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Type | Label | Description |
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Statement | ||
Theorem | mirmid 28301 | Point inversion preserves midpoints. (Contributed by Thierry Arnoux, 12-Dec-2019.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β πΊDimTarskiGβ₯2) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ π = ((pInvGβπΊ)βπ) & β’ (π β π β π) β β’ (π β ((πβπ΄)(midGβπΊ)(πβπ΅)) = (πβ(π΄(midGβπΊ)π΅))) | ||
Theorem | lmieu 28302* | Uniqueness of the line mirror point. Theorem 10.2 of [Schwabhauser] p. 88. (Contributed by Thierry Arnoux, 1-Dec-2019.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β πΊDimTarskiGβ₯2) & β’ πΏ = (LineGβπΊ) & β’ (π β π· β ran πΏ) & β’ (π β π΄ β π) β β’ (π β β!π β π ((π΄(midGβπΊ)π) β π· β§ (π·(βGβπΊ)(π΄πΏπ) β¨ π΄ = π))) | ||
Theorem | lmif 28303 | Line mirror as a function. (Contributed by Thierry Arnoux, 11-Dec-2019.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β πΊDimTarskiGβ₯2) & β’ π = ((lInvGβπΊ)βπ·) & β’ πΏ = (LineGβπΊ) & β’ (π β π· β ran πΏ) β β’ (π β π:πβΆπ) | ||
Theorem | lmicl 28304 | Closure of the line mirror. (Contributed by Thierry Arnoux, 11-Dec-2019.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β πΊDimTarskiGβ₯2) & β’ π = ((lInvGβπΊ)βπ·) & β’ πΏ = (LineGβπΊ) & β’ (π β π· β ran πΏ) & β’ (π β π΄ β π) β β’ (π β (πβπ΄) β π) | ||
Theorem | islmib 28305 | Property of the line mirror. (Contributed by Thierry Arnoux, 11-Dec-2019.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β πΊDimTarskiGβ₯2) & β’ π = ((lInvGβπΊ)βπ·) & β’ πΏ = (LineGβπΊ) & β’ (π β π· β ran πΏ) & β’ (π β π΄ β π) & β’ (π β π΅ β π) β β’ (π β (π΅ = (πβπ΄) β ((π΄(midGβπΊ)π΅) β π· β§ (π·(βGβπΊ)(π΄πΏπ΅) β¨ π΄ = π΅)))) | ||
Theorem | lmicom 28306 | The line mirroring function is an involution. Theorem 10.4 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β πΊDimTarskiGβ₯2) & β’ π = ((lInvGβπΊ)βπ·) & β’ πΏ = (LineGβπΊ) & β’ (π β π· β ran πΏ) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β (πβπ΄) = π΅) β β’ (π β (πβπ΅) = π΄) | ||
Theorem | lmilmi 28307 | Line mirroring is an involution. Theorem 10.5 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β πΊDimTarskiGβ₯2) & β’ π = ((lInvGβπΊ)βπ·) & β’ πΏ = (LineGβπΊ) & β’ (π β π· β ran πΏ) & β’ (π β π΄ β π) β β’ (π β (πβ(πβπ΄)) = π΄) | ||
Theorem | lmireu 28308* | Any point has a unique antecedent through line mirroring. Theorem 10.6 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β πΊDimTarskiGβ₯2) & β’ π = ((lInvGβπΊ)βπ·) & β’ πΏ = (LineGβπΊ) & β’ (π β π· β ran πΏ) & β’ (π β π΄ β π) β β’ (π β β!π β π (πβπ) = π΄) | ||
Theorem | lmieq 28309 | Equality deduction for line mirroring. Theorem 10.7 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β πΊDimTarskiGβ₯2) & β’ π = ((lInvGβπΊ)βπ·) & β’ πΏ = (LineGβπΊ) & β’ (π β π· β ran πΏ) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β (πβπ΄) = (πβπ΅)) β β’ (π β π΄ = π΅) | ||
Theorem | lmiinv 28310 | 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.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β πΊDimTarskiGβ₯2) & β’ π = ((lInvGβπΊ)βπ·) & β’ πΏ = (LineGβπΊ) & β’ (π β π· β ran πΏ) & β’ (π β π΄ β π) β β’ (π β ((πβπ΄) = π΄ β π΄ β π·)) | ||
Theorem | lmicinv 28311 | The mirroring line is an invariant. (Contributed by Thierry Arnoux, 8-Aug-2020.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β πΊDimTarskiGβ₯2) & β’ π = ((lInvGβπΊ)βπ·) & β’ πΏ = (LineGβπΊ) & β’ (π β π· β ran πΏ) & β’ (π β π΄ β π) & β’ (π β π΄ β π·) β β’ (π β (πβπ΄) = π΄) | ||
Theorem | lmimid 28312 | If we have a right angle, then the mirror point is the point inversion. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β πΊDimTarskiGβ₯2) & β’ π = ((lInvGβπΊ)βπ·) & β’ πΏ = (LineGβπΊ) & β’ (π β π· β ran πΏ) & β’ (π β π΄ β π) & β’ π = ((pInvGβπΊ)βπ΅) & β’ (π β β¨βπ΄π΅πΆββ© β (βGβπΊ)) & β’ (π β π΄ β π·) & β’ (π β π΅ β π·) & β’ (π β πΆ β π) & β’ (π β π΄ β π΅) β β’ (π β (πβπΆ) = (πβπΆ)) | ||
Theorem | lmif1o 28313 | The line mirroring function π is a bijection. Theorem 10.9 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β πΊDimTarskiGβ₯2) & β’ π = ((lInvGβπΊ)βπ·) & β’ πΏ = (LineGβπΊ) & β’ (π β π· β ran πΏ) β β’ (π β π:πβ1-1-ontoβπ) | ||
