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Theorem List for Metamath Proof Explorer - 25401-25500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremragflat3 25401 Right angle and colinearity. Theorem 8.9 of [Schwabhauser] p. 58. (Contributed by Thierry Arnoux, 4-Sep-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))       (𝜑 → (𝐴 = 𝐵𝐶 = 𝐵))

Theoremragcgr 25402 Right angle and colinearity. Theorem 8.10 of [Schwabhauser] p. 58. (Contributed by Thierry Arnoux, 4-Sep-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &    = (cgrG‘𝐺)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)       (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))

Theoremmotrag 25403 Right angles are preserved by motions. (Contributed by Thierry Arnoux, 16-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐹 ∈ (𝐺Ismt𝐺))    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))       (𝜑 → ⟨“(𝐹𝐴)(𝐹𝐵)(𝐹𝐶)”⟩ ∈ (∟G‘𝐺))

Theoremragncol 25404 Right angle implies non-colinearity. A consequence of theorem 8.9 of [Schwabhauser] p. 58. (Contributed by Thierry Arnoux, 1-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   (𝜑𝐴𝐵)    &   (𝜑𝐶𝐵)       (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))

Theoremperpln1 25405 Derive a line from perpendicularity. (Contributed by Thierry Arnoux, 27-Nov-2019.)
𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴(⟂G‘𝐺)𝐵)       (𝜑𝐴 ∈ ran 𝐿)

Theoremperpln2 25406 Derive a line from perpendicularity. (Contributed by Thierry Arnoux, 27-Nov-2019.)
𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴(⟂G‘𝐺)𝐵)       (𝜑𝐵 ∈ ran 𝐿)

Theoremisperp 25407* Property for 2 lines A, B to be perpendicular. Item (ii) of definition 8.11 of [Schwabhauser] p. 59. (Contributed by Thierry Arnoux, 16-Oct-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴 ∈ ran 𝐿)    &   (𝜑𝐵 ∈ ran 𝐿)       (𝜑 → (𝐴(⟂G‘𝐺)𝐵 ↔ ∃𝑥 ∈ (𝐴𝐵)∀𝑢𝐴𝑣𝐵 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))

Theoremperpcom 25408 The "perpendicular" relation commutes. Theorem 8.12 of [Schwabhauser] p. 59. (Contributed by Thierry Arnoux, 16-Oct-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴 ∈ ran 𝐿)    &   (𝜑𝐵 ∈ ran 𝐿)    &   (𝜑𝐴(⟂G‘𝐺)𝐵)       (𝜑𝐵(⟂G‘𝐺)𝐴)

Theoremperpneq 25409 Two perpendicular lines are different. Theorem 8.14 of [Schwabhauser] p. 59. (Contributed by Thierry Arnoux, 18-Oct-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴 ∈ ran 𝐿)    &   (𝜑𝐵 ∈ ran 𝐿)    &   (𝜑𝐴(⟂G‘𝐺)𝐵)       (𝜑𝐴𝐵)

Theoremisperp2 25410* Property for 2 lines A, B, intersecting at a point X to be perpendicular. Item (i) of definition 8.13 of [Schwabhauser] p. 59. (Contributed by Thierry Arnoux, 16-Oct-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴 ∈ ran 𝐿)    &   (𝜑𝐵 ∈ ran 𝐿)    &   (𝜑𝑋 ∈ (𝐴𝐵))       (𝜑 → (𝐴(⟂G‘𝐺)𝐵 ↔ ∀𝑢𝐴𝑣𝐵 ⟨“𝑢𝑋𝑣”⟩ ∈ (∟G‘𝐺)))

Theoremisperp2d 25411 One direction of isperp2 25410. (Contributed by Thierry Arnoux, 10-Nov-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴 ∈ ran 𝐿)    &   (𝜑𝐵 ∈ ran 𝐿)    &   (𝜑𝑋 ∈ (𝐴𝐵))    &   (𝜑𝑈𝐴)    &   (𝜑𝑉𝐵)    &   (𝜑𝐴(⟂G‘𝐺)𝐵)       (𝜑 → ⟨“𝑈𝑋𝑉”⟩ ∈ (∟G‘𝐺))

