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Theorem List for Metamath Proof Explorer - 27501-27600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxclmi 27501 Declare the constant for the line mirroring function.
class lInvG
 
Definitiondf-mid 27502* Define the midpoint operation. Definition 10.1 of [Schwabhauser] p. 88. See ismidb 27506, midbtwn 27507, and midcgr 27508. (Contributed by Thierry Arnoux, 9-Jun-2019.)
midG = (𝑔 ∈ V ↦ (π‘Ž ∈ (Baseβ€˜π‘”), 𝑏 ∈ (Baseβ€˜π‘”) ↦ (β„©π‘š ∈ (Baseβ€˜π‘”)𝑏 = (((pInvGβ€˜π‘”)β€˜π‘š)β€˜π‘Ž))))
 
Definitiondf-lmi 27503* Define the line mirroring function. Definition 10.3 of [Schwabhauser] p. 89. See islmib 27515. (Contributed by Thierry Arnoux, 1-Dec-2019.)
lInvG = (𝑔 ∈ V ↦ (π‘š ∈ ran (LineGβ€˜π‘”) ↦ (π‘Ž ∈ (Baseβ€˜π‘”) ↦ (℩𝑏 ∈ (Baseβ€˜π‘”)((π‘Ž(midGβ€˜π‘”)𝑏) ∈ π‘š ∧ (π‘š(βŸ‚Gβ€˜π‘”)(π‘Ž(LineGβ€˜π‘”)𝑏) ∨ π‘Ž = 𝑏))))))
 
Theoremmidf 27504 Midpoint as a function. (Contributed by Thierry Arnoux, 1-Dec-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    β‡’   (πœ‘ β†’ (midGβ€˜πΊ):(𝑃 Γ— 𝑃)βŸΆπ‘ƒ)
 
Theoremmidcl 27505 Closure of the midpoint. (Contributed by Thierry Arnoux, 1-Dec-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    β‡’   (πœ‘ β†’ (𝐴(midGβ€˜πΊ)𝐡) ∈ 𝑃)
 
Theoremismidb 27506 Property of the midpoint. (Contributed by Thierry Arnoux, 1-Dec-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   π‘† = (pInvGβ€˜πΊ)    &   (πœ‘ β†’ 𝑀 ∈ 𝑃)    β‡’   (πœ‘ β†’ (𝐡 = ((π‘†β€˜π‘€)β€˜π΄) ↔ (𝐴(midGβ€˜πΊ)𝐡) = 𝑀))
 
Theoremmidbtwn 27507 Betweenness of midpoint. (Contributed by Thierry Arnoux, 7-Dec-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    β‡’   (πœ‘ β†’ (𝐴(midGβ€˜πΊ)𝐡) ∈ (𝐴𝐼𝐡))
 
Theoremmidcgr 27508 Congruence of midpoint. (Contributed by Thierry Arnoux, 7-Dec-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ (𝐴(midGβ€˜πΊ)𝐡) = 𝐢)    β‡’   (πœ‘ β†’ (𝐢 βˆ’ 𝐴) = (𝐢 βˆ’ 𝐡))
 
Theoremmidid 27509 Midpoint of a null segment. (Contributed by Thierry Arnoux, 7-Dec-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    β‡’   (πœ‘ β†’ (𝐴(midGβ€˜πΊ)𝐴) = 𝐴)
 
Theoremmidcom 27510 Commutativity rule for the midpoint. (Contributed by Thierry Arnoux, 2-Dec-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    β‡’   (πœ‘ β†’ (𝐴(midGβ€˜πΊ)𝐡) = (𝐡(midGβ€˜πΊ)𝐴))
 
Theoremmirmid 27511 Point inversion preserves midpoints. (Contributed by Thierry Arnoux, 12-Dec-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   π‘† = ((pInvGβ€˜πΊ)β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ 𝑃)    β‡’   (πœ‘ β†’ ((π‘†β€˜π΄)(midGβ€˜πΊ)(π‘†β€˜π΅)) = (π‘†β€˜(𝐴(midGβ€˜πΊ)𝐡)))
 
Theoremlmieu 27512* Uniqueness of the line mirror point. Theorem 10.2 of [Schwabhauser] p. 88. (Contributed by Thierry Arnoux, 1-Dec-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    &   πΏ = (LineGβ€˜πΊ)    &   (πœ‘ β†’ 𝐷 ∈ ran 𝐿)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    β‡’   (πœ‘ β†’ βˆƒ!𝑏 ∈ 𝑃 ((𝐴(midGβ€˜πΊ)𝑏) ∈ 𝐷 ∧ (𝐷(βŸ‚Gβ€˜πΊ)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))
 
Theoremlmif 27513 Line mirror as a function. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    &   π‘€ = ((lInvGβ€˜πΊ)β€˜π·)    &   πΏ = (LineGβ€˜πΊ)    &   (πœ‘ β†’ 𝐷 ∈ ran 𝐿)    β‡’   (πœ‘ β†’ 𝑀:π‘ƒβŸΆπ‘ƒ)
 
Theoremlmicl 27514 Closure of the line mirror. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    &   π‘€ = ((lInvGβ€˜πΊ)β€˜π·)    &   πΏ = (LineGβ€˜πΊ)    &   (πœ‘ β†’ 𝐷 ∈ ran 𝐿)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    β‡’   (πœ‘ β†’ (π‘€β€˜π΄) ∈ 𝑃)
 
Theoremislmib 27515 Property of the line mirror. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    &   π‘€ = ((lInvGβ€˜πΊ)β€˜π·)    &   πΏ = (LineGβ€˜πΊ)    &   (πœ‘ β†’ 𝐷 ∈ ran 𝐿)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    β‡’   (πœ‘ β†’ (𝐡 = (π‘€β€˜π΄) ↔ ((𝐴(midGβ€˜πΊ)𝐡) ∈ 𝐷 ∧ (𝐷(βŸ‚Gβ€˜πΊ)(𝐴𝐿𝐡) ∨ 𝐴 = 𝐡))))
 
Theoremlmicom 27516 The line mirroring function is an involution. Theorem 10.4 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    &   π‘€ = ((lInvGβ€˜πΊ)β€˜π·)    &   πΏ = (LineGβ€˜πΊ)    &   (πœ‘ β†’ 𝐷 ∈ ran 𝐿)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ (π‘€β€˜π΄) = 𝐡)    β‡’   (πœ‘ β†’ (π‘€β€˜π΅) = 𝐴)
 
Theoremlmilmi 27517 Line mirroring is an involution. Theorem 10.5 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    &   π‘€ = ((lInvGβ€˜πΊ)β€˜π·)    &   πΏ = (LineGβ€˜πΊ)    &   (πœ‘ β†’ 𝐷 ∈ ran 𝐿)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    β‡’   (πœ‘ β†’ (π‘€β€˜(π‘€β€˜π΄)) = 𝐴)
 