Theorem | lmiisolem 28314 | Lemma for lmiiso 28315. (Contributed by Thierry Arnoux, 14-Dec-2019.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β πΊDimTarskiGβ₯2) & β’ π = ((lInvGβπΊ)βπ·) & β’ πΏ = (LineGβπΊ) & β’ (π β π· β ran πΏ) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ π = ((pInvGβπΊ)βπ) & β’ π = ((π΄(midGβπΊ)(πβπ΄))(midGβπΊ)(π΅(midGβπΊ)(πβπ΅))) β β’ (π β ((πβπ΄) β (πβπ΅)) = (π΄ β π΅)) | ||
Theorem | lmiiso 28315 | 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.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β πΊDimTarskiGβ₯2) & β’ π = ((lInvGβπΊ)βπ·) & β’ πΏ = (LineGβπΊ) & β’ (π β π· β ran πΏ) & β’ (π β π΄ β π) & β’ (π β π΅ β π) β β’ (π β ((πβπ΄) β (πβπ΅)) = (π΄ β π΅)) | ||
Theorem | lmimot 28316 | Line mirroring is a motion of the geometric space. Theorem 10.11 of [Schwabhauser] p. 90. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β πΊDimTarskiGβ₯2) & β’ π = ((lInvGβπΊ)βπ·) & β’ πΏ = (LineGβπΊ) & β’ (π β π· β ran πΏ) β β’ (π β π β (πΊIsmtπΊ)) | ||
Theorem | hypcgrlem1 28317 | Lemma for hypcgr 28319, case where triangles share a cathetus. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β πΊDimTarskiGβ₯2) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β β¨βπ΄π΅πΆββ© β (βGβπΊ)) & β’ (π β β¨βπ·πΈπΉββ© β (βGβπΊ)) & β’ (π β (π΄ β π΅) = (π· β πΈ)) & β’ (π β (π΅ β πΆ) = (πΈ β πΉ)) & β’ (π β π΅ = πΈ) & β’ π = ((lInvGβπΊ)β((π΄(midGβπΊ)π·)(LineGβπΊ)π΅)) & β’ (π β πΆ = πΉ) β β’ (π β (π΄ β πΆ) = (π· β πΉ)) | ||
Theorem | hypcgrlem2 28318 | Lemma for hypcgr 28319, case where triangles share one vertex π΅. (Contributed by Thierry Arnoux, 16-Dec-2019.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β πΊDimTarskiGβ₯2) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β β¨βπ΄π΅πΆββ© β (βGβπΊ)) & β’ (π β β¨βπ·πΈπΉββ© β (βGβπΊ)) & β’ (π β (π΄ β π΅) = (π· β πΈ)) & β’ (π β (π΅ β πΆ) = (πΈ β πΉ)) & β’ (π β π΅ = πΈ) & β’ π = ((lInvGβπΊ)β((πΆ(midGβπΊ)πΉ)(LineGβπΊ)π΅)) β β’ (π β (π΄ β πΆ) = (π· β πΉ)) | ||
Theorem | hypcgr 28319 | 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.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β πΊDimTarskiGβ₯2) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β β¨βπ΄π΅πΆββ© β (βGβπΊ)) & β’ (π β β¨βπ·πΈπΉββ© β (βGβπΊ)) & β’ (π β (π΄ β π΅) = (π· β πΈ)) & β’ (π β (π΅ β πΆ) = (πΈ β πΉ)) β β’ (π β (π΄ β πΆ) = (π· β πΉ)) | ||
Theorem | lmiopp 28320* | 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.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ πΏ = (LineGβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β πΊDimTarskiGβ₯2) & β’ (π β π· β ran πΏ) & β’ π = {β¨π, πβ© β£ ((π β (π β π·) β§ π β (π β π·)) β§ βπ‘ β π· π‘ β (ππΌπ))} & β’ π = ((lInvGβπΊ)βπ·) & β’ (π β π΄ β π) & β’ (π β Β¬ π΄ β π·) β β’ (π β π΄π(πβπ΄)) | ||
Theorem | lnperpex 28321* | Existence of a perpendicular to a line πΏ at a given point π΄. Theorem 10.15 of [Schwabhauser] p. 92. (Contributed by Thierry Arnoux, 2-Aug-2020.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ πΏ = (LineGβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β πΊDimTarskiGβ₯2) & β’ (π β π· β ran πΏ) & β’ π = {β¨π, πβ© β£ ((π β (π β π·) β§ π β (π β π·)) β§ βπ‘ β π· π‘ β (ππΌπ))} & β’ (π β π΄ β π·) & β’ (π β π β π) & β’ (π β Β¬ π β π·) β β’ (π β βπ β π (π·(βGβπΊ)(ππΏπ΄) β§ π((hpGβπΊ)βπ·)π)) | ||
Theorem | trgcopy 28322* | 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.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ πΏ = (LineGβπΊ) & β’ πΎ = (hlGβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β Β¬ (π΄ β (π΅πΏπΆ) β¨ π΅ = πΆ)) & β’ (π β Β¬ (π· β (πΈπΏπΉ) β¨ πΈ = πΉ)) & β’ (π β (π΄ β π΅) = (π· β πΈ)) β β’ (π β βπ β π (β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βπ·πΈπββ© β§ π((hpGβπΊ)β(π·πΏπΈ))πΉ)) | ||
