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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | midbtwn 28801 | 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 28802 | 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 28803 | 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 28804 | 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 28805 | 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 28806* | 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 28807 | 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 28808 | 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 28809 | 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 28810 | 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 28811 | 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 28812* | 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 28813 | 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 28814 | 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 28815 | 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 28816 | 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 28817 | 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 28818 | Lemma for lmiiso 28819. (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 28819 | 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 28820 | 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 28821 | Lemma for hypcgr 28823, 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 28822 | Lemma for hypcgr 28823, 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 28823 | 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 28824* | 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 28825* | 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 28826* | 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 28827* | Lemma for trgcopyeu 28828. (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 28828* | 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 28829 | Declare the constant for the congruence between angles relation. |
class cgrA | ||
Definition | df-cgra 28830* | Define the congruence relation between angles. As for triangles we use "words of points". See iscgra 28831 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 28831* | 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 28852 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 28832* | A special version of iscgra 28831 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 28833 | 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 28834 | 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 28835 | 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 28836 | 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 28837 | 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 28838 | 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 28839 | 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 28840 | 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 28841 | 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 28842 | 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 28843 | 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 28844 | 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 28845 | 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 28846 | 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 28847 | 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 28848 | 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 28849 | 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 28850 | 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 28851 | 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 28852* | 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 28830 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 28853 | 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 28854 | 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 28855* | 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 28856 | 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 28857 | Extend class relation with the geometrical "point in angle" relation. |
class inA | ||
Syntax | cleag 28858 | Extend class relation with the "angle less than" relation. |
class ≤∠ | ||
Definition | df-inag 28859* | 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 28860* | 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 28861 | 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 28862 | 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 28863 | 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”〉) | ||
Theorem | inagne1 28864 | Deduce inequality from the in-angle relation. (Contributed by Thierry Arnoux, 29-Oct-2021.) |
⊢ 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 → 𝐴 ≠ 𝐵) | ||
Theorem | inagne2 28865 | Deduce inequality from the in-angle relation. (Contributed by Thierry Arnoux, 29-Oct-2021.) |
