → 𝑋 ≠ 𝐵) & ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) ⇒ ⊢ (𝜑 → 𝑋(inA‘𝐺)〈“𝐴𝐵𝐶”〉) | ||
Theorem | inagswap 26800 | Swap the order of the half lines delimiting the angle. Theorem 11.24 of [Schwabhauser] p. 101. (Contributed by Thierry Arnoux, 15-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋(inA‘𝐺)〈“𝐴𝐵𝐶”〉) ⇒ ⊢ (𝜑 → 𝑋(inA‘𝐺)〈“𝐶𝐵𝐴”〉) |
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