Theorem | trgcopyeulem 28323* | Lemma for trgcopyeu 28324. (Contributed by Thierry Arnoux, 8-Aug-2020.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ πΏ = (LineGβπΊ) & β’ πΎ = (hlGβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β Β¬ (π΄ β (π΅πΏπΆ) β¨ π΅ = πΆ)) & β’ (π β Β¬ (π· β (πΈπΏπΉ) β¨ πΈ = πΉ)) & β’ (π β (π΄ β π΅) = (π· β πΈ)) & β’ π = {β¨π, πβ© β£ ((π β (π β (π·πΏπΈ)) β§ π β (π β (π·πΏπΈ))) β§ βπ‘ β (π·πΏπΈ)π‘ β (ππΌπ))} & β’ (π β π β π) & β’ (π β π β π) & β’ (π β β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βπ·πΈπββ©) & β’ (π β β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βπ·πΈπββ©) & β’ (π β π((hpGβπΊ)β(π·πΏπΈ))πΉ) & β’ (π β π((hpGβπΊ)β(π·πΏπΈ))πΉ) β β’ (π β π = π) | ||
Theorem | trgcopyeu 28324* | 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.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ πΏ = (LineGβπΊ) & β’ πΎ = (hlGβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β Β¬ (π΄ β (π΅πΏπΆ) β¨ π΅ = πΆ)) & β’ (π β Β¬ (π· β (πΈπΏπΉ) β¨ πΈ = πΉ)) & β’ (π β (π΄ β π΅) = (π· β πΈ)) β β’ (π β β!π β π (β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βπ·πΈπββ© β§ π((hpGβπΊ)β(π·πΏπΈ))πΉ)) | ||
Syntax | ccgra 28325 | Declare the constant for the congruence between angles relation. |
class cgrA | ||
Definition | df-cgra 28326* | Define the congruence relation between angles. As for triangles we use "words of points". See iscgra 28327 for a more human readable version. (Contributed by Thierry Arnoux, 30-Jul-2020.) |
β’ cgrA = (π β V β¦ {β¨π, πβ© β£ [(Baseβπ) / π][(hlGβπ) / π]((π β (π βm (0..^3)) β§ π β (π βm (0..^3))) β§ βπ₯ β π βπ¦ β π (π(cgrGβπ)β¨βπ₯(πβ1)π¦ββ© β§ π₯(πβ(πβ1))(πβ0) β§ π¦(πβ(πβ1))(πβ2)))}) | ||
Theorem | iscgra 28327* | 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 28348 shows that those definitions are indeed equivalent. (Contributed by Thierry Arnoux, 31-Jul-2020.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ πΎ = (hlGβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) β β’ (π β (β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ© β βπ₯ β π βπ¦ β π (β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βπ₯πΈπ¦ββ© β§ π₯(πΎβπΈ)π· β§ π¦(πΎβπΈ)πΉ))) | ||
Theorem | iscgra1 28328* | A special version of iscgra 28327 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.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ πΎ = (hlGβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ β = (distβπΊ) & β’ (π β π΄ β π΅) & β’ (π β (π΄ β π΅) = (π· β πΈ)) β β’ (π β (β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ© β βπ₯ β π (β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βπ·πΈπ₯ββ© β§ π₯(πΎβπΈ)πΉ))) | ||
Theorem | iscgrad 28329 | Sufficient conditions for angle congruence, deduction version. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ πΎ = (hlGβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βππΈπββ©) & β’ (π β π(πΎβπΈ)π·) & β’ (π β π(πΎβπΈ)πΉ) β β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) | ||
Theorem | cgrane1 28330 | Angles imply inequality. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ πΎ = (hlGβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) β β’ (π β π΄ β π΅) | ||
Theorem | cgrane2 28331 | Angles imply inequality. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ πΎ = (hlGβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) β β’ (π β π΅ β πΆ) | ||
Theorem | cgrane3 28332 | Angles imply inequality. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ πΎ = (hlGβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) β β’ (π β πΈ β π·) | ||
Theorem | cgrane4 28333 | Angles imply inequality. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ πΎ = (hlGβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) β β’ (π β πΈ β πΉ) | ||
Theorem | cgrahl1 28334 | 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.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ πΎ = (hlGβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) & β’ (π β π β π) & β’ (π β π(πΎβπΈ)π·) β β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βππΈπΉββ©) | ||
Theorem | cgrahl2 28335 | 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.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ πΎ = (hlGβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) & β’ (π β π β π) & β’ (π β π(πΎβπΈ)πΉ) β β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπββ©) | ||