Theoremlmireu 27518* Any point has a unique antecedent through line mirroring. Theorem 10.6 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    &   π‘€ = ((lInvGβ€˜πΊ)β€˜π·)    &   πΏ = (LineGβ€˜πΊ)    &   (πœ‘ β†’ 𝐷 ∈ ran 𝐿)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    β‡’   (πœ‘ β†’ βˆƒ!𝑏 ∈ 𝑃 (π‘€β€˜π‘) = 𝐴)
 
Theoremlmieq 27519 Equality deduction for line mirroring. Theorem 10.7 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    &   π‘€ = ((lInvGβ€˜πΊ)β€˜π·)    &   πΏ = (LineGβ€˜πΊ)    &   (πœ‘ β†’ 𝐷 ∈ ran 𝐿)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ (π‘€β€˜π΄) = (π‘€β€˜π΅))    β‡’   (πœ‘ β†’ 𝐴 = 𝐡)
 
Theoremlmiinv 27520 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 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    &   π‘€ = ((lInvGβ€˜πΊ)β€˜π·)    &   πΏ = (LineGβ€˜πΊ)    &   (πœ‘ β†’ 𝐷 ∈ ran 𝐿)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    β‡’   (πœ‘ β†’ ((π‘€β€˜π΄) = 𝐴 ↔ 𝐴 ∈ 𝐷))
 
Theoremlmicinv 27521 The mirroring line is an invariant. (Contributed by Thierry Arnoux, 8-Aug-2020.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    &   π‘€ = ((lInvGβ€˜πΊ)β€˜π·)    &   πΏ = (LineGβ€˜πΊ)    &   (πœ‘ β†’ 𝐷 ∈ ran 𝐿)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐴 ∈ 𝐷)    β‡’   (πœ‘ β†’ (π‘€β€˜π΄) = 𝐴)
 
Theoremlmimid 27522 If we have a right angle, then the mirror point is the point inversion. (Contributed by Thierry Arnoux, 15-Dec-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    &   π‘€ = ((lInvGβ€˜πΊ)β€˜π·)    &   πΏ = (LineGβ€˜πΊ)    &   (πœ‘ β†’ 𝐷 ∈ ran 𝐿)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   π‘† = ((pInvGβ€˜πΊ)β€˜π΅)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ© ∈ (∟Gβ€˜πΊ))    &   (πœ‘ β†’ 𝐴 ∈ 𝐷)    &   (πœ‘ β†’ 𝐡 ∈ 𝐷)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐴 β‰  𝐡)    β‡’   (πœ‘ β†’ (π‘€β€˜πΆ) = (π‘†β€˜πΆ))
 
Theoremlmif1o 27523 The line mirroring function 𝑀 is a bijection. Theorem 10.9 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    &   π‘€ = ((lInvGβ€˜πΊ)β€˜π·)    &   πΏ = (LineGβ€˜πΊ)    &   (πœ‘ β†’ 𝐷 ∈ ran 𝐿)    β‡’   (πœ‘ β†’ 𝑀:𝑃–1-1-onto→𝑃)
 
Theoremlmiisolem 27524 Lemma for lmiiso 27525. (Contributed by Thierry Arnoux, 14-Dec-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    &   π‘€ = ((lInvGβ€˜πΊ)β€˜π·)    &   πΏ = (LineGβ€˜πΊ)    &   (πœ‘ β†’ 𝐷 ∈ ran 𝐿)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   π‘† = ((pInvGβ€˜πΊ)β€˜π‘)    &   π‘ = ((𝐴(midGβ€˜πΊ)(π‘€β€˜π΄))(midGβ€˜πΊ)(𝐡(midGβ€˜πΊ)(π‘€β€˜π΅)))    β‡’   (πœ‘ β†’ ((π‘€β€˜π΄) βˆ’ (π‘€β€˜π΅)) = (𝐴 βˆ’ 𝐡))
 
Theoremlmiiso 27525 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 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    &   π‘€ = ((lInvGβ€˜πΊ)β€˜π·)    &   πΏ = (LineGβ€˜πΊ)    &   (πœ‘ β†’ 𝐷 ∈ ran 𝐿)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    β‡’   (πœ‘ β†’ ((π‘€β€˜π΄) βˆ’ (π‘€β€˜π΅)) = (𝐴 βˆ’ 𝐡))
 
Theoremlmimot 27526 Line mirroring is a motion of the geometric space. Theorem 10.11 of [Schwabhauser] p. 90. (Contributed by Thierry Arnoux, 15-Dec-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    &   π‘€ = ((lInvGβ€˜πΊ)β€˜π·)    &   πΏ = (LineGβ€˜πΊ)    &   (πœ‘ β†’ 𝐷 ∈ ran 𝐿)    β‡’   (πœ‘ β†’ 𝑀 ∈ (𝐺Ismt𝐺))
 
Theoremhypcgrlem1 27527 Lemma for hypcgr 27529, case where triangles share a cathetus. (Contributed by Thierry Arnoux, 15-Dec-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ© ∈ (∟Gβ€˜πΊ))    &   (πœ‘ β†’ βŸ¨β€œπ·πΈπΉβ€βŸ© ∈ (∟Gβ€˜πΊ))    &   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸))    &   (πœ‘ β†’ (𝐡 βˆ’ 𝐢) = (𝐸 βˆ’ 𝐹))    &   (πœ‘ β†’ 𝐡 = 𝐸)    &   π‘† = ((lInvGβ€˜πΊ)β€˜((𝐴(midGβ€˜πΊ)𝐷)(LineGβ€˜πΊ)𝐡))    &   (πœ‘ β†’ 𝐢 = 𝐹)    β‡’   (πœ‘ β†’ (𝐴 βˆ’ 𝐢) = (𝐷 βˆ’ 𝐹))
 
Theoremhypcgrlem2 27528 Lemma for hypcgr 27529, case where triangles share one vertex 𝐡. (Contributed by Thierry Arnoux, 16-Dec-2019.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ© ∈ (∟Gβ€˜πΊ))    &   (πœ‘ β†’ βŸ¨β€œπ·πΈπΉβ€βŸ© ∈ (∟Gβ€˜πΊ))    &   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸))    &   (πœ‘ β†’ (𝐡 βˆ’ 𝐢) = (𝐸 βˆ’ 𝐹))    &   (πœ‘ β†’ 𝐡 = 𝐸)    &   π‘† = ((lInvGβ€˜πΊ)β€˜((𝐢(midGβ€˜πΊ)𝐹)(LineGβ€˜πΊ)𝐡))    β‡’   (πœ‘ β†’ (𝐴 βˆ’ 𝐢) = (𝐷 βˆ’ 𝐹))
 