⊢ 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 → 𝐶 ≠ 𝐵) | ||
Theorem | inagne3 28866 | Deduce inequality from the in-angle relation. (Contributed by Thierry Arnoux, 29-Oct-2021.) |
⊢ 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 ≠ 𝐵) | ||
Theorem | inaghl 28867 | The "point lie in angle" relation is independent of the points chosen on the half lines starting from 𝐵. Theorem 11.25 of [Schwabhauser] p. 101. (Contributed by Thierry Arnoux, 27-Sep-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 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝐷(𝐾‘𝐵)𝐴) & ⊢ (𝜑 → 𝐹(𝐾‘𝐵)𝐶) & ⊢ (𝜑 → 𝑌(𝐾‘𝐵)𝑋) ⇒ ⊢ (𝜑 → 𝑌(inA‘𝐺)〈“𝐷𝐵𝐹”〉) | ||
Definition | df-leag 28868* | Definition of the geometrical "angle less than" relation. Definition 11.27 of [Schwabhauser] p. 102. (Contributed by Thierry Arnoux, 7-Oct-2020.) |
⊢ ≤∠ = (𝑔 ∈ V ↦ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ ((Base‘𝑔) ↑m (0..^3)) ∧ 𝑏 ∈ ((Base‘𝑔) ↑m (0..^3))) ∧ ∃𝑥 ∈ (Base‘𝑔)(𝑥(inA‘𝑔)〈“(𝑏‘0)(𝑏‘1)(𝑏‘2)”〉 ∧ 〈“(𝑎‘0)(𝑎‘1)(𝑎‘2)”〉(cgrA‘𝑔)〈“(𝑏‘0)(𝑏‘1)𝑥”〉))}) | ||
Theorem | isleag 28869* | Geometrical "less than" property for angles. Definition 11.27 of [Schwabhauser] p. 102. (Contributed by Thierry Arnoux, 7-Oct-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) ⇒ ⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉(≤∠‘𝐺)〈“𝐷𝐸𝐹”〉 ↔ ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)〈“𝐷𝐸𝐹”〉 ∧ 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑥”〉))) | ||
Theorem | isleagd 28870 | Sufficient condition for "less than" angle relation, deduction version (Contributed by Thierry Arnoux, 12-Oct-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ ≤ = (≤∠‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑋(inA‘𝐺)〈“𝐷𝐸𝐹”〉) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑋”〉) ⇒ ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ≤ 〈“𝐷𝐸𝐹”〉) | ||
Theorem | leagne1 28871 | Deduce inequality from the less-than angle relation. (Contributed by Thierry Arnoux, 25-Feb-2023.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(≤∠‘𝐺)〈“𝐷𝐸𝐹”〉) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐵) | ||
Theorem | leagne2 28872 | Deduce inequality from the less-than angle relation. (Contributed by Thierry Arnoux, 25-Feb-2023.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(≤∠‘𝐺)〈“𝐷𝐸𝐹”〉) ⇒ ⊢ (𝜑 → 𝐶 ≠ 𝐵) | ||
Theorem | leagne3 28873 | Deduce inequality from the less-than angle relation. (Contributed by Thierry Arnoux, 25-Feb-2023.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(≤∠‘𝐺)〈“𝐷𝐸𝐹”〉) ⇒ ⊢ (𝜑 → 𝐷 ≠ 𝐸) | ||
Theorem | leagne4 28874 | Deduce inequality from the less-than angle relation. (Contributed by Thierry Arnoux, 25-Feb-2023.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(≤∠‘𝐺)〈“𝐷𝐸𝐹”〉) ⇒ ⊢ (𝜑 → 𝐹 ≠ 𝐸) | ||
Theorem | cgrg3col4 28875* | Lemma 11.28 of [Schwabhauser] p. 102. Extend a congruence of three points with a fourth colinear point. (Contributed by Thierry Arnoux, 8-Oct-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ (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 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝐷𝐸𝐹”〉) & ⊢ (𝜑 → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶)) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 〈“𝐴𝐵𝐶𝑋”〉(cgrG‘𝐺)〈“𝐷𝐸𝐹𝑦”〉) | ||
Theorem | tgsas1 28876 | First congruence theorem: SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then third sides are equal in length. Theorem 11.49 of [Schwabhauser] p. 107. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (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 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) & ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹)) ⇒ ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐹 − 𝐷)) | ||
Theorem | tgsas 28877 | First congruence theorem: SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent. Theorem 11.49 of [Schwabhauser] p. 107. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (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 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) & ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹)) ⇒ ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝐷𝐸𝐹”〉) | ||
Theorem | tgsas2 28878 | First congruence theorem: SAS. Theorem 11.49 of [Schwabhauser] p. 107. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (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 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) & ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹)) & ⊢ (𝜑 → 𝐴 ≠ 𝐶) ⇒ ⊢ (𝜑 → 〈“𝐶𝐴𝐵”〉(cgrA‘𝐺)〈“𝐹𝐷𝐸”〉) | ||
Theorem | tgsas3 28879 | First congruence theorem: SAS. Theorem 11.49 of [Schwabhauser] p. 107. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (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 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) & ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹)) & ⊢ (𝜑 → 𝐴 ≠ 𝐶) ⇒ ⊢ (𝜑 → 〈“𝐵𝐶𝐴”〉(cgrA‘𝐺)〈“𝐸𝐹𝐷”〉) | ||