Theorem | cgracgr 28336 | 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.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ πΎ = (hlGβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) & β’ (π β π β π) & β’ β = (distβπΊ) & β’ (π β π β π) & β’ (π β π(πΎβπ΅)π΄) & β’ (π β π(πΎβπ΅)πΆ) & β’ (π β (π΅ β π) = (πΈ β π·)) & β’ (π β (π΅ β π) = (πΈ β πΉ)) β β’ (π β (π β π) = (π· β πΉ)) | ||
Theorem | cgraid 28337 | Angle congruence is reflexive. Theorem 11.6 of [Schwabhauser] p. 96. (Contributed by Thierry Arnoux, 31-Jul-2020.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ πΎ = (hlGβπΊ) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π΄ β π΅) & β’ (π β π΅ β πΆ) β β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ΄π΅πΆββ©) | ||
Theorem | cgraswap 28338 | Swap rays in a congruence relation. Theorem 11.9 of [Schwabhauser] p. 96. (Contributed by Thierry Arnoux, 5-Mar-2020.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ πΎ = (hlGβπΊ) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π΄ β π΅) & β’ (π β π΅ β πΆ) β β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπΆπ΅π΄ββ©) | ||
Theorem | cgrcgra 28339 | Triangle congruence implies angle congruence. This is a portion of CPCTC, focusing on a specific angle. (Contributed by Arnoux, 2-Aug-2020.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ πΎ = (hlGβπΊ) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β π΄ β π΅) & β’ (π β π΅ β πΆ) & β’ (π β β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βπ·πΈπΉββ©) β β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) | ||
Theorem | cgracom 28340 | Angle congruence commutes. Theorem 11.7 of [Schwabhauser] p. 97. (Contributed by Thierry Arnoux, 5-Mar-2020.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ πΎ = (hlGβπΊ) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) β β’ (π β β¨βπ·πΈπΉββ©(cgrAβπΊ)β¨βπ΄π΅πΆββ©) | ||
Theorem | cgratr 28341 | Angle congruence is transitive. Theorem 11.8 of [Schwabhauser] p. 97. (Contributed by Thierry Arnoux, 5-Mar-2020.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ πΎ = (hlGβπΊ) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) & β’ (π β π» β π) & β’ (π β π β π) & β’ (π β π½ β π) & β’ (π β β¨βπ·πΈπΉββ©(cgrAβπΊ)β¨βπ»ππ½ββ©) β β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ»ππ½ββ©) | ||
Theorem | flatcgra 28342 | Flat angles are congruent. (Contributed by Thierry Arnoux, 13-Feb-2023.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ β = (distβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β π΅ β (π΄πΌπΆ)) & β’ (π β πΈ β (π·πΌπΉ)) & β’ (π β π΄ β π΅) & β’ (π β πΆ β π΅) & β’ (π β π· β πΈ) & β’ (π β πΉ β πΈ) β β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) | ||
Theorem | cgraswaplr 28343 | Swap both side of angle congruence. (Contributed by Thierry Arnoux, 5-Oct-2020.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ β = (distβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) β β’ (π β β¨βπΆπ΅π΄ββ©(cgrAβπΊ)β¨βπΉπΈπ·ββ©) | ||
Theorem | cgrabtwn 28344 | Angle congruence preserves flat angles. Part of Theorem 11.21 of [Schwabhauser] p. 97. (Contributed by Thierry Arnoux, 9-Aug-2020.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ β = (distβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) & β’ (π β π΅ β (π΄πΌπΆ)) β β’ (π β πΈ β (π·πΌπΉ)) | ||
Theorem | cgrahl 28345 | Angle congruence preserves null angles. Part of Theorem 11.21 of [Schwabhauser] p. 97. (Contributed by Thierry Arnoux, 9-Aug-2020.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ β = (distβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) & β’ πΎ = (hlGβπΊ) & β’ (π β π΄(πΎβπ΅)πΆ) β β’ (π β π·(πΎβπΈ)πΉ) | ||
Theorem | cgracol 28346 | Angle congruence preserves colinearity. (Contributed by Thierry Arnoux, 9-Aug-2020.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ β = (distβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) & β’ πΏ = (LineGβπΊ) & β’ (π β (πΆ β (π΄πΏπ΅) β¨ π΄ = π΅)) β β’ (π β (πΉ β (π·πΏπΈ) β¨ π· = πΈ)) | ||
Theorem | cgrancol 28347 | Angle congruence preserves non-colinearity. (Contributed by Thierry Arnoux, 9-Aug-2020.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ β = (distβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) & β’ πΏ = (LineGβπΊ) & β’ (π β Β¬ (πΆ β (π΄πΏπ΅) β¨ π΄ = π΅)) β β’ (π β Β¬ (πΉ β (π·πΏπΈ) β¨ π· = πΈ)) | ||