Theoremhypcgr 27529 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 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ© ∈ (∟Gβ€˜πΊ))    &   (πœ‘ β†’ βŸ¨β€œπ·πΈπΉβ€βŸ© ∈ (∟Gβ€˜πΊ))    &   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸))    &   (πœ‘ β†’ (𝐡 βˆ’ 𝐢) = (𝐸 βˆ’ 𝐹))    β‡’   (πœ‘ β†’ (𝐴 βˆ’ 𝐢) = (𝐷 βˆ’ 𝐹))
 
Theoremlmiopp 27530* 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 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   πΏ = (LineGβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    &   (πœ‘ β†’ 𝐷 ∈ ran 𝐿)    &   π‘‚ = {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ (𝑃 βˆ– 𝐷) ∧ 𝑏 ∈ (𝑃 βˆ– 𝐷)) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (π‘ŽπΌπ‘))}    &   π‘€ = ((lInvGβ€˜πΊ)β€˜π·)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ Β¬ 𝐴 ∈ 𝐷)    β‡’   (πœ‘ β†’ 𝐴𝑂(π‘€β€˜π΄))
 
Theoremlnperpex 27531* 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 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   πΏ = (LineGβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐺DimTarskiGβ‰₯2)    &   (πœ‘ β†’ 𝐷 ∈ ran 𝐿)    &   π‘‚ = {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ (𝑃 βˆ– 𝐷) ∧ 𝑏 ∈ (𝑃 βˆ– 𝐷)) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (π‘ŽπΌπ‘))}    &   (πœ‘ β†’ 𝐴 ∈ 𝐷)    &   (πœ‘ β†’ 𝑄 ∈ 𝑃)    &   (πœ‘ β†’ Β¬ 𝑄 ∈ 𝐷)    β‡’   (πœ‘ β†’ βˆƒπ‘ ∈ 𝑃 (𝐷(βŸ‚Gβ€˜πΊ)(𝑝𝐿𝐴) ∧ 𝑝((hpGβ€˜πΊ)β€˜π·)𝑄))
 
Theoremtrgcopy 27532* 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 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   πΏ = (LineGβ€˜πΊ)    &   πΎ = (hlGβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ Β¬ (𝐴 ∈ (𝐡𝐿𝐢) ∨ 𝐡 = 𝐢))    &   (πœ‘ β†’ Β¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))    &   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸))    β‡’   (πœ‘ β†’ βˆƒπ‘“ ∈ 𝑃 (βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπ‘“β€βŸ© ∧ 𝑓((hpGβ€˜πΊ)β€˜(𝐷𝐿𝐸))𝐹))
 
Theoremtrgcopyeulem 27533* Lemma for trgcopyeu 27534. (Contributed by Thierry Arnoux, 8-Aug-2020.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   πΏ = (LineGβ€˜πΊ)    &   πΎ = (hlGβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ Β¬ (𝐴 ∈ (𝐡𝐿𝐢) ∨ 𝐡 = 𝐢))    &   (πœ‘ β†’ Β¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))    &   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸))    &   π‘‚ = {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ (𝑃 βˆ– (𝐷𝐿𝐸)) ∧ 𝑏 ∈ (𝑃 βˆ– (𝐷𝐿𝐸))) ∧ βˆƒπ‘‘ ∈ (𝐷𝐿𝐸)𝑑 ∈ (π‘ŽπΌπ‘))}    &   (πœ‘ β†’ 𝑋 ∈ 𝑃)    &   (πœ‘ β†’ π‘Œ ∈ 𝑃)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπ‘‹β€βŸ©)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπ‘Œβ€βŸ©)    &   (πœ‘ β†’ 𝑋((hpGβ€˜πΊ)β€˜(𝐷𝐿𝐸))𝐹)    &   (πœ‘ β†’ π‘Œ((hpGβ€˜πΊ)β€˜(𝐷𝐿𝐸))𝐹)    β‡’   (πœ‘ β†’ 𝑋 = π‘Œ)
 
Theoremtrgcopyeu 27534* 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 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   πΏ = (LineGβ€˜πΊ)    &   πΎ = (hlGβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ Β¬ (𝐴 ∈ (𝐡𝐿𝐢) ∨ 𝐡 = 𝐢))    &   (πœ‘ β†’ Β¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))    &   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸))    β‡’   (πœ‘ β†’ βˆƒ!𝑓 ∈ 𝑃 (βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπ‘“β€βŸ© ∧ 𝑓((hpGβ€˜πΊ)β€˜(𝐷𝐿𝐸))𝐹))
 
16.2.16  Congruence of angles
 
Syntaxccgra 27535 Declare the constant for the congruence between angles relation.
class cgrA
 
Definitiondf-cgra 27536* Define the congruence relation between angles. As for triangles we use "words of points". See iscgra 27537 for a more human readable version. (Contributed by Thierry Arnoux, 30-Jul-2020.)
cgrA = (𝑔 ∈ V ↦ {βŸ¨π‘Ž, π‘βŸ© ∣ [(Baseβ€˜π‘”) / 𝑝][(hlGβ€˜π‘”) / π‘˜]((π‘Ž ∈ (𝑝 ↑m (0..^3)) ∧ 𝑏 ∈ (𝑝 ↑m (0..^3))) ∧ βˆƒπ‘₯ ∈ 𝑝 βˆƒπ‘¦ ∈ 𝑝 (π‘Ž(cgrGβ€˜π‘”)βŸ¨β€œπ‘₯(π‘β€˜1)π‘¦β€βŸ© ∧ π‘₯(π‘˜β€˜(π‘β€˜1))(π‘β€˜0) ∧ 𝑦(π‘˜β€˜(π‘β€˜1))(π‘β€˜2)))})
 
Theoremiscgra 27537* 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 27558 shows that those definitions are indeed equivalent. (Contributed by Thierry Arnoux, 31-Jul-2020.)
𝑃 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   πΎ = (hlGβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    β‡’   (πœ‘ β†’ (βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ© ↔ βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 (βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ‘₯πΈπ‘¦β€βŸ© ∧ π‘₯(πΎβ€˜πΈ)𝐷 ∧ 𝑦(πΎβ€˜πΈ)𝐹)))
 
Theoremiscgra1 27538* A special version of iscgra 27537 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 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   πΎ = (hlGβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &    βˆ’ = (distβ€˜πΊ)    &   (πœ‘ β†’ 𝐴 β‰  𝐡)    &   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸))    β‡’   (πœ‘ β†’ (βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ© ↔ βˆƒπ‘₯ ∈ 𝑃 (βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπ‘₯β€βŸ© ∧ π‘₯(πΎβ€˜πΈ)𝐹)))
 
Theoremiscgrad 27539 Sufficient conditions for angle congruence, deduction version. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   πΎ = (hlGβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ 𝑋 ∈ 𝑃)    &   (πœ‘ β†’ π‘Œ ∈ 𝑃)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ‘‹πΈπ‘Œβ€βŸ©)    &   (πœ‘ β†’ 𝑋(πΎβ€˜πΈ)𝐷)    &   (πœ‘ β†’ π‘Œ(πΎβ€˜πΈ)𝐹)    β‡’   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)
 