Theoremragperp 25412 Deduce that two lines are perpendicular from a right angle statement. One direction of theorem 8.13 of [Schwabhauser] p. 59. (Contributed by Thierry Arnoux, 20-Oct-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴 ∈ ran 𝐿)    &   (𝜑𝐵 ∈ ran 𝐿)    &   (𝜑𝑋 ∈ (𝐴𝐵))    &   (𝜑𝑈𝐴)    &   (𝜑𝑉𝐵)    &   (𝜑𝑈𝑋)    &   (𝜑𝑉𝑋)    &   (𝜑 → ⟨“𝑈𝑋𝑉”⟩ ∈ (∟G‘𝐺))       (𝜑𝐴(⟂G‘𝐺)𝐵)

Theoremfootex 25413* Lemma for foot 25414: existence part. (Contributed by Thierry Arnoux, 19-Oct-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴 ∈ ran 𝐿)    &   (𝜑𝐶𝑃)    &   (𝜑 → ¬ 𝐶𝐴)       (𝜑 → ∃𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)

Theoremfoot 25414* From a point 𝐶 outside of a line 𝐴, there exists a unique point 𝑥 on 𝐴 such that (𝐶𝐿𝑥) is perpendicular to 𝐴. That point is called the foot from 𝐶 on 𝐴. Theorem 8.18 of [Schwabhauser] p. 60. (Contributed by Thierry Arnoux, 19-Oct-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴 ∈ ran 𝐿)    &   (𝜑𝐶𝑃)    &   (𝜑 → ¬ 𝐶𝐴)       (𝜑 → ∃!𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)

Theoremfootne 25415 Uniqueness of the foot point. (Contributed by Thierry Arnoux, 28-Feb-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴 ∈ ran 𝐿)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝑃)    &   (𝜑 → (𝑋𝐿𝑌)(⟂G‘𝐺)𝐴)       (𝜑 → ¬ 𝑌𝐴)

Theoremfooteq 25416 Uniqueness of the foot point. (Contributed by Thierry Arnoux, 1-Mar-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴 ∈ ran 𝐿)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   (𝜑𝑍𝑃)    &   (𝜑 → (𝑋𝐿𝑍)(⟂G‘𝐺)𝐴)    &   (𝜑 → (𝑌𝐿𝑍)(⟂G‘𝐺)𝐴)       (𝜑𝑋 = 𝑌)

Theoremhlperpnel 25417 A point on a half-line which is perpendicular to a line cannot be on that line. (Contributed by Thierry Arnoux, 1-Mar-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴 ∈ ran 𝐿)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝑈𝐴)    &   (𝜑𝑉𝑃)    &   (𝜑𝑊𝑃)    &   (𝜑𝐴(⟂G‘𝐺)(𝑈𝐿𝑉))    &   (𝜑𝑉(𝐾𝑈)𝑊)       (𝜑 → ¬ 𝑊𝐴)

Theoremperprag 25418 Deduce a right angle from perpendicular lines. (Contributed by Thierry Arnoux, 10-Nov-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶 ∈ (𝐴𝐿𝐵))    &   (𝜑𝐷𝑃)    &   (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐶𝐿𝐷))       (𝜑 → ⟨“𝐴𝐶𝐷”⟩ ∈ (∟G‘𝐺))

TheoremperpdragALT 25419 Deduce a right angle from perpendicular lines. (Contributed by Thierry Arnoux, 12-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝐷)    &   (𝜑𝐵𝐷)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷(⟂G‘𝐺)(𝐵𝐿𝐶))       (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))

Theoremperpdrag 25420 Deduce a right angle from perpendicular lines. (Contributed by Thierry Arnoux, 12-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝐷)    &   (𝜑𝐵𝐷)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷(⟂G‘𝐺)(𝐵𝐿𝐶))       (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))

Theoremcolperp 25421 Deduce a perpendicularity from perpendicularity and colinearity. (Contributed by Thierry Arnoux, 8-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)𝐷)    &   (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    &   (𝜑𝐴𝐶)       (𝜑 → (𝐴𝐿𝐶)(⟂G‘𝐺)𝐷)