Theorem | dfcgra2 28348* | 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 28326 is indeed equivalent to the textbook's definition. (Contributed by Thierry Arnoux, 2-Aug-2020.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ β = (distβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) β β’ (π β (β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ© β ((π΄ β π΅ β§ πΆ β π΅) β§ (π· β πΈ β§ πΉ β πΈ) β§ βπ β π βπ β π βπ β π βπ β π (((π΄ β (π΅πΌπ) β§ (π΄ β π) = (πΈ β π·)) β§ (πΆ β (π΅πΌπ) β§ (πΆ β π) = (πΈ β πΉ))) β§ ((π· β (πΈπΌπ) β§ (π· β π) = (π΅ β π΄)) β§ (πΉ β (πΈπΌπ) β§ (πΉ β π) = (π΅ β πΆ))) β§ (π β π) = (π β π))))) | ||
Theorem | sacgr 28349 | 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.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ β = (distβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) & β’ (π β π΅ β (π΄πΌπ)) & β’ (π β πΈ β (π·πΌπ)) & β’ (π β π΅ β π) & β’ (π β πΈ β π) β β’ (π β β¨βππ΅πΆββ©(cgrAβπΊ)β¨βππΈπΉββ©) | ||
Theorem | oacgr 28350 | 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.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ β = (distβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β π΅ β (π΄πΌπ·)) & β’ (π β π΅ β (πΆπΌπΉ)) & β’ (π β π΅ β π΄) & β’ (π β π΅ β πΆ) & β’ (π β π΅ β π·) & β’ (π β π΅ β πΉ) β β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·π΅πΉββ©) | ||
Theorem | acopy 28351* | 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.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ β = (distβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ πΏ = (LineGβπΊ) & β’ (π β Β¬ (π΄ β (π΅πΏπΆ) β¨ π΅ = πΆ)) & β’ (π β Β¬ (π· β (πΈπΏπΉ) β¨ πΈ = πΉ)) β β’ (π β βπ β π (β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπββ© β§ π((hpGβπΊ)β(π·πΏπΈ))πΉ)) | ||
Theorem | acopyeu 28352 | 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 π and π both fulfill the conditions, then they are on the same half-line. (Contributed by Thierry Arnoux, 9-Aug-2020.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ β = (distβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ πΏ = (LineGβπΊ) & β’ (π β Β¬ (π΄ β (π΅πΏπΆ) β¨ π΅ = πΆ)) & β’ (π β Β¬ (π· β (πΈπΏπΉ) β¨ πΈ = πΉ)) & β’ (π β π β π) & β’ (π β π β π) & β’ πΎ = (hlGβπΊ) & β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπββ©) & β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπββ©) & β’ (π β π((hpGβπΊ)β(π·πΏπΈ))πΉ) & β’ (π β π((hpGβπΊ)β(π·πΏπΈ))πΉ) β β’ (π β π(πΎβπΈ)π) | ||
Syntax | cinag 28353 | Extend class relation with the geometrical "point in angle" relation. |
class inA | ||
Syntax | cleag 28354 | Extend class relation with the "angle less than" relation. |
class β€β | ||
Definition | df-inag 28355* | Definition of the geometrical "in angle" relation. (Contributed by Thierry Arnoux, 15-Aug-2020.) |
β’ inA = (π β V β¦ {β¨π, π‘β© β£ ((π β (Baseβπ) β§ π‘ β ((Baseβπ) βm (0..^3))) β§ (((π‘β0) β (π‘β1) β§ (π‘β2) β (π‘β1) β§ π β (π‘β1)) β§ βπ₯ β (Baseβπ)(π₯ β ((π‘β0)(Itvβπ)(π‘β2)) β§ (π₯ = (π‘β1) β¨ π₯((hlGβπ)β(π‘β1))π))))}) | ||
Theorem | isinag 28356* | Property for point π to lie in the angle β¨βπ΄π΅πΆββ©. Definition 11.23 of [Schwabhauser] p. 101. (Contributed by Thierry Arnoux, 15-Aug-2020.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ πΎ = (hlGβπΊ) & β’ (π β π β π) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β πΊ β π) β β’ (π β (π(inAβπΊ)β¨βπ΄π΅πΆββ© β ((π΄ β π΅ β§ πΆ β π΅ β§ π β π΅) β§ βπ₯ β π (π₯ β (π΄πΌπΆ) β§ (π₯ = π΅ β¨ π₯(πΎβπ΅)π))))) | ||
Theorem | isinagd 28357 | Sufficient conditions for in-angle relation, deduction version. (Contributed by Thierry Arnoux, 20-Oct-2020.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ πΎ = (hlGβπΊ) & β’ (π β π β π) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β πΊ β π) & β’ (π β π β π) & β’ (π β π΄ β π΅) & β’ (π β πΆ β π΅) & β’ (π β π β π΅) & β’ (π β π β (π΄πΌπΆ)) & β’ (π β (π = π΅ β¨ π(πΎβπ΅)π)) β β’ (π β π(inAβπΊ)β¨βπ΄π΅πΆββ©) | ||