Theoremcgrane1 27540 Angles imply inequality. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   πΎ = (hlGβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)    β‡’   (πœ‘ β†’ 𝐴 β‰  𝐡)
 
Theoremcgrane2 27541 Angles imply inequality. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   πΎ = (hlGβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)    β‡’   (πœ‘ β†’ 𝐡 β‰  𝐢)
 
Theoremcgrane3 27542 Angles imply inequality. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   πΎ = (hlGβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)    β‡’   (πœ‘ β†’ 𝐸 β‰  𝐷)
 
Theoremcgrane4 27543 Angles imply inequality. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   πΎ = (hlGβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)    β‡’   (πœ‘ β†’ 𝐸 β‰  𝐹)
 
Theoremcgrahl1 27544 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 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   πΎ = (hlGβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)    &   (πœ‘ β†’ 𝑋 ∈ 𝑃)    &   (πœ‘ β†’ 𝑋(πΎβ€˜πΈ)𝐷)    β‡’   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ‘‹πΈπΉβ€βŸ©)
 
Theoremcgrahl2 27545 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 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   πΎ = (hlGβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)    &   (πœ‘ β†’ 𝑋 ∈ 𝑃)    &   (πœ‘ β†’ 𝑋(πΎβ€˜πΈ)𝐹)    β‡’   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπ‘‹β€βŸ©)
 
Theoremcgracgr 27546 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 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   πΎ = (hlGβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)    &   (πœ‘ β†’ 𝑋 ∈ 𝑃)    &    βˆ’ = (distβ€˜πΊ)    &   (πœ‘ β†’ π‘Œ ∈ 𝑃)    &   (πœ‘ β†’ 𝑋(πΎβ€˜π΅)𝐴)    &   (πœ‘ β†’ π‘Œ(πΎβ€˜π΅)𝐢)    &   (πœ‘ β†’ (𝐡 βˆ’ 𝑋) = (𝐸 βˆ’ 𝐷))    &   (πœ‘ β†’ (𝐡 βˆ’ π‘Œ) = (𝐸 βˆ’ 𝐹))    β‡’   (πœ‘ β†’ (𝑋 βˆ’ π‘Œ) = (𝐷 βˆ’ 𝐹))
 
Theoremcgraid 27547 Angle congruence is reflexive. Theorem 11.6 of [Schwabhauser] p. 96. (Contributed by Thierry Arnoux, 31-Jul-2020.)
𝑃 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   πΎ = (hlGβ€˜πΊ)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐴 β‰  𝐡)    &   (πœ‘ β†’ 𝐡 β‰  𝐢)    β‡’   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ©)
 
Theoremcgraswap 27548 Swap rays in a congruence relation. Theorem 11.9 of [Schwabhauser] p. 96. (Contributed by Thierry Arnoux, 5-Mar-2020.)
𝑃 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   πΎ = (hlGβ€˜πΊ)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐴 β‰  𝐡)    &   (πœ‘ β†’ 𝐡 β‰  𝐢)    β‡’   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΆπ΅π΄β€βŸ©)
 
Theoremcgrcgra 27549 Triangle congruence implies angle congruence. This is a portion of CPCTC, focusing on a specific angle. (Contributed by Arnoux, 2-Aug-2020.)
𝑃 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   πΎ = (hlGβ€˜πΊ)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ 𝐴 β‰  𝐡)    &   (πœ‘ β†’ 𝐡 β‰  𝐢)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)    β‡’   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)
 
Theoremcgracom 27550 Angle congruence commutes. Theorem 11.7 of [Schwabhauser] p. 97. (Contributed by Thierry Arnoux, 5-Mar-2020.)
𝑃 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   πΎ = (hlGβ€˜πΊ)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)    β‡’   (πœ‘ β†’ βŸ¨β€œπ·πΈπΉβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ©)
 
Theoremcgratr 27551 Angle congruence is transitive. Theorem 11.8 of [Schwabhauser] p. 97. (Contributed by Thierry Arnoux, 5-Mar-2020.)
𝑃 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   πΎ = (hlGβ€˜πΊ)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)    &   (πœ‘ β†’ 𝐻 ∈ 𝑃)    &   (πœ‘ β†’ π‘ˆ ∈ 𝑃)    &   (πœ‘ β†’ 𝐽 ∈ 𝑃)    &   (πœ‘ β†’ βŸ¨β€œπ·πΈπΉβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ»π‘ˆπ½β€βŸ©)    β‡’   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ»π‘ˆπ½β€βŸ©)
 
Theoremflatcgra 27552 Flat angles are congruent. (Contributed by Thierry Arnoux, 13-Feb-2023.)
𝑃 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ (𝐴𝐼𝐢))    &   (πœ‘ β†’ 𝐸 ∈ (𝐷𝐼𝐹))    &   (πœ‘ β†’ 𝐴 β‰  𝐡)    &   (πœ‘ β†’ 𝐢 β‰  𝐡)    &   (πœ‘ β†’ 𝐷 β‰  𝐸)    &   (πœ‘ β†’ 𝐹 β‰  𝐸)    β‡’   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)
 
Theoremcgraswaplr 27553 Swap both side of angle congruence. (Contributed by Thierry Arnoux, 5-Oct-2020.)
𝑃 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)    β‡’   (πœ‘ β†’ βŸ¨β€œπΆπ΅π΄β€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΉπΈπ·β€βŸ©)
 
Theoremcgrabtwn 27554 Angle congruence preserves flat angles. Part of Theorem 11.21 of [Schwabhauser] p. 97. (Contributed by Thierry Arnoux, 9-Aug-2020.)
𝑃 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)    &   (πœ‘ β†’ 𝐡 ∈ (𝐴𝐼𝐢))    β‡’   (πœ‘ β†’ 𝐸 ∈ (𝐷𝐼𝐹))
 
Theoremcgrahl 27555 Angle congruence preserves null angles. Part of Theorem 11.21 of [Schwabhauser] p. 97. (Contributed by Thierry Arnoux, 9-Aug-2020.)
𝑃 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)    &   πΎ = (hlGβ€˜πΊ)    &   (πœ‘ β†’ 𝐴(πΎβ€˜π΅)𝐢)    β‡’   (πœ‘ β†’ 𝐷(πΎβ€˜πΈ)𝐹)
 
Theoremcgracol 27556 Angle congruence preserves colinearity. (Contributed by Thierry Arnoux, 9-Aug-2020.)
𝑃 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)    &   πΏ = (LineGβ€˜πΊ)    &   (πœ‘ β†’ (𝐢 ∈ (𝐴𝐿𝐡) ∨ 𝐴 = 𝐡))    β‡’   (πœ‘ β†’ (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
 
