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Theorem List for Metamath Proof Explorer - 25701-25800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlnoppnhpg 25701* 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 25702* 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 25703* 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 25704* 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 25705* 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 25706* 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 25707* 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 25708* 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 25709 Declare the constant for the midpoint operation.
class midG

Syntaxclmi 25710 Declare the constant for the line mirroring function.
class lInvG

Definitiondf-mid 25711* Define the midpoint operation. Definition 10.1 of [Schwabhauser] p. 88. See ismidb 25715, midbtwn 25716, and midcgr 25717. (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 25712* Define the line mirroring function. Definition 10.3 of [Schwabhauser] p. 89. See islmib 25724. (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 25713 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 25714 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 25715 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 25716 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 25717 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 25718 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 25719 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 25720 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 25721* 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 25722 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 25723 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 25724 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 25725 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 25726 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 25727* 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 25728 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 25729 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 25730 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 25731 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 25732 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 25733 Lemma for lmiiso 25734. (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 25734 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 25735 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 25736 Lemma for hypcgr 25738, 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 25737 Lemma for hypcgr 25738, 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 25738 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 25739* 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 25740* 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 25741* 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 25742* Lemma for trgcopyeu 25743. (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 25743* 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 25744 Declare the constant for the congruence between angles relation.
class cgrA

Definitiondf-cgra 25745* Define the congruence relation bewteen angles. As for triangles we use "words of points". See iscgra 25746 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)))})

Theoremiscgra 25746* 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 25766 shows that those definitions are indeed equivalent. (Contributed by Thierry Arnoux, 31-Jul-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)       (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑥𝑃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)))

Theoremiscgra1 25747* A special version of iscgra 25746 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &    = (dist‘𝐺)    &   (𝜑𝐴𝐵)    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))       (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)))

Theoremiscgrad 25748 Sufficient conditions for angle congruence, deduction version. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑋𝐸𝑌”⟩)    &   (𝜑𝑋(𝐾𝐸)𝐷)    &   (𝜑𝑌(𝐾𝐸)𝐹)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)

Theoremcgrane1 25749 Angles imply inequality. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)       (𝜑𝐴𝐵)

Theoremcgrane2 25750 Angles imply inequality. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)       (𝜑𝐵𝐶)

Theoremcgrane3 25751 Angles imply inequality. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)       (𝜑𝐸𝐷)

Theoremcgrane4 25752 Angles imply inequality. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)       (𝜑𝐸𝐹)

Theoremcgrahl1 25753 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑𝑋𝑃)    &   (𝜑𝑋(𝐾𝐸)𝐷)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑋𝐸𝐹”⟩)

Theoremcgrahl2 25754 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑𝑋𝑃)    &   (𝜑𝑋(𝐾𝐸)𝐹)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑋”⟩)

Theoremcgracgr 25755 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑𝑋𝑃)    &    = (dist‘𝐺)    &   (𝜑𝑌𝑃)    &   (𝜑𝑋(𝐾𝐵)𝐴)    &   (𝜑𝑌(𝐾𝐵)𝐶)    &   (𝜑 → (𝐵 𝑋) = (𝐸 𝐷))    &   (𝜑 → (𝐵 𝑌) = (𝐸 𝐹))       (𝜑 → (𝑋 𝑌) = (𝐷 𝐹))

Theoremcgraid 25756 Angle congruence is reflexive. Theorem 11.6 of [Schwabhauser] p. 96. (Contributed by Thierry Arnoux, 31-Jul-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)

Theoremcgraswap 25757 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)    &   (𝜑𝐺 ∈ TarskiG)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐶𝐵𝐴”⟩)

Theoremcgrcgra 25758 Triangle congruence implies angle congruence. (Contributed by Thierry Arnoux, 2-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)

Theoremcgracom 25759 Angle congruence commutes. Theorem 11.7 of [Schwabhauser] p. 97. (Contributed by Thierry Arnoux, 5-Mar-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)       (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)

Theoremcgratr 25760 Angle congruence is transitive. Theorem 11.8 of [Schwabhauser] p. 97. (Contributed by Thierry Arnoux, 5-Mar-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑𝐻𝑃)    &   (𝜑𝑈𝑃)    &   (𝜑𝐽𝑃)    &   (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐻𝑈𝐽”⟩)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐻𝑈𝐽”⟩)

Theoremcgraswaplr 25761 Swap both side of angle congruence. (Contributed by Thierry Arnoux, 5-Oct-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (dist‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)       (𝜑 → ⟨“𝐶𝐵𝐴”⟩(cgrA‘𝐺)⟨“𝐹𝐸𝐷”⟩)

Theoremcgrabtwn 25762 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑𝐵 ∈ (𝐴𝐼𝐶))       (𝜑𝐸 ∈ (𝐷𝐼𝐹))