Theorem | inagflat 28358 | Any point lies in a flat angle. (Contributed by Thierry Arnoux, 13-Feb-2023.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ πΎ = (hlGβπΊ) & β’ (π β π β π) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β πΊ β TarskiG) & β’ (π β π β π) & β’ (π β π΄ β π΅) & β’ (π β πΆ β π΅) & β’ (π β π β π΅) & β’ (π β π΅ β (π΄πΌπΆ)) β β’ (π β π(inAβπΊ)β¨βπ΄π΅πΆββ©) | ||
Theorem | inagswap 28359 | Swap the order of the half lines delimiting the angle. Theorem 11.24 of [Schwabhauser] p. 101. (Contributed by Thierry Arnoux, 15-Aug-2020.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ πΎ = (hlGβπΊ) & β’ (π β π β π) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β πΊ β TarskiG) & β’ (π β π(inAβπΊ)β¨βπ΄π΅πΆββ©) β β’ (π β π(inAβπΊ)β¨βπΆπ΅π΄ββ©) | ||
Theorem | inagne1 28360 | Deduce inequality from the in-angle relation. (Contributed by Thierry Arnoux, 29-Oct-2021.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ πΎ = (hlGβπΊ) & β’ (π β π β π) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β πΊ β TarskiG) & β’ (π β π(inAβπΊ)β¨βπ΄π΅πΆββ©) β β’ (π β π΄ β π΅) | ||
Theorem | inagne2 28361 | Deduce inequality from the in-angle relation. (Contributed by Thierry Arnoux, 29-Oct-2021.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ πΎ = (hlGβπΊ) & β’ (π β π β π) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β πΊ β TarskiG) & β’ (π β π(inAβπΊ)β¨βπ΄π΅πΆββ©) β β’ (π β πΆ β π΅) | ||
Theorem | inagne3 28362 | Deduce inequality from the in-angle relation. (Contributed by Thierry Arnoux, 29-Oct-2021.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ πΎ = (hlGβπΊ) & β’ (π β π β π) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β πΊ β TarskiG) & β’ (π β π(inAβπΊ)β¨βπ΄π΅πΆββ©) β β’ (π β π β π΅) | ||
Theorem | inaghl 28363 | The "point lie in angle" relation is independent of the points chosen on the half lines starting from π΅. Theorem 11.25 of [Schwabhauser] p. 101. (Contributed by Thierry Arnoux, 27-Sep-2020.) |
β’ π = (BaseβπΊ) & β’ πΌ = (ItvβπΊ) & β’ πΎ = (hlGβπΊ) & β’ (π β π β π) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β πΊ β TarskiG) & β’ (π β π(inAβπΊ)β¨βπ΄π΅πΆββ©) & β’ (π β π· β π) & β’ (π β πΉ β π) & β’ (π β π β π) & β’ (π β π·(πΎβπ΅)π΄) & β’ (π β πΉ(πΎβπ΅)πΆ) & β’ (π β π(πΎβπ΅)π) β β’ (π β π(inAβπΊ)β¨βπ·π΅πΉββ©) | ||
Definition | df-leag 28364* | Definition of the geometrical "angle less than" relation. Definition 11.27 of [Schwabhauser] p. 102. (Contributed by Thierry Arnoux, 7-Oct-2020.) |
β’ β€β = (π β V β¦ {β¨π, πβ© β£ ((π β ((Baseβπ) βm (0..^3)) β§ π β ((Baseβπ) βm (0..^3))) β§ βπ₯ β (Baseβπ)(π₯(inAβπ)β¨β(πβ0)(πβ1)(πβ2)ββ© β§ β¨β(πβ0)(πβ1)(πβ2)ββ©(cgrAβπ)β¨β(πβ0)(πβ1)π₯ββ©))}) | ||
Theorem | isleag 28365* | Geometrical "less than" property for angles. Definition 11.27 of [Schwabhauser] p. 102. (Contributed by Thierry Arnoux, 7-Oct-2020.) |
β’ π = (BaseβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) β β’ (π β (β¨βπ΄π΅πΆββ©(β€β βπΊ)β¨βπ·πΈπΉββ© β βπ₯ β π (π₯(inAβπΊ)β¨βπ·πΈπΉββ© β§ β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπ₯ββ©))) | ||
Theorem | isleagd 28366 | Sufficient condition for "less than" angle relation, deduction version (Contributed by Thierry Arnoux, 12-Oct-2020.) |
β’ π = (BaseβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ β€ = (β€β βπΊ) & β’ (π β π β π) & β’ (π β π(inAβπΊ)β¨βπ·πΈπΉββ©) & β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπββ©) β β’ (π β β¨βπ΄π΅πΆββ© β€ β¨βπ·πΈπΉββ©) | ||
Theorem | leagne1 28367 | Deduce inequality from the less-than angle relation. (Contributed by Thierry Arnoux, 25-Feb-2023.) |
β’ π = (BaseβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β β¨βπ΄π΅πΆββ©(β€β βπΊ)β¨βπ·πΈπΉββ©) β β’ (π β π΄ β π΅) | ||
Theorem | leagne2 28368 | Deduce inequality from the less-than angle relation. (Contributed by Thierry Arnoux, 25-Feb-2023.) |
β’ π = (BaseβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β β¨βπ΄π΅πΆββ©(β€β βπΊ)β¨βπ·πΈπΉββ©) β β’ (π β πΆ β π΅) | ||
Theorem | leagne3 28369 | Deduce inequality from the less-than angle relation. (Contributed by Thierry Arnoux, 25-Feb-2023.) |
β’ π = (BaseβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β β¨βπ΄π΅πΆββ©(β€β βπΊ)β¨βπ·πΈπΉββ©) β β’ (π β π· β πΈ) | ||