Theoremcolperpexlem1 25422 Lemma for colperp 25421. First part of lemma 8.20 of [Schwabhauser] p. 62. (Contributed by Thierry Arnoux, 27-Oct-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑆 = (pInvG‘𝐺)    &   𝑀 = (𝑆𝐴)    &   𝑁 = (𝑆𝐵)    &   𝐾 = (𝑆𝑄)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝑄𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → (𝐾‘(𝑀𝐶)) = (𝑁𝐶))       (𝜑 → ⟨“𝐵𝐴𝑄”⟩ ∈ (∟G‘𝐺))

Theoremcolperpexlem2 25423 Lemma for colperpex 25425. Second part of lemma 8.20 of [Schwabhauser] p. 62. (Contributed by Thierry Arnoux, 10-Nov-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑆 = (pInvG‘𝐺)    &   𝑀 = (𝑆𝐴)    &   𝑁 = (𝑆𝐵)    &   𝐾 = (𝑆𝑄)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝑄𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → (𝐾‘(𝑀𝐶)) = (𝑁𝐶))    &   (𝜑𝐵𝐶)       (𝜑𝐴𝑄)

Theoremcolperpexlem3 25424* Lemma for colperpex 25425. Case 1 of theorem 8.21 of [Schwabhauser] p. 63. (Contributed by Thierry Arnoux, 20-Nov-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐴𝐵)    &   (𝜑 → ¬ 𝐶 ∈ (𝐴𝐿𝐵))       (𝜑 → ∃𝑝𝑃 ((𝐴𝐿𝑝)(⟂G‘𝐺)(𝐴𝐿𝐵) ∧ ∃𝑡𝑃 ((𝑡 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑡 ∈ (𝐶𝐼𝑝))))

Theoremcolperpex 25425* In dimension 2 and above, on a line (𝐴𝐿𝐵) there is always a perpendicular 𝑃 from 𝐴 on a given plane (here given by 𝐶, in case 𝐶 does not lie on the line). Theorem 8.21 of [Schwabhauser] p. 63. (Contributed by Thierry Arnoux, 20-Nov-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐴𝐵)    &   (𝜑𝐺DimTarskiG≥2)       (𝜑 → ∃𝑝𝑃 ((𝐴𝐿𝑝)(⟂G‘𝐺)(𝐴𝐿𝐵) ∧ ∃𝑡𝑃 ((𝑡 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑡 ∈ (𝐶𝐼𝑝))))

Theoremmideulem2 25426 Lemma for opphllem 25427, which is itself used for mideu 25430. (Contributed by Thierry Arnoux, 19-Feb-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐴𝐵)    &   (𝜑𝑄𝑃)    &   (𝜑𝑂𝑃)    &   (𝜑𝑇𝑃)    &   (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵))    &   (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂))    &   (𝜑𝑇 ∈ (𝐴𝐿𝐵))    &   (𝜑𝑇 ∈ (𝑄𝐼𝑂))    &   (𝜑𝑅𝑃)    &   (𝜑𝑅 ∈ (𝐵𝐼𝑄))    &   (𝜑 → (𝐴 𝑂) = (𝐵 𝑅))    &   (𝜑𝑋𝑃)    &   (𝜑𝑋 ∈ (𝑇𝐼𝐵))    &   (𝜑𝑋 ∈ (𝑅𝐼𝑂))    &   (𝜑𝑍𝑃)    &   (𝜑𝑋 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑍))    &   (𝜑 → (𝑋 𝑍) = (𝑋 𝑅))    &   (𝜑𝑀𝑃)    &   (𝜑𝑅 = ((𝑆𝑀)‘𝑍))       (𝜑𝐵 = 𝑀)