Theoremcgrahl 25763 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴(𝐾𝐵)𝐶)       (𝜑𝐷(𝐾𝐸)𝐹)

Theoremcgracol 25764 Angle congruence preserves colinearity. (Contributed by Thierry Arnoux, 9-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (dist‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))       (𝜑 → (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))

Theoremcgrancol 25765 Angle congruence preserves non-colinearity. (Contributed by Thierry Arnoux, 9-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (dist‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))       (𝜑 → ¬ (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))

Theoremdfcgra2 25766* 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 25745 is indeed equivalent with the textbook's definition. (Contributed by Thierry Arnoux, 2-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (dist‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)       (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸) ∧ ∃𝑎𝑃𝑐𝑃𝑑𝑃𝑓𝑃 (((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)))))

Theoremsacgr 25767 Supplementary angles of congruent angles are themselves congruent. Theorem 11.13 of [Schwabhauser] p. 98. (Contributed by Thierry Arnoux, 30-Sep-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (dist‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑𝐵 ∈ (𝐴𝐼𝑋))    &   (𝜑𝐸 ∈ (𝐷𝐼𝑌))    &   (𝜑𝐵𝑋)    &   (𝜑𝐸𝑌)       (𝜑 → ⟨“𝑋𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑌𝐸𝐹”⟩)

Theoremoacgr 25768 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑𝐵 ∈ (𝐴𝐼𝐷))    &   (𝜑𝐵 ∈ (𝐶𝐼𝐹))    &   (𝜑𝐵𝐴)    &   (𝜑𝐵𝐶)    &   (𝜑𝐵𝐷)    &   (𝜑𝐵𝐹)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐵𝐹”⟩)

Theoremacopy 25769* 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))       (𝜑 → ∃𝑓𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))

Theoremacopyeu 25770 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑋”⟩)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑌”⟩)    &   (𝜑𝑋((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)    &   (𝜑𝑌((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)       (𝜑𝑋(𝐾𝐸)𝑌)

15.2.17  Angle Comparisons

Syntaxcinag 25771 Extend class relation with the geometrical "point in angle" relation.
class inA

Syntaxcleag 25772 Extend class relation with the "angle less than" relation.
class

Definitiondf-inag 25773* 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))))})

Theoremisinag 25774* Property for point 𝑋 to lie in the angle ⟨“𝐴𝐵𝐶”⟩ Defnition 11.23 of [Schwabhauser] p. 101. (Contributed by Thierry Arnoux, 15-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝑋𝑃)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐺𝑉)       (𝜑 → (𝑋(inA‘𝐺)⟨“𝐴𝐵𝐶”⟩ ↔ ((𝐴𝐵𝐶𝐵𝑋𝐵) ∧ ∃𝑥𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵𝑥(𝐾𝐵)𝑋)))))

Theoreminagswap 25775 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)    &   (𝜑𝑋𝑃)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋(inA‘𝐺)⟨“𝐴𝐵𝐶”⟩)       (𝜑𝑋(inA‘𝐺)⟨“𝐶𝐵𝐴”⟩)

Theoreminaghl 25776 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)    &   (𝜑𝑋𝑃)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋(inA‘𝐺)⟨“𝐴𝐵𝐶”⟩)    &   (𝜑𝐷𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝐷(𝐾𝐵)𝐴)    &   (𝜑𝐹(𝐾𝐵)𝐶)    &   (𝜑𝑌(𝐾𝐵)𝑋)       (𝜑𝑌(inA‘𝐺)⟨“𝐷𝐵𝐹”⟩)

Definitiondf-leag 25777* 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”⟩))})

Theoremisleag 25778* Geometrical "less than" property for angles. Definition 11.27 of [Schwabhauser] p. 102. (Contributed by Thierry Arnoux, 7-Oct-2020.)
𝑃 = (Base‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)       (𝜑 → (⟨“𝐴𝐵𝐶”⟩(≤𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑥𝑃 (𝑥(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑥”⟩)))

Theoremcgrg3col4 25779* 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝑋𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))       (𝜑 → ∃𝑦𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)

15.2.18  Congruence Theorems

Theoremtgsas1 25780 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))       (𝜑 → (𝐶 𝐴) = (𝐹 𝐷))

Theoremtgsas 25781 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))       (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)

Theoremtgsas2 25782 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))    &   (𝜑𝐴𝐶)       (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩)

Theoremtgsas3 25783 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))    &   (𝜑𝐴𝐶)       (𝜑 → ⟨“𝐵𝐶𝐴”⟩(cgrA‘𝐺)⟨“𝐸𝐹𝐷”⟩)

Theoremtgasa1 25784 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   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))

Theoremtgasa 25785 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   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”⟩)