Theorem | leagne4 28370 | Deduce inequality from the less-than angle relation. (Contributed by Thierry Arnoux, 25-Feb-2023.) |
β’ π = (BaseβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β β¨βπ΄π΅πΆββ©(β€β βπΊ)β¨βπ·πΈπΉββ©) β β’ (π β πΉ β πΈ) | ||
Theorem | cgrg3col4 28371* | 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.) |
β’ π = (BaseβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ πΏ = (LineGβπΊ) & β’ (π β π β π) & β’ (π β β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βπ·πΈπΉββ©) & β’ (π β (π β (π΄πΏπΆ) β¨ π΄ = πΆ)) β β’ (π β βπ¦ β π β¨βπ΄π΅πΆπββ©(cgrGβπΊ)β¨βπ·πΈπΉπ¦ββ©) | ||
Theorem | tgsas1 28372 | 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.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β (π΄ β π΅) = (π· β πΈ)) & β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) & β’ (π β (π΅ β πΆ) = (πΈ β πΉ)) β β’ (π β (πΆ β π΄) = (πΉ β π·)) | ||
Theorem | tgsas 28373 | 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.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β (π΄ β π΅) = (π· β πΈ)) & β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) & β’ (π β (π΅ β πΆ) = (πΈ β πΉ)) β β’ (π β β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βπ·πΈπΉββ©) | ||
Theorem | tgsas2 28374 | First congruence theorem: SAS. Theorem 11.49 of [Schwabhauser] p. 107. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β (π΄ β π΅) = (π· β πΈ)) & β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) & β’ (π β (π΅ β πΆ) = (πΈ β πΉ)) & β’ (π β π΄ β πΆ) β β’ (π β β¨βπΆπ΄π΅ββ©(cgrAβπΊ)β¨βπΉπ·πΈββ©) | ||
Theorem | tgsas3 28375 | First congruence theorem: SAS. Theorem 11.49 of [Schwabhauser] p. 107. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β (π΄ β π΅) = (π· β πΈ)) & β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) & β’ (π β (π΅ β πΆ) = (πΈ β πΉ)) & β’ (π β π΄ β πΆ) β β’ (π β β¨βπ΅πΆπ΄ββ©(cgrAβπΊ)β¨βπΈπΉπ·ββ©) | ||
Theorem | tgasa1 28376 | 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.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ πΏ = (LineGβπΊ) & β’ (π β Β¬ (πΆ β (π΄πΏπ΅) β¨ π΄ = π΅)) & β’ (π β (π΄ β π΅) = (π· β πΈ)) & β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) & β’ (π β β¨βπΆπ΄π΅ββ©(cgrAβπΊ)β¨βπΉπ·πΈββ©) β β’ (π β (π΅ β πΆ) = (πΈ β πΉ)) | ||
Theorem | tgasa 28377 | 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.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ πΏ = (LineGβπΊ) & β’ (π β Β¬ (πΆ β (π΄πΏπ΅) β¨ π΄ = π΅)) & β’ (π β (π΄ β π΅) = (π· β πΈ)) & β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) & β’ (π β β¨βπΆπ΄π΅ββ©(cgrAβπΊ)β¨βπΉπ·πΈββ©) β β’ (π β β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βπ·πΈπΉββ©) | ||
Theorem | tgsss1 28378 | 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.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β (π΄ β π΅) = (π· β πΈ)) & β’ (π β (π΅ β πΆ) = (πΈ β πΉ)) & β’ (π β (πΆ β π΄) = (πΉ β π·)) & β’ (π β π΄ β π΅) & β’ (π β π΅ β πΆ) & β’ (π β πΆ β π΄) β β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) | ||
Theorem | tgsss2 28379 | Third congruence theorem: SSS. Theorem 11.51 of [Schwabhauser] p. 109. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β (π΄ β π΅) = (π· β πΈ)) & β’ (π β (π΅ β πΆ) = (πΈ β πΉ)) & β’ (π β (πΆ β π΄) = (πΉ β π·)) & β’ (π β π΄ β π΅) & β’ (π β π΅ β πΆ) & β’ (π β πΆ β π΄) β β’ (π β β¨βπΆπ΄π΅ββ©(cgrAβπΊ)β¨βπΉπ·πΈββ©) | ||
Theorem | tgsss3 28380 | Third congruence theorem: SSS. Theorem 11.51 of [Schwabhauser] p. 109. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β (π΄ β π΅) = (π· β πΈ)) & β’ (π β (π΅ β πΆ) = (πΈ β πΉ)) & β’ (π β (πΆ β π΄) = (πΉ β π·)) & β’ (π β π΄ β π΅) & β’ (π β π΅ β πΆ) & β’ (π β πΆ β π΄) β β’ (π β β¨βπ΅πΆπ΄ββ©(cgrAβπΊ)β¨βπΈπΉπ·ββ©) | ||
Theorem | dfcgrg2 28381 | 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 28029, already covers that part: see trgcgr 28034. 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 28034. (Contributed by Thierry Arnoux, 18-Jan-2023.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΈ β π) & β’ (π β πΉ β π) & β’ (π β π΄ β π΅) & β’ (π β π΅ β πΆ) & β’ (π β πΆ β π΄) β β’ (π β (β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βπ·πΈπΉββ© β (((π΄ β π΅) = (π· β πΈ) β§ (π΅ β πΆ) = (πΈ β πΉ) β§ (πΆ β π΄) = (πΉ β π·)) β§ (β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ© β§ β¨βπΆπ΄π΅ββ©(cgrAβπΊ)β¨βπΉπ·πΈββ© β§ β¨βπ΅πΆπ΄ββ©(cgrAβπΊ)β¨βπΈπΉπ·ββ©)))) | ||