Theoremopphllem 25427* Lemma 8.24 of [Schwabhauser] p. 66. This is used later for mideulem 25428 and later for opphl 25446. (Contributed by Thierry Arnoux, 21-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐴𝐵)    &   (𝜑𝑄𝑃)    &   (𝜑𝑂𝑃)    &   (𝜑𝑇𝑃)    &   (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵))    &   (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂))    &   (𝜑𝑇 ∈ (𝐴𝐿𝐵))    &   (𝜑𝑇 ∈ (𝑄𝐼𝑂))    &   (𝜑𝑅𝑃)    &   (𝜑𝑅 ∈ (𝐵𝐼𝑄))    &   (𝜑 → (𝐴 𝑂) = (𝐵 𝑅))       (𝜑 → ∃𝑥𝑃 (𝐵 = ((𝑆𝑥)‘𝐴) ∧ 𝑂 = ((𝑆𝑥)‘𝑅)))

Theoremmideulem 25428* Lemma for mideu 25430. We can assume mideulem.9 "without loss of generality" (Contributed by Thierry Arnoux, 25-Nov-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐴𝐵)    &   (𝜑𝑄𝑃)    &   (𝜑𝑂𝑃)    &   (𝜑𝑇𝑃)    &   (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵))    &   (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂))    &   (𝜑𝑇 ∈ (𝐴𝐿𝐵))    &   (𝜑𝑇 ∈ (𝑄𝐼𝑂))    &   (𝜑 → (𝐴 𝑂)(≤G‘𝐺)(𝐵 𝑄))       (𝜑 → ∃𝑥𝑃 𝐵 = ((𝑆𝑥)‘𝐴))

Theoremmidex 25429* Existence of the midpoint, part Theorem 8.22 of [Schwabhauser] p. 64. Note that this proof requires a construction in 2 dimensions or more, i.e. it does not prove the existence of a midpoint in dimension 1, for a geometry restricted to a line. (Contributed by Thierry Arnoux, 25-Nov-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐺DimTarskiG≥2)       (𝜑 → ∃𝑥𝑃 𝐵 = ((𝑆𝑥)‘𝐴))

Theoremmideu 25430* Existence and uniqueness of the midpoint, Theorem 8.22 of [Schwabhauser] p. 64. (Contributed by Thierry Arnoux, 25-Nov-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐺DimTarskiG≥2)       (𝜑 → ∃!𝑥𝑃 𝐵 = ((𝑆𝑥)‘𝐴))

15.2.14  Half-planes

Theoremislnopp 25431* The property for two points 𝐴 and 𝐵 to lie on the opposite sides of a set 𝐷 Definition 9.1 of [Schwabhauser] p. 67. (Contributed by Thierry Arnoux, 19-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))))

Theoremislnoppd 25432* Deduce that 𝐴 and 𝐵 lie on opposite sides of line 𝐿. (Contributed by Thierry Arnoux, 16-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝐷)    &   (𝜑 → ¬ 𝐴𝐷)    &   (𝜑 → ¬ 𝐵𝐷)    &   (𝜑𝐶 ∈ (𝐴𝐼𝐵))       (𝜑𝐴𝑂𝐵)

Theoremoppne1 25433* 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 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐴𝑂𝐵)       (𝜑 → ¬ 𝐴𝐷)

Theoremoppne2 25434* 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 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐴𝑂𝐵)       (𝜑 → ¬ 𝐵𝐷)

Theoremoppne3 25435* 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 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐴𝑂𝐵)       (𝜑𝐴𝐵)

Theoremoppcom 25436* 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 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐴𝑂𝐵)       (𝜑𝐵𝑂𝐴)

Theoremopptgdim2 25437* 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 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐴𝑂𝐵)       (𝜑𝐺DimTarskiG≥2)

Theoremoppnid 25438* 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 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)       (𝜑 → ¬ 𝐴𝑂𝐴)

Theoremopphllem1 25439* Lemma for opphl 25446. (Contributed by Thierry Arnoux, 20-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑆 = ((pInvG‘𝐺)‘𝑀)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝑅𝐷)    &   (𝜑𝐴𝑂𝐶)    &   (𝜑𝑀𝐷)    &   (𝜑𝐴 = (𝑆𝐶))    &   (𝜑𝐴𝑅)    &   (𝜑𝐵𝑅)    &   (𝜑𝐵 ∈ (𝑅𝐼𝐴))       (𝜑𝐵𝑂𝐶)