Theoremtgsss1 25786 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))    &   (𝜑 → (𝐶 𝐴) = (𝐹 𝐷))    &   (𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)    &   (𝜑𝐶𝐴)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)

Theoremtgsss2 25787 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))    &   (𝜑 → (𝐶 𝐴) = (𝐹 𝐷))    &   (𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)    &   (𝜑𝐶𝐴)       (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩)

Theoremtgsss3 25788 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))    &   (𝜑 → (𝐶 𝐴) = (𝐹 𝐷))    &   (𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)    &   (𝜑𝐶𝐴)       (𝜑 → ⟨“𝐵𝐶𝐴”⟩(cgrA‘𝐺)⟨“𝐸𝐹𝐷”⟩)

Theoremisoas 25789 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐵”⟩)       (𝜑 → (𝐴 𝐵) = (𝐴 𝐶))

15.2.19  Equilateral triangles

Syntaxceqlg 25790 Declare the class of equilateral triangles.
class eqltrG

Definitiondf-eqlg 25791* 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)”⟩})

Theoremiseqlg 25792 Property of a triangle being equilateral. (Contributed by Thierry Arnoux, 5-Oct-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)       (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ∈ (eqltrG‘𝐺) ↔ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐵𝐶𝐴”⟩))

Theoremiseqlgd 25793 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐵 𝐶))    &   (𝜑 → (𝐵 𝐶) = (𝐶 𝐴))    &   (𝜑 → (𝐶 𝐴) = (𝐴 𝐵))       (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (eqltrG‘𝐺))

15.3  Properties of geometries

15.3.1  Isomorphisms between geometries

Theoremf1otrgds 25794* Convenient lemma for f1otrg 25796. (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)    &   (𝜑𝐹:𝐵1-1-onto𝑃)    &   ((𝜑 ∧ (𝑒𝐵𝑓𝐵)) → (𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)))    &   ((𝜑 ∧ (𝑒𝐵𝑓𝐵𝑔𝐵)) → (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑒)𝐼(𝐹𝑓))))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐸𝑌) = ((𝐹𝑋)𝐷(𝐹𝑌)))

Theoremf1otrgitv 25795* Convenient lemma for f1otrg 25796. (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)    &   (𝜑𝐹:𝐵1-1-onto𝑃)    &   ((𝜑 ∧ (𝑒𝐵𝑓𝐵)) → (𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)))    &   ((𝜑 ∧ (𝑒𝐵𝑓𝐵𝑔𝐵)) → (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑒)𝐼(𝐹𝑓))))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → (𝑍 ∈ (𝑋𝐽𝑌) ↔ (𝐹𝑍) ∈ ((𝐹𝑋)𝐼(𝐹𝑌))))

Theoremf1otrg 25796* 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)    &   (𝜑𝐹:𝐵1-1-onto𝑃)    &   ((𝜑 ∧ (𝑒𝐵𝑓𝐵)) → (𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)))    &   ((𝜑 ∧ (𝑒𝐵𝑓𝐵𝑔𝐵)) → (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑒)𝐼(𝐹𝑓))))    &   (𝜑𝐻𝑉)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑 → (LineG‘𝐻) = (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐽𝑦) ∨ 𝑥 ∈ (𝑧𝐽𝑦) ∨ 𝑦 ∈ (𝑥𝐽𝑧))}))       (𝜑𝐻 ∈ TarskiG)

Theoremf1otrge 25797* 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)    &   (𝜑𝐹:𝐵1-1-onto𝑃)    &   ((𝜑 ∧ (𝑒𝐵𝑓𝐵)) → (𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)))    &   ((𝜑 ∧ (𝑒𝐵𝑓𝐵𝑔𝐵)) → (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑒)𝐼(𝐹𝑓))))    &   (𝜑𝐻𝑉)    &   (𝜑𝐺 ∈ TarskiGE)       (𝜑𝐻 ∈ TarskiGE)

15.4  Geometry in Hilbert spaces

Syntaxcttg 25798 Function to convert an algebraic structure to a Tarski geometry.
class toTG

Definitiondf-ttg 25799* 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(-g𝑤)𝑥) = (𝑘( ·𝑠𝑤)(𝑦(-g𝑤)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))})⟩))

Theoremttgval 25800* Define a function to augment a subcomplex Hilbert space with betweenness and a line definition. (Contributed by Thierry Arnoux, 25-Mar-2019.)
𝐺 = (toTG‘𝐻)    &   𝐵 = (Base‘𝐻)    &    = (-g𝐻)    &    · = ( ·𝑠𝐻)    &   𝐼 = (Itv‘𝐺)       (𝐻𝑉 → (𝐺 = ((𝐻 sSet ⟨(Itv‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})⟩) ∧ 𝐼 = (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})))

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