Theoremcgrancol 27557 Angle congruence preserves non-colinearity. (Contributed by Thierry Arnoux, 9-Aug-2020.)
𝑃 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)    &   πΏ = (LineGβ€˜πΊ)    &   (πœ‘ β†’ Β¬ (𝐢 ∈ (𝐴𝐿𝐡) ∨ 𝐴 = 𝐡))    β‡’   (πœ‘ β†’ Β¬ (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
 
Theoremdfcgra2 27558* 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 27536 is indeed equivalent to the textbook's definition. (Contributed by Thierry Arnoux, 2-Aug-2020.)
𝑃 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    β‡’   (πœ‘ β†’ (βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ© ↔ ((𝐴 β‰  𝐡 ∧ 𝐢 β‰  𝐡) ∧ (𝐷 β‰  𝐸 ∧ 𝐹 β‰  𝐸) ∧ βˆƒπ‘Ž ∈ 𝑃 βˆƒπ‘ ∈ 𝑃 βˆƒπ‘‘ ∈ 𝑃 βˆƒπ‘“ ∈ 𝑃 (((𝐴 ∈ (π΅πΌπ‘Ž) ∧ (𝐴 βˆ’ π‘Ž) = (𝐸 βˆ’ 𝐷)) ∧ (𝐢 ∈ (𝐡𝐼𝑐) ∧ (𝐢 βˆ’ 𝑐) = (𝐸 βˆ’ 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 βˆ’ 𝑑) = (𝐡 βˆ’ 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 βˆ’ 𝑓) = (𝐡 βˆ’ 𝐢))) ∧ (π‘Ž βˆ’ 𝑐) = (𝑑 βˆ’ 𝑓)))))
 
Theoremsacgr 27559 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 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ 𝑋 ∈ 𝑃)    &   (πœ‘ β†’ π‘Œ ∈ 𝑃)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)    &   (πœ‘ β†’ 𝐡 ∈ (𝐴𝐼𝑋))    &   (πœ‘ β†’ 𝐸 ∈ (π·πΌπ‘Œ))    &   (πœ‘ β†’ 𝐡 β‰  𝑋)    &   (πœ‘ β†’ 𝐸 β‰  π‘Œ)    β‡’   (πœ‘ β†’ βŸ¨β€œπ‘‹π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ‘ŒπΈπΉβ€βŸ©)
 
Theoremoacgr 27560 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 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ (𝐴𝐼𝐷))    &   (πœ‘ β†’ 𝐡 ∈ (𝐢𝐼𝐹))    &   (πœ‘ β†’ 𝐡 β‰  𝐴)    &   (πœ‘ β†’ 𝐡 β‰  𝐢)    &   (πœ‘ β†’ 𝐡 β‰  𝐷)    &   (πœ‘ β†’ 𝐡 β‰  𝐹)    β‡’   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·π΅πΉβ€βŸ©)
 
Theoremacopy 27561* 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 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   πΏ = (LineGβ€˜πΊ)    &   (πœ‘ β†’ Β¬ (𝐴 ∈ (𝐡𝐿𝐢) ∨ 𝐡 = 𝐢))    &   (πœ‘ β†’ Β¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))    β‡’   (πœ‘ β†’ βˆƒπ‘“ ∈ 𝑃 (βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπ‘“β€βŸ© ∧ 𝑓((hpGβ€˜πΊ)β€˜(𝐷𝐿𝐸))𝐹))
 
Theoremacopyeu 27562 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 π‘Œ both fulfill the conditions, then they are on the same half-line. (Contributed by Thierry Arnoux, 9-Aug-2020.)
𝑃 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   πΏ = (LineGβ€˜πΊ)    &   (πœ‘ β†’ Β¬ (𝐴 ∈ (𝐡𝐿𝐢) ∨ 𝐡 = 𝐢))    &   (πœ‘ β†’ Β¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))    &   (πœ‘ β†’ 𝑋 ∈ 𝑃)    &   (πœ‘ β†’ π‘Œ ∈ 𝑃)    &   πΎ = (hlGβ€˜πΊ)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπ‘‹β€βŸ©)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπ‘Œβ€βŸ©)    &   (πœ‘ β†’ 𝑋((hpGβ€˜πΊ)β€˜(𝐷𝐿𝐸))𝐹)    &   (πœ‘ β†’ π‘Œ((hpGβ€˜πΊ)β€˜(𝐷𝐿𝐸))𝐹)    β‡’   (πœ‘ β†’ 𝑋(πΎβ€˜πΈ)π‘Œ)
 
16.2.17  Angle Comparisons
 
Syntaxcinag 27563 Extend class relation with the geometrical "point in angle" relation.
class inA
 
Syntaxcleag 27564 Extend class relation with the "angle less than" relation.
class β‰€βˆ 
 
Definitiondf-inag 27565* Definition of the geometrical "in angle" relation. (Contributed by Thierry Arnoux, 15-Aug-2020.)
inA = (𝑔 ∈ V ↦ {βŸ¨π‘, π‘‘βŸ© ∣ ((𝑝 ∈ (Baseβ€˜π‘”) ∧ 𝑑 ∈ ((Baseβ€˜π‘”) ↑m (0..^3))) ∧ (((π‘‘β€˜0) β‰  (π‘‘β€˜1) ∧ (π‘‘β€˜2) β‰  (π‘‘β€˜1) ∧ 𝑝 β‰  (π‘‘β€˜1)) ∧ βˆƒπ‘₯ ∈ (Baseβ€˜π‘”)(π‘₯ ∈ ((π‘‘β€˜0)(Itvβ€˜π‘”)(π‘‘β€˜2)) ∧ (π‘₯ = (π‘‘β€˜1) ∨ π‘₯((hlGβ€˜π‘”)β€˜(π‘‘β€˜1))𝑝))))})
 
Theoremisinag 27566* Property for point 𝑋 to lie in the angle βŸ¨β€œπ΄π΅πΆβ€βŸ©. Definition 11.23 of [Schwabhauser] p. 101. (Contributed by Thierry Arnoux, 15-Aug-2020.)
𝑃 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   πΎ = (hlGβ€˜πΊ)    &   (πœ‘ β†’ 𝑋 ∈ 𝑃)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐺 ∈ 𝑉)    β‡’   (πœ‘ β†’ (𝑋(inAβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ© ↔ ((𝐴 β‰  𝐡 ∧ 𝐢 β‰  𝐡 ∧ 𝑋 β‰  𝐡) ∧ βˆƒπ‘₯ ∈ 𝑃 (π‘₯ ∈ (𝐴𝐼𝐢) ∧ (π‘₯ = 𝐡 ∨ π‘₯(πΎβ€˜π΅)𝑋)))))
 