Theorem | tgasa1 28880 | Second congruence theorem: ASA. (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent. Theorem 11.50 of [Schwabhauser] p. 108. (Contributed by Thierry Arnoux, 15-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (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 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)) & ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸)) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) & ⊢ (𝜑 → 〈“𝐶𝐴𝐵”〉(cgrA‘𝐺)〈“𝐹𝐷𝐸”〉) ⇒ ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹)) | ||
Theorem | tgasa 28881 | Second congruence theorem: ASA. (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent. Theorem 11.50 of [Schwabhauser] p. 108. (Contributed by Thierry Arnoux, 15-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (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 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)) & ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸)) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) & ⊢ (𝜑 → 〈“𝐶𝐴𝐵”〉(cgrA‘𝐺)〈“𝐹𝐷𝐸”〉) ⇒ ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝐷𝐸𝐹”〉) | ||
Theorem | tgsss1 28882 | Third congruence theorem: SSS (Side-Side-Side): If the three pairs of sides of two triangles are equal in length, then the triangles are congruent. Theorem 11.51 of [Schwabhauser] p. 109. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (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 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) & ⊢ (𝜑 → 𝐶 ≠ 𝐴) ⇒ ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) | ||
Theorem | tgsss2 28883 | Third congruence theorem: SSS. Theorem 11.51 of [Schwabhauser] p. 109. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (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 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) & ⊢ (𝜑 → 𝐶 ≠ 𝐴) ⇒ ⊢ (𝜑 → 〈“𝐶𝐴𝐵”〉(cgrA‘𝐺)〈“𝐹𝐷𝐸”〉) | ||
Theorem | tgsss3 28884 | Third congruence theorem: SSS. Theorem 11.51 of [Schwabhauser] p. 109. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (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 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) & ⊢ (𝜑 → 𝐶 ≠ 𝐴) ⇒ ⊢ (𝜑 → 〈“𝐵𝐶𝐴”〉(cgrA‘𝐺)〈“𝐸𝐹𝐷”〉) | ||
Theorem | dfcgrg2 28885 | Congruence for two triangles can also be defined as congruence of sides and angles (6 parts). This is often the actual textbook definition of triangle congruence, see for example https://en.wikipedia.org/wiki/Congruence_(geometry)#Congruence_of_triangles. With this definition, the "SSS" congruence theorem has an additional part, namely, that triangle congruence implies congruence of the sides (which means equality of the lengths). Because our development of elementary geometry strives to closely follow Schwabhaeuser's, our original definition of shape congruence, df-cgrg 28533, already covers that part: see trgcgr 28538. This theorem is also named "CPCTC", which stands for "Corresponding Parts of Congruent Triangles are Congruent", see https://en.wikipedia.org/wiki/Congruence_(geometry)#CPCTC 28538. (Contributed by Thierry Arnoux, 18-Jan-2023.) |
⊢ 𝑃 = (Base‘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 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) & ⊢ (𝜑 → 𝐶 ≠ 𝐴) ⇒ ⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝐷𝐸𝐹”〉 ↔ (((𝐴 − 𝐵) = (𝐷 − 𝐸) ∧ (𝐵 − 𝐶) = (𝐸 − 𝐹) ∧ (𝐶 − 𝐴) = (𝐹 − 𝐷)) ∧ (〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉 ∧ 〈“𝐶𝐴𝐵”〉(cgrA‘𝐺)〈“𝐹𝐷𝐸”〉 ∧ 〈“𝐵𝐶𝐴”〉(cgrA‘𝐺)〈“𝐸𝐹𝐷”〉)))) | ||
Theorem | isoas 28886 | Congruence theorem for isocele triangles: if two angles of a triangle are congruent, then the corresponding sides also are. (Contributed by Thierry Arnoux, 5-Oct-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)) & ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐴𝐶𝐵”〉) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐴 − 𝐶)) | ||
Syntax | ceqlg 28887 | Declare the class of equilateral triangles. |
class eqltrG | ||
Definition | df-eqlg 28888* | Define the class of equilateral triangles. (Contributed by Thierry Arnoux, 27-Nov-2019.) |
⊢ eqltrG = (𝑔 ∈ V ↦ {𝑥 ∈ ((Base‘𝑔) ↑m (0..^3)) ∣ 𝑥(cgrG‘𝑔)〈“(𝑥‘1)(𝑥‘2)(𝑥‘0)”〉}) | ||
Theorem | iseqlg 28889 | Property of a triangle being equilateral. (Contributed by Thierry Arnoux, 5-Oct-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) ⇒ ⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉 ∈ (eqltrG‘𝐺) ↔ 〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝐵𝐶𝐴”〉)) | ||
Theorem | iseqlgd 28890 | Condition for a triangle to be equilateral. (Contributed by Thierry Arnoux, 5-Oct-2020.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐵 − 𝐶)) & ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐶 − 𝐴)) & ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐴 − 𝐵)) ⇒ ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (eqltrG‘𝐺)) | ||