Theoremopphllem2 25440* Lemma for opphl 25446. 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 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑆 = ((pInvG‘𝐺)‘𝑀)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝑅𝐷)    &   (𝜑𝐴𝑂𝐶)    &   (𝜑𝑀𝐷)    &   (𝜑𝐴 = (𝑆𝐶))    &   (𝜑𝐴𝑅)    &   (𝜑𝐵𝑅)    &   (𝜑 → (𝐴 ∈ (𝑅𝐼𝐵) ∨ 𝐵 ∈ (𝑅𝐼𝐴)))       (𝜑𝐵𝑂𝐶)

Theoremopphllem3 25441* Lemma for opphl 25446: 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 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐺 ∈ TarskiG)    &   𝐾 = (hlG‘𝐺)    &   𝑁 = ((pInvG‘𝐺)‘𝑀)    &   (𝜑𝐴𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝑅𝐷)    &   (𝜑𝑆𝐷)    &   (𝜑𝑀𝑃)    &   (𝜑𝐴𝑂𝐶)    &   (𝜑𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))    &   (𝜑𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))    &   (𝜑𝑅𝑆)    &   (𝜑 → (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴))    &   (𝜑𝑈𝑃)    &   (𝜑 → (𝑁𝑅) = 𝑆)       (𝜑 → (𝑈(𝐾𝑅)𝐴 ↔ (𝑁𝑈)(𝐾𝑆)𝐶))

Theoremopphllem4 25442* Lemma for opphl 25446. (Contributed by Thierry Arnoux, 22-Feb-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐺 ∈ TarskiG)    &   𝐾 = (hlG‘𝐺)    &   𝑁 = ((pInvG‘𝐺)‘𝑀)    &   (𝜑𝐴𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝑅𝐷)    &   (𝜑𝑆𝐷)    &   (𝜑𝑀𝑃)    &   (𝜑𝐴𝑂𝐶)    &   (𝜑𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))    &   (𝜑𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))    &   (𝜑𝑅𝑆)    &   (𝜑 → (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴))    &   (𝜑𝑈𝑃)    &   (𝜑 → (𝑁𝑅) = 𝑆)    &   (𝜑𝑉𝑃)    &   (𝜑𝑈(𝐾𝑅)𝐴)    &   (𝜑𝑉(𝐾𝑆)𝐶)       (𝜑𝑈𝑂𝑉)

Theoremopphllem5 25443* 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 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐺 ∈ TarskiG)    &   𝐾 = (hlG‘𝐺)    &   𝑁 = ((pInvG‘𝐺)‘𝑀)    &   (𝜑𝐴𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝑅𝐷)    &   (𝜑𝑆𝐷)    &   (𝜑𝑀𝑃)    &   (𝜑𝐴𝑂𝐶)    &   (𝜑𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))    &   (𝜑𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))    &   (𝜑𝑈𝑃)    &   (𝜑𝑉𝑃)    &   (𝜑𝑈(𝐾𝑅)𝐴)    &   (𝜑𝑉(𝐾𝑆)𝐶)       (𝜑𝑈𝑂𝑉)

Theoremopphllem6 25444* 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 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐺 ∈ TarskiG)    &   𝐾 = (hlG‘𝐺)    &   𝑁 = ((pInvG‘𝐺)‘𝑀)    &   (𝜑𝐴𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝑅𝐷)    &   (𝜑𝑆𝐷)    &   (𝜑𝑀𝑃)    &   (𝜑𝐴𝑂𝐶)    &   (𝜑𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))    &   (𝜑𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))    &   (𝜑𝑈𝑃)    &   (𝜑 → (𝑁𝑅) = 𝑆)       (𝜑 → (𝑈(𝐾𝑅)𝐴 ↔ (𝑁𝑈)(𝐾𝑆)𝐶))

Theoremoppperpex 25445* Restating colperpex 25425 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 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐺 ∈ TarskiG)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝐷)    &   (𝜑𝐶𝑃)    &   (𝜑 → ¬ 𝐶𝐷)    &   (𝜑𝐺DimTarskiG≥2)       (𝜑 → ∃𝑝𝑃 ((𝐴𝐿𝑝)(⟂G‘𝐺)𝐷𝐶𝑂𝑝))