Theoremisinagd 27567 Sufficient conditions for in-angle relation, deduction version. (Contributed by Thierry Arnoux, 20-Oct-2020.)
𝑃 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   πΎ = (hlGβ€˜πΊ)    &   (πœ‘ β†’ 𝑋 ∈ 𝑃)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐺 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑃)    &   (πœ‘ β†’ 𝐴 β‰  𝐡)    &   (πœ‘ β†’ 𝐢 β‰  𝐡)    &   (πœ‘ β†’ 𝑋 β‰  𝐡)    &   (πœ‘ β†’ π‘Œ ∈ (𝐴𝐼𝐢))    &   (πœ‘ β†’ (π‘Œ = 𝐡 ∨ π‘Œ(πΎβ€˜π΅)𝑋))    β‡’   (πœ‘ β†’ 𝑋(inAβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ©)
 
Theoreminagflat 27568 Any point lies in a flat angle. (Contributed by Thierry Arnoux, 13-Feb-2023.)
𝑃 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   πΎ = (hlGβ€˜πΊ)    &   (πœ‘ β†’ 𝑋 ∈ 𝑃)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝑋 ∈ 𝑃)    &   (πœ‘ β†’ 𝐴 β‰  𝐡)    &   (πœ‘ β†’ 𝐢 β‰  𝐡)    &   (πœ‘ β†’ 𝑋 β‰  𝐡)    &   (πœ‘ β†’ 𝐡 ∈ (𝐴𝐼𝐢))    β‡’   (πœ‘ β†’ 𝑋(inAβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ©)
 
Theoreminagswap 27569 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 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   πΎ = (hlGβ€˜πΊ)    &   (πœ‘ β†’ 𝑋 ∈ 𝑃)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝑋(inAβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ©)    β‡’   (πœ‘ β†’ 𝑋(inAβ€˜πΊ)βŸ¨β€œπΆπ΅π΄β€βŸ©)
 
Theoreminagne1 27570 Deduce inequality from the in-angle relation. (Contributed by Thierry Arnoux, 29-Oct-2021.)
𝑃 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   πΎ = (hlGβ€˜πΊ)    &   (πœ‘ β†’ 𝑋 ∈ 𝑃)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝑋(inAβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ©)    β‡’   (πœ‘ β†’ 𝐴 β‰  𝐡)
 
Theoreminagne2 27571 Deduce inequality from the in-angle relation. (Contributed by Thierry Arnoux, 29-Oct-2021.)
𝑃 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   πΎ = (hlGβ€˜πΊ)    &   (πœ‘ β†’ 𝑋 ∈ 𝑃)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝑋(inAβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ©)    β‡’   (πœ‘ β†’ 𝐢 β‰  𝐡)
 
Theoreminagne3 27572 Deduce inequality from the in-angle relation. (Contributed by Thierry Arnoux, 29-Oct-2021.)
𝑃 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   πΎ = (hlGβ€˜πΊ)    &   (πœ‘ β†’ 𝑋 ∈ 𝑃)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝑋(inAβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ©)    β‡’   (πœ‘ β†’ 𝑋 β‰  𝐡)
 
Theoreminaghl 27573 The "point lie in angle" relation is independent of the points chosen on the half lines starting from 𝐡. Theorem 11.25 of [Schwabhauser] p. 101. (Contributed by Thierry Arnoux, 27-Sep-2020.)
𝑃 = (Baseβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   πΎ = (hlGβ€˜πΊ)    &   (πœ‘ β†’ 𝑋 ∈ 𝑃)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝑋(inAβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ©)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ π‘Œ ∈ 𝑃)    &   (πœ‘ β†’ 𝐷(πΎβ€˜π΅)𝐴)    &   (πœ‘ β†’ 𝐹(πΎβ€˜π΅)𝐢)    &   (πœ‘ β†’ π‘Œ(πΎβ€˜π΅)𝑋)    β‡’   (πœ‘ β†’ π‘Œ(inAβ€˜πΊ)βŸ¨β€œπ·π΅πΉβ€βŸ©)
 
Definitiondf-leag 27574* Definition of the geometrical "angle less than" relation. Definition 11.27 of [Schwabhauser] p. 102. (Contributed by Thierry Arnoux, 7-Oct-2020.)
β‰€βˆ  = (𝑔 ∈ V ↦ {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ ((Baseβ€˜π‘”) ↑m (0..^3)) ∧ 𝑏 ∈ ((Baseβ€˜π‘”) ↑m (0..^3))) ∧ βˆƒπ‘₯ ∈ (Baseβ€˜π‘”)(π‘₯(inAβ€˜π‘”)βŸ¨β€œ(π‘β€˜0)(π‘β€˜1)(π‘β€˜2)β€βŸ© ∧ βŸ¨β€œ(π‘Žβ€˜0)(π‘Žβ€˜1)(π‘Žβ€˜2)β€βŸ©(cgrAβ€˜π‘”)βŸ¨β€œ(π‘β€˜0)(π‘β€˜1)π‘₯β€βŸ©))})
 
Theoremisleag 27575* Geometrical "less than" property for angles. Definition 11.27 of [Schwabhauser] p. 102. (Contributed by Thierry Arnoux, 7-Oct-2020.)
𝑃 = (Baseβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    β‡’   (πœ‘ β†’ (βŸ¨β€œπ΄π΅πΆβ€βŸ©(β‰€βˆ β€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ© ↔ βˆƒπ‘₯ ∈ 𝑃 (π‘₯(inAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ© ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπ‘₯β€βŸ©)))
 
Theoremisleagd 27576 Sufficient condition for "less than" angle relation, deduction version (Contributed by Thierry Arnoux, 12-Oct-2020.)
𝑃 = (Baseβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &    ≀ = (β‰€βˆ β€˜πΊ)    &   (πœ‘ β†’ 𝑋 ∈ 𝑃)    &   (πœ‘ β†’ 𝑋(inAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπ‘‹β€βŸ©)    β‡’   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ© ≀ βŸ¨β€œπ·πΈπΉβ€βŸ©)
 
Theoremleagne1 27577 Deduce inequality from the less-than angle relation. (Contributed by Thierry Arnoux, 25-Feb-2023.)
𝑃 = (Baseβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(β‰€βˆ β€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)    β‡’   (πœ‘ β†’ 𝐴 β‰  𝐡)
 
Theoremleagne2 27578 Deduce inequality from the less-than angle relation. (Contributed by Thierry Arnoux, 25-Feb-2023.)
𝑃 = (Baseβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(β‰€βˆ β€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)    β‡’   (πœ‘ β†’ 𝐢 β‰  𝐡)
 
Theoremleagne3 27579 Deduce inequality from the less-than angle relation. (Contributed by Thierry Arnoux, 25-Feb-2023.)
𝑃 = (Baseβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(β‰€βˆ β€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)    β‡’   (πœ‘ β†’ 𝐷 β‰  𝐸)
 
Theoremleagne4 27580 Deduce inequality from the less-than angle relation. (Contributed by Thierry Arnoux, 25-Feb-2023.)
𝑃 = (Baseβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(β‰€βˆ β€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)    β‡’   (πœ‘ β†’ 𝐹 β‰  𝐸)
 