Theorem | f1otrgds 28891* | Convenient lemma for f1otrg 28893. (Contributed by Thierry Arnoux, 19-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐷 = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐵 = (Base‘𝐻) & ⊢ 𝐸 = (dist‘𝐻) & ⊢ 𝐽 = (Itv‘𝐻) & ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝑃) & ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵)) → (𝑒𝐸𝑓) = ((𝐹‘𝑒)𝐷(𝐹‘𝑓))) & ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹‘𝑔) ∈ ((𝐹‘𝑒)𝐼(𝐹‘𝑓)))) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋𝐸𝑌) = ((𝐹‘𝑋)𝐷(𝐹‘𝑌))) | ||
Theorem | f1otrgitv 28892* | Convenient lemma for f1otrg 28893. (Contributed by Thierry Arnoux, 19-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐷 = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐵 = (Base‘𝐻) & ⊢ 𝐸 = (dist‘𝐻) & ⊢ 𝐽 = (Itv‘𝐻) & ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝑃) & ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵)) → (𝑒𝐸𝑓) = ((𝐹‘𝑒)𝐷(𝐹‘𝑓))) & ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹‘𝑔) ∈ ((𝐹‘𝑒)𝐼(𝐹‘𝑓)))) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐽𝑌) ↔ (𝐹‘𝑍) ∈ ((𝐹‘𝑋)𝐼(𝐹‘𝑌)))) | ||
Theorem | f1otrg 28893* | A bijection between bases which conserves distances and intervals conserves also geometries. (Contributed by Thierry Arnoux, 23-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐷 = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐵 = (Base‘𝐻) & ⊢ 𝐸 = (dist‘𝐻) & ⊢ 𝐽 = (Itv‘𝐻) & ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝑃) & ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵)) → (𝑒𝐸𝑓) = ((𝐹‘𝑒)𝐷(𝐹‘𝑓))) & ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹‘𝑔) ∈ ((𝐹‘𝑒)𝐼(𝐹‘𝑓)))) & ⊢ (𝜑 → 𝐻 ∈ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → (LineG‘𝐻) = (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐽𝑦) ∨ 𝑥 ∈ (𝑧𝐽𝑦) ∨ 𝑦 ∈ (𝑥𝐽𝑧))})) ⇒ ⊢ (𝜑 → 𝐻 ∈ TarskiG) | ||
Theorem | f1otrge 28894* | A bijection between bases which conserves distances and intervals conserves also the property of being a Euclidean geometry. (Contributed by Thierry Arnoux, 23-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐷 = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐵 = (Base‘𝐻) & ⊢ 𝐸 = (dist‘𝐻) & ⊢ 𝐽 = (Itv‘𝐻) & ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝑃) & ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵)) → (𝑒𝐸𝑓) = ((𝐹‘𝑒)𝐷(𝐹‘𝑓))) & ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹‘𝑔) ∈ ((𝐹‘𝑒)𝐼(𝐹‘𝑓)))) & ⊢ (𝜑 → 𝐻 ∈ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ TarskiGE) ⇒ ⊢ (𝜑 → 𝐻 ∈ TarskiGE) | ||
Syntax | cttg 28895 | Function to convert an algebraic structure to a Tarski geometry. |
class toTG | ||
Definition | df-ttg 28896* | Define a function converting a subcomplex Hilbert space to a Tarski Geometry. It does so by equipping the structure with a betweenness operation. Note that because the scalar product is applied over the interval (0[,]1), only spaces whose scalar field is a superset of that interval can be considered. (Contributed by Thierry Arnoux, 24-Mar-2019.) |
⊢ toTG = (𝑤 ∈ V ↦ ⦋(𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ ∃𝑘 ∈ (0[,]1)(𝑧(-g‘𝑤)𝑥) = (𝑘( ·𝑠 ‘𝑤)(𝑦(-g‘𝑤)𝑥))}) / 𝑖⦌((𝑤 sSet 〈(Itv‘ndx), 𝑖〉) sSet 〈(LineG‘ndx), (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})〉)) | ||
Theorem | ttgval 28897* | Define a function to augment a subcomplex Hilbert space with betweenness and a line definition. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Proof shortened by AV, 9-Nov-2024.) |
⊢ 𝐺 = (toTG‘𝐻) & ⊢ 𝐵 = (Base‘𝐻) & ⊢ − = (-g‘𝐻) & ⊢ · = ( ·𝑠 ‘𝐻) & ⊢ 𝐼 = (Itv‘𝐺) ⇒ ⊢ (𝐻 ∈ 𝑉 → (𝐺 = ((𝐻 sSet 〈(Itv‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})〉) sSet 〈(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})〉) ∧ 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))}))) | ||
Theorem | ttgvalOLD 28898* | Obsolete version of ttgval 28897 as of 9-Nov-2024. Define a function to augment a subcomplex Hilbert space with betweenness and a line definition. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐺 = (toTG‘𝐻) & ⊢ 𝐵 = (Base‘𝐻) & ⊢ − = (-g‘𝐻) & ⊢ · = ( ·𝑠 ‘𝐻) & ⊢ 𝐼 = (Itv‘𝐺) ⇒ ⊢ (𝐻 ∈ 𝑉 → (𝐺 = ((𝐻 sSet 〈(Itv‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})〉) sSet 〈(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})〉) ∧ 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))}))) | ||
Theorem | ttglem 28899 | Lemma for ttgbas 28901, ttgvsca 28906 etc. (Contributed by Thierry Arnoux, 15-Apr-2019.) (Revised by AV, 29-Oct-2024.) |
⊢ 𝐺 = (toTG‘𝐻) & ⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ (𝐸‘ndx) ≠ (LineG‘ndx) & ⊢ (𝐸‘ndx) ≠ (Itv‘ndx) ⇒ ⊢ (𝐸‘𝐻) = (𝐸‘𝐺) | ||
Theorem | ttglemOLD 28900 | Obsolete version of ttglem 28899 as of 29-Oct-2024. Lemma for ttgbas 28901 and ttgvsca 28906. (Contributed by Thierry Arnoux, 15-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐺 = (toTG‘𝐻) & ⊢ 𝐸 = Slot 𝑁 & ⊢ 𝑁 ∈ ℕ & ⊢ 𝑁 < ;16 ⇒ ⊢ (𝐸‘𝐻) = (𝐸‘𝐺) |
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