Theoremopphl 25446* If two points 𝐴 and 𝐶 lie on the opposite side of a line 𝐷 then any point of the half line (𝑅 𝐴) also lies opposite to 𝐶. Theorem 9.5 of [Schwabhauser] p. 69. (Contributed by Thierry Arnoux, 3-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐺 ∈ TarskiG)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐴𝑂𝐶)    &   (𝜑𝑅𝐷)    &   (𝜑𝐴(𝐾𝑅)𝐵)       (𝜑𝐵𝑂𝐶)

Theoremoutpasch 25447* 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝑅𝑃)    &   (𝜑𝑄𝑃)    &   (𝜑𝐶 ∈ (𝐴𝐼𝑅))    &   (𝜑𝑄 ∈ (𝐵𝐼𝐶))       (𝜑 → ∃𝑥𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)))

Theoremhlpasch 25448* 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝑋𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐴𝐵)    &   (𝜑𝐶(𝐾𝐵)𝐷)    &   (𝜑𝐴 ∈ (𝑋𝐼𝐶))       (𝜑 → ∃𝑒𝑃 (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))

Syntaxchpg 25449 "Belong to the same open half-plane" relation for points in a geometry.
class hpG

Definitiondf-hpg 25450* Define the open half plane relation for a geometry 𝐺. Definition 9.7 of [Schwabhauser] p. 71. See hpgbr 25452 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]𝑐𝑝 (((𝑎 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝑖𝑐)) ∧ ((𝑏 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝑖𝑐)))}))

Theoremishpg 25451* 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 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)       (𝜑 → ((hpG‘𝐺)‘𝐷) = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (𝑎𝑂𝑐𝑏𝑂𝑐)})

Theoremhpgbr 25452* 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 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))

Theoremhpgne1 25453* 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 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐴((hpG‘𝐺)‘𝐷)𝐵)       (𝜑 → ¬ 𝐴𝐷)

Theoremhpgne2 25454* 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 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐴((hpG‘𝐺)‘𝐷)𝐵)       (𝜑 → ¬ 𝐵𝐷)

Theoremlnopp2hpgb 25455* 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 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐴𝑂𝐶)       (𝜑 → (𝐵𝑂𝐶𝐴((hpG‘𝐺)‘𝐷)𝐵))

Theoremlnoppnhpg 25456* 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 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐴𝑂𝐵)       (𝜑 → ¬ 𝐴((hpG‘𝐺)‘𝐷)𝐵)

Theoremhpgerlem 25457* 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑 → ¬ 𝐴𝐷)       (𝜑 → ∃𝑐𝑃 𝐴𝑂𝑐)

Theoremhpgid 25458* 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑 → ¬ 𝐴𝐷)       (𝜑𝐴((hpG‘𝐺)‘𝐷)𝐴)

Theoremhpgcom 25459* 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐵𝑃)    &   (𝜑𝐴((hpG‘𝐺)‘𝐷)𝐵)       (𝜑𝐵((hpG‘𝐺)‘𝐷)𝐴)

Theoremhpgtr 25460* 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐵𝑃)    &   (𝜑𝐴((hpG‘𝐺)‘𝐷)𝐵)    &   (𝜑𝐶𝑃)    &   (𝜑𝐵((hpG‘𝐺)‘𝐷)𝐶)       (𝜑𝐴((hpG‘𝐺)‘𝐷)𝐶)

Theoremcolopp 25461* 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝐷)    &   (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))       (𝜑 → (𝐴𝑂𝐵 ↔ (𝐶 ∈ (𝐴𝐼𝐵) ∧ ¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷)))

Theoremcolhp 25462* 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝐷)    &   (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    &   𝐾 = (hlG‘𝐺)       (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ (𝐴(𝐾𝐶)𝐵 ∧ ¬ 𝐴𝐷)))