Theoremcgrg3col4 27581* 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 = (Baseβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   πΏ = (LineGβ€˜πΊ)    &   (πœ‘ β†’ 𝑋 ∈ 𝑃)    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)    &   (πœ‘ β†’ (𝑋 ∈ (𝐴𝐿𝐢) ∨ 𝐴 = 𝐢))    β‡’   (πœ‘ β†’ βˆƒπ‘¦ ∈ 𝑃 βŸ¨β€œπ΄π΅πΆπ‘‹β€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉπ‘¦β€βŸ©)
 
16.2.18  Congruence Theorems
 
Theoremtgsas1 27582 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 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸))    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)    &   (πœ‘ β†’ (𝐡 βˆ’ 𝐢) = (𝐸 βˆ’ 𝐹))    β‡’   (πœ‘ β†’ (𝐢 βˆ’ 𝐴) = (𝐹 βˆ’ 𝐷))
 
Theoremtgsas 27583 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 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸))    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)    &   (πœ‘ β†’ (𝐡 βˆ’ 𝐢) = (𝐸 βˆ’ 𝐹))    β‡’   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)
 
Theoremtgsas2 27584 First congruence theorem: SAS. Theorem 11.49 of [Schwabhauser] p. 107. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸))    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)    &   (πœ‘ β†’ (𝐡 βˆ’ 𝐢) = (𝐸 βˆ’ 𝐹))    &   (πœ‘ β†’ 𝐴 β‰  𝐢)    β‡’   (πœ‘ β†’ βŸ¨β€œπΆπ΄π΅β€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΉπ·πΈβ€βŸ©)
 
Theoremtgsas3 27585 First congruence theorem: SAS. Theorem 11.49 of [Schwabhauser] p. 107. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸))    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)    &   (πœ‘ β†’ (𝐡 βˆ’ 𝐢) = (𝐸 βˆ’ 𝐹))    &   (πœ‘ β†’ 𝐴 β‰  𝐢)    β‡’   (πœ‘ β†’ βŸ¨β€œπ΅πΆπ΄β€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΈπΉπ·β€βŸ©)
 
Theoremtgasa1 27586 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 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   πΏ = (LineGβ€˜πΊ)    &   (πœ‘ β†’ Β¬ (𝐢 ∈ (𝐴𝐿𝐡) ∨ 𝐴 = 𝐡))    &   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸))    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)    &   (πœ‘ β†’ βŸ¨β€œπΆπ΄π΅β€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΉπ·πΈβ€βŸ©)    β‡’   (πœ‘ β†’ (𝐡 βˆ’ 𝐢) = (𝐸 βˆ’ 𝐹))
 
Theoremtgasa 27587 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 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   πΏ = (LineGβ€˜πΊ)    &   (πœ‘ β†’ Β¬ (𝐢 ∈ (𝐴𝐿𝐡) ∨ 𝐴 = 𝐡))    &   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸))    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)    &   (πœ‘ β†’ βŸ¨β€œπΆπ΄π΅β€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΉπ·πΈβ€βŸ©)    β‡’   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)
 
Theoremtgsss1 27588 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 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸))    &   (πœ‘ β†’ (𝐡 βˆ’ 𝐢) = (𝐸 βˆ’ 𝐹))    &   (πœ‘ β†’ (𝐢 βˆ’ 𝐴) = (𝐹 βˆ’ 𝐷))    &   (πœ‘ β†’ 𝐴 β‰  𝐡)    &   (πœ‘ β†’ 𝐡 β‰  𝐢)    &   (πœ‘ β†’ 𝐢 β‰  𝐴)    β‡’   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ©)
 
Theoremtgsss2 27589 Third congruence theorem: SSS. Theorem 11.51 of [Schwabhauser] p. 109. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸))    &   (πœ‘ β†’ (𝐡 βˆ’ 𝐢) = (𝐸 βˆ’ 𝐹))    &   (πœ‘ β†’ (𝐢 βˆ’ 𝐴) = (𝐹 βˆ’ 𝐷))    &   (πœ‘ β†’ 𝐴 β‰  𝐡)    &   (πœ‘ β†’ 𝐡 β‰  𝐢)    &   (πœ‘ β†’ 𝐢 β‰  𝐴)    β‡’   (πœ‘ β†’ βŸ¨β€œπΆπ΄π΅β€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΉπ·πΈβ€βŸ©)
 
Theoremtgsss3 27590 Third congruence theorem: SSS. Theorem 11.51 of [Schwabhauser] p. 109. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸))    &   (πœ‘ β†’ (𝐡 βˆ’ 𝐢) = (𝐸 βˆ’ 𝐹))    &   (πœ‘ β†’ (𝐢 βˆ’ 𝐴) = (𝐹 βˆ’ 𝐷))    &   (πœ‘ β†’ 𝐴 β‰  𝐡)    &   (πœ‘ β†’ 𝐡 β‰  𝐢)    &   (πœ‘ β†’ 𝐢 β‰  𝐴)    β‡’   (πœ‘ β†’ βŸ¨β€œπ΅πΆπ΄β€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΈπΉπ·β€βŸ©)
 
Theoremdfcgrg2 27591 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 27239, already covers that part: see trgcgr 27244. 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 27244. (Contributed by Thierry Arnoux, 18-Jan-2023.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ 𝐷 ∈ 𝑃)    &   (πœ‘ β†’ 𝐸 ∈ 𝑃)    &   (πœ‘ β†’ 𝐹 ∈ 𝑃)    &   (πœ‘ β†’ 𝐴 β‰  𝐡)    &   (πœ‘ β†’ 𝐡 β‰  𝐢)    &   (πœ‘ β†’ 𝐢 β‰  𝐴)    β‡’   (πœ‘ β†’ (βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ© ↔ (((𝐴 βˆ’ 𝐡) = (𝐷 βˆ’ 𝐸) ∧ (𝐡 βˆ’ 𝐢) = (𝐸 βˆ’ 𝐹) ∧ (𝐢 βˆ’ 𝐴) = (𝐹 βˆ’ 𝐷)) ∧ (βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ·πΈπΉβ€βŸ© ∧ βŸ¨β€œπΆπ΄π΅β€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΉπ·πΈβ€βŸ© ∧ βŸ¨β€œπ΅πΆπ΄β€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπΈπΉπ·β€βŸ©))))
 
Theoremisoas 27592 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 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   πΏ = (LineGβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ Β¬ (𝐢 ∈ (𝐴𝐿𝐡) ∨ 𝐴 = 𝐡))    &   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrAβ€˜πΊ)βŸ¨β€œπ΄πΆπ΅β€βŸ©)    β‡’   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐴 βˆ’ 𝐢))
 