Theoremhphl 25463* 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝐷)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑 → ¬ 𝐵𝐷)    &   (𝜑𝐵(𝐾𝐴)𝐶)       (𝜑𝐵((hpG‘𝐺)‘𝐷)𝐶)

15.2.15  Midpoints and Line Mirroring

Syntaxcmid 25464 Declare the constant for the midpoint operation.
class midG

Syntaxclmi 25465 Declare the constant for the line mirroring function.
class lInvG

Definitiondf-mid 25466* Define the midpoint operation. Definition 10.1 of [Schwabhauser] p. 88. See ismidb 25470, midbtwn 25471, and midcgr 25472. (Contributed by Thierry Arnoux, 9-Jun-2019.)
midG = (𝑔 ∈ V ↦ (𝑎 ∈ (Base‘𝑔), 𝑏 ∈ (Base‘𝑔) ↦ (𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎))))

Definitiondf-lmi 25467* Define the line mirroring function. Definition 10.3 of [Schwabhauser] p. 89. See islmib 25479. (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))))))

Theoremmidf 25468 Midpoint as a function. (Contributed by Thierry Arnoux, 1-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)       (𝜑 → (midG‘𝐺):(𝑃 × 𝑃)⟶𝑃)

Theoremmidcl 25469 Closure of the midpoint. (Contributed by Thierry Arnoux, 1-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑 → (𝐴(midG‘𝐺)𝐵) ∈ 𝑃)

Theoremismidb 25470 Property of the midpoint. (Contributed by Thierry Arnoux, 1-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝑀𝑃)       (𝜑 → (𝐵 = ((𝑆𝑀)‘𝐴) ↔ (𝐴(midG‘𝐺)𝐵) = 𝑀))

Theoremmidbtwn 25471 Betweenness of midpoint. (Contributed by Thierry Arnoux, 7-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑 → (𝐴(midG‘𝐺)𝐵) ∈ (𝐴𝐼𝐵))

Theoremmidcgr 25472 Congruence of midpoint. (Contributed by Thierry Arnoux, 7-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑 → (𝐴(midG‘𝐺)𝐵) = 𝐶)       (𝜑 → (𝐶 𝐴) = (𝐶 𝐵))

Theoremmidid 25473 Midpoint of a null segment. (Contributed by Thierry Arnoux, 7-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑 → (𝐴(midG‘𝐺)𝐴) = 𝐴)

Theoremmidcom 25474 Commutativity rule for the midpoint. (Contributed by Thierry Arnoux, 2-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑 → (𝐴(midG‘𝐺)𝐵) = (𝐵(midG‘𝐺)𝐴))

Theoremmirmid 25475 Point inversion preserves midpoints. (Contributed by Thierry Arnoux, 12-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   𝑆 = ((pInvG‘𝐺)‘𝑀)    &   (𝜑𝑀𝑃)       (𝜑 → ((𝑆𝐴)(midG‘𝐺)(𝑆𝐵)) = (𝑆‘(𝐴(midG‘𝐺)𝐵)))

Theoremlmieu 25476* 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)       (𝜑 → ∃!𝑏𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))

Theoremlmif 25477 Line mirror as a function. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)       (𝜑𝑀:𝑃𝑃)

Theoremlmicl 25478 Closure of the line mirror. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)       (𝜑 → (𝑀𝐴) ∈ 𝑃)

Theoremislmib 25479 Property of the line mirror. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑 → (𝐵 = (𝑀𝐴) ↔ ((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))))

Theoremlmicom 25480 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑 → (𝑀𝐴) = 𝐵)       (𝜑 → (𝑀𝐵) = 𝐴)

Theoremlmilmi 25481 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)       (𝜑 → (𝑀‘(𝑀𝐴)) = 𝐴)

Theoremlmireu 25482* 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)       (𝜑 → ∃!𝑏𝑃 (𝑀𝑏) = 𝐴)

Theoremlmieq 25483 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑 → (𝑀𝐴) = (𝑀𝐵))       (𝜑𝐴 = 𝐵)

Theoremlmiinv 25484 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)       (𝜑 → ((𝑀𝐴) = 𝐴𝐴𝐷))