16.2.19  Equilateral triangles
 
Syntaxceqlg 27593 Declare the class of equilateral triangles.
class eqltrG
 
Definitiondf-eqlg 27594* Define the class of equilateral triangles. (Contributed by Thierry Arnoux, 27-Nov-2019.)
eqltrG = (𝑔 ∈ V ↦ {π‘₯ ∈ ((Baseβ€˜π‘”) ↑m (0..^3)) ∣ π‘₯(cgrGβ€˜π‘”)βŸ¨β€œ(π‘₯β€˜1)(π‘₯β€˜2)(π‘₯β€˜0)β€βŸ©})
 
Theoremiseqlg 27595 Property of a triangle being equilateral. (Contributed by Thierry Arnoux, 5-Oct-2020.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   πΏ = (LineGβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    β‡’   (πœ‘ β†’ (βŸ¨β€œπ΄π΅πΆβ€βŸ© ∈ (eqltrGβ€˜πΊ) ↔ βŸ¨β€œπ΄π΅πΆβ€βŸ©(cgrGβ€˜πΊ)βŸ¨β€œπ΅πΆπ΄β€βŸ©))
 
Theoremiseqlgd 27596 Condition for a triangle to be equilateral. (Contributed by Thierry Arnoux, 5-Oct-2020.)
𝑃 = (Baseβ€˜πΊ)    &    βˆ’ = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   πΏ = (LineGβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ 𝐴 ∈ 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝑃)    &   (πœ‘ β†’ 𝐢 ∈ 𝑃)    &   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝐡 βˆ’ 𝐢))    &   (πœ‘ β†’ (𝐡 βˆ’ 𝐢) = (𝐢 βˆ’ 𝐴))    &   (πœ‘ β†’ (𝐢 βˆ’ 𝐴) = (𝐴 βˆ’ 𝐡))    β‡’   (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ© ∈ (eqltrGβ€˜πΊ))
 
16.3  Properties of geometries
 
16.3.1  Isomorphisms between geometries
 
Theoremf1otrgds 27597* Convenient lemma for f1otrg 27599. (Contributed by Thierry Arnoux, 19-Mar-2019.)
𝑃 = (Baseβ€˜πΊ)    &   π· = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   π΅ = (Baseβ€˜π»)    &   πΈ = (distβ€˜π»)    &   π½ = (Itvβ€˜π»)    &   (πœ‘ β†’ 𝐹:𝐡–1-1-onto→𝑃)    &   ((πœ‘ ∧ (𝑒 ∈ 𝐡 ∧ 𝑓 ∈ 𝐡)) β†’ (𝑒𝐸𝑓) = ((πΉβ€˜π‘’)𝐷(πΉβ€˜π‘“)))    &   ((πœ‘ ∧ (𝑒 ∈ 𝐡 ∧ 𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ (𝑔 ∈ (𝑒𝐽𝑓) ↔ (πΉβ€˜π‘”) ∈ ((πΉβ€˜π‘’)𝐼(πΉβ€˜π‘“))))    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ (π‘‹πΈπ‘Œ) = ((πΉβ€˜π‘‹)𝐷(πΉβ€˜π‘Œ)))
 
Theoremf1otrgitv 27598* Convenient lemma for f1otrg 27599. (Contributed by Thierry Arnoux, 19-Mar-2019.)
𝑃 = (Baseβ€˜πΊ)    &   π· = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   π΅ = (Baseβ€˜π»)    &   πΈ = (distβ€˜π»)    &   π½ = (Itvβ€˜π»)    &   (πœ‘ β†’ 𝐹:𝐡–1-1-onto→𝑃)    &   ((πœ‘ ∧ (𝑒 ∈ 𝐡 ∧ 𝑓 ∈ 𝐡)) β†’ (𝑒𝐸𝑓) = ((πΉβ€˜π‘’)𝐷(πΉβ€˜π‘“)))    &   ((πœ‘ ∧ (𝑒 ∈ 𝐡 ∧ 𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ (𝑔 ∈ (𝑒𝐽𝑓) ↔ (πΉβ€˜π‘”) ∈ ((πΉβ€˜π‘’)𝐼(πΉβ€˜π‘“))))    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑍 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑍 ∈ (π‘‹π½π‘Œ) ↔ (πΉβ€˜π‘) ∈ ((πΉβ€˜π‘‹)𝐼(πΉβ€˜π‘Œ))))
 
Theoremf1otrg 27599* A bijection between bases which conserves distances and intervals conserves also geometries. (Contributed by Thierry Arnoux, 23-Mar-2019.)
𝑃 = (Baseβ€˜πΊ)    &   π· = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   π΅ = (Baseβ€˜π»)    &   πΈ = (distβ€˜π»)    &   π½ = (Itvβ€˜π»)    &   (πœ‘ β†’ 𝐹:𝐡–1-1-onto→𝑃)    &   ((πœ‘ ∧ (𝑒 ∈ 𝐡 ∧ 𝑓 ∈ 𝐡)) β†’ (𝑒𝐸𝑓) = ((πΉβ€˜π‘’)𝐷(πΉβ€˜π‘“)))    &   ((πœ‘ ∧ (𝑒 ∈ 𝐡 ∧ 𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ (𝑔 ∈ (𝑒𝐽𝑓) ↔ (πΉβ€˜π‘”) ∈ ((πΉβ€˜π‘’)𝐼(πΉβ€˜π‘“))))    &   (πœ‘ β†’ 𝐻 ∈ 𝑉)    &   (πœ‘ β†’ 𝐺 ∈ TarskiG)    &   (πœ‘ β†’ (LineGβ€˜π») = (π‘₯ ∈ 𝐡, 𝑦 ∈ (𝐡 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝐡 ∣ (𝑧 ∈ (π‘₯𝐽𝑦) ∨ π‘₯ ∈ (𝑧𝐽𝑦) ∨ 𝑦 ∈ (π‘₯𝐽𝑧))}))    β‡’   (πœ‘ β†’ 𝐻 ∈ TarskiG)
 
Theoremf1otrge 27600* 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 = (Baseβ€˜πΊ)    &   π· = (distβ€˜πΊ)    &   πΌ = (Itvβ€˜πΊ)    &   π΅ = (Baseβ€˜π»)    &   πΈ = (distβ€˜π»)    &   π½ = (Itvβ€˜π»)    &   (πœ‘ β†’ 𝐹:𝐡–1-1-onto→𝑃)    &   ((πœ‘ ∧ (𝑒 ∈ 𝐡 ∧ 𝑓 ∈ 𝐡)) β†’ (𝑒𝐸𝑓) = ((πΉβ€˜π‘’)𝐷(πΉβ€˜π‘“)))    &   ((πœ‘ ∧ (𝑒 ∈ 𝐡 ∧ 𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ (𝑔 ∈ (𝑒𝐽𝑓) ↔ (πΉβ€˜π‘”) ∈ ((πΉβ€˜π‘’)𝐼(πΉβ€˜π‘“))))    &   (πœ‘ β†’ 𝐻 ∈ 𝑉)    &   (πœ‘ β†’ 𝐺 ∈ TarskiGE)    β‡’   (πœ‘ β†’ 𝐻 ∈ TarskiGE)
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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