Theoremlmicinv 25485 The mirroring line is an invariant. (Contributed by Thierry Arnoux, 8-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐴𝐷)       (𝜑 → (𝑀𝐴) = 𝐴)

Theoremlmimid 25486 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   𝑆 = ((pInvG‘𝐺)‘𝐵)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   (𝜑𝐴𝐷)    &   (𝜑𝐵𝐷)    &   (𝜑𝐶𝑃)    &   (𝜑𝐴𝐵)       (𝜑 → (𝑀𝐶) = (𝑆𝐶))

Theoremlmif1o 25487 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)       (𝜑𝑀:𝑃1-1-onto𝑃)

Theoremlmiisolem 25488 Lemma for lmiiso 25489. (Contributed by Thierry Arnoux, 14-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   𝑆 = ((pInvG‘𝐺)‘𝑍)    &   𝑍 = ((𝐴(midG‘𝐺)(𝑀𝐴))(midG‘𝐺)(𝐵(midG‘𝐺)(𝑀𝐵)))       (𝜑 → ((𝑀𝐴) (𝑀𝐵)) = (𝐴 𝐵))

Theoremlmiiso 25489 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑 → ((𝑀𝐴) (𝑀𝐵)) = (𝐴 𝐵))

Theoremlmimot 25490 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)       (𝜑𝑀 ∈ (𝐺Ismt𝐺))

Theoremhypcgrlem1 25491 Lemma for hypcgr 25493, case where triangles share a cathetus. (Contributed by Thierry Arnoux, 15-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))    &   (𝜑𝐵 = 𝐸)    &   𝑆 = ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵))    &   (𝜑𝐶 = 𝐹)       (𝜑 → (𝐴 𝐶) = (𝐷 𝐹))

Theoremhypcgrlem2 25492 Lemma for hypcgr 25493, case where triangles share one vertex 𝐵. (Contributed by Thierry Arnoux, 16-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))    &   (𝜑𝐵 = 𝐸)    &   𝑆 = ((lInvG‘𝐺)‘((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵))       (𝜑 → (𝐴 𝐶) = (𝐷 𝐹))

Theoremhypcgr 25493 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))       (𝜑 → (𝐴 𝐶) = (𝐷 𝐹))

Theoremlmiopp 25494* 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐷 ∈ ran 𝐿)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   (𝜑𝐴𝑃)    &   (𝜑 → ¬ 𝐴𝐷)       (𝜑𝐴𝑂(𝑀𝐴))

Theoremlnperpex 25495* 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐷 ∈ ran 𝐿)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐴𝐷)    &   (𝜑𝑄𝑃)    &   (𝜑 → ¬ 𝑄𝐷)       (𝜑 → ∃𝑝𝑃 (𝐷(⟂G‘𝐺)(𝑝𝐿𝐴) ∧ 𝑝((hpG‘𝐺)‘𝐷)𝑄))

Theoremtrgcopy 25496* 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))       (𝜑 → ∃𝑓𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))

Theoremtrgcopyeulem 25497* Lemma for trgcopyeu 25498. (Contributed by Thierry Arnoux, 8-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (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))}    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑋”⟩)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑌”⟩)    &   (𝜑𝑋((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)    &   (𝜑𝑌((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)       (𝜑𝑋 = 𝑌)

Theoremtrgcopyeu 25498* 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))       (𝜑 → ∃!𝑓𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))

15.2.16  Congruence of angles

Syntaxccgra 25499 Declare the constant for the congruence between angles relation.
class cgrA

Definitiondf-cgra 25500* Define the congruence relation bewteen angles. As for triangles we use "words of points". See iscgra 25501 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 ∈ (𝑝𝑚 (0..^3)) ∧ 𝑏 ∈ (𝑝𝑚 (0..^3))) ∧ ∃𝑥𝑝𝑦𝑝 (𝑎(cgrG‘𝑔)⟨“𝑥(𝑏‘1)𝑦”⟩ ∧ 𝑥(𝑘‘(𝑏‘1))(𝑏‘0) ∧ 𝑦(𝑘‘(𝑏‘1))(𝑏‘2)))})

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