Theorem | isoas 28382 | 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.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ πΏ = (LineGβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β Β¬ (πΆ β (π΄πΏπ΅) β¨ π΄ = π΅)) & β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ΄πΆπ΅ββ©) β β’ (π β (π΄ β π΅) = (π΄ β πΆ)) | ||
Syntax | ceqlg 28383 | Declare the class of equilateral triangles. |
class eqltrG | ||
Definition | df-eqlg 28384* | Define the class of equilateral triangles. (Contributed by Thierry Arnoux, 27-Nov-2019.) |
β’ eqltrG = (π β V β¦ {π₯ β ((Baseβπ) βm (0..^3)) β£ π₯(cgrGβπ)β¨β(π₯β1)(π₯β2)(π₯β0)ββ©}) | ||
Theorem | iseqlg 28385 | Property of a triangle being equilateral. (Contributed by Thierry Arnoux, 5-Oct-2020.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ πΏ = (LineGβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) β β’ (π β (β¨βπ΄π΅πΆββ© β (eqltrGβπΊ) β β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βπ΅πΆπ΄ββ©)) | ||
Theorem | iseqlgd 28386 | Condition for a triangle to be equilateral. (Contributed by Thierry Arnoux, 5-Oct-2020.) |
β’ π = (BaseβπΊ) & β’ β = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ πΏ = (LineGβπΊ) & β’ (π β πΊ β TarskiG) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β (π΄ β π΅) = (π΅ β πΆ)) & β’ (π β (π΅ β πΆ) = (πΆ β π΄)) & β’ (π β (πΆ β π΄) = (π΄ β π΅)) β β’ (π β β¨βπ΄π΅πΆββ© β (eqltrGβπΊ)) | ||
Theorem | f1otrgds 28387* | Convenient lemma for f1otrg 28389. (Contributed by Thierry Arnoux, 19-Mar-2019.) |
β’ π = (BaseβπΊ) & β’ π· = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ π΅ = (Baseβπ») & β’ πΈ = (distβπ») & β’ π½ = (Itvβπ») & β’ (π β πΉ:π΅β1-1-ontoβπ) & β’ ((π β§ (π β π΅ β§ π β π΅)) β (ππΈπ) = ((πΉβπ)π·(πΉβπ))) & β’ ((π β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β (π β (ππ½π) β (πΉβπ) β ((πΉβπ)πΌ(πΉβπ)))) & β’ (π β π β π΅) & β’ (π β π β π΅) β β’ (π β (ππΈπ) = ((πΉβπ)π·(πΉβπ))) | ||
Theorem | f1otrgitv 28388* | Convenient lemma for f1otrg 28389. (Contributed by Thierry Arnoux, 19-Mar-2019.) |
β’ π = (BaseβπΊ) & β’ π· = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ π΅ = (Baseβπ») & β’ πΈ = (distβπ») & β’ π½ = (Itvβπ») & β’ (π β πΉ:π΅β1-1-ontoβπ) & β’ ((π β§ (π β π΅ β§ π β π΅)) β (ππΈπ) = ((πΉβπ)π·(πΉβπ))) & β’ ((π β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β (π β (ππ½π) β (πΉβπ) β ((πΉβπ)πΌ(πΉβπ)))) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β π΅) β β’ (π β (π β (ππ½π) β (πΉβπ) β ((πΉβπ)πΌ(πΉβπ)))) | ||
Theorem | f1otrg 28389* | A bijection between bases which conserves distances and intervals conserves also geometries. (Contributed by Thierry Arnoux, 23-Mar-2019.) |
β’ π = (BaseβπΊ) & β’ π· = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ π΅ = (Baseβπ») & β’ πΈ = (distβπ») & β’ π½ = (Itvβπ») & β’ (π β πΉ:π΅β1-1-ontoβπ) & β’ ((π β§ (π β π΅ β§ π β π΅)) β (ππΈπ) = ((πΉβπ)π·(πΉβπ))) & β’ ((π β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β (π β (ππ½π) β (πΉβπ) β ((πΉβπ)πΌ(πΉβπ)))) & β’ (π β π» β π) & β’ (π β πΊ β TarskiG) & β’ (π β (LineGβπ») = (π₯ β π΅, π¦ β (π΅ β {π₯}) β¦ {π§ β π΅ β£ (π§ β (π₯π½π¦) β¨ π₯ β (π§π½π¦) β¨ π¦ β (π₯π½π§))})) β β’ (π β π» β TarskiG) | ||
Theorem | f1otrge 28390* | 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.) |
β’ π = (BaseβπΊ) & β’ π· = (distβπΊ) & β’ πΌ = (ItvβπΊ) & β’ π΅ = (Baseβπ») & β’ πΈ = (distβπ») & β’ π½ = (Itvβπ») & β’ (π β πΉ:π΅β1-1-ontoβπ) & β’ ((π β§ (π β π΅ β§ π β π΅)) β (ππΈπ) = ((πΉβπ)π·(πΉβπ))) & β’ ((π β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β (π β (ππ½π) β (πΉβπ) β ((πΉβπ)πΌ(πΉβπ)))) & β’ (π β π» β π) & β’ (π β πΊ β TarskiGE) β β’ (π β π» β TarskiGE) | ||
Syntax | cttg 28391 | Function to convert an algebraic structure to a Tarski geometry. |
class toTG | ||
Definition | df-ttg 28392* | 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.) |
β’ toTG = (π€ β V β¦ β¦(π₯ β (Baseβπ€), π¦ β (Baseβπ€) β¦ {π§ β (Baseβπ€) β£ βπ β (0[,]1)(π§(-gβπ€)π₯) = (π( Β·π βπ€)(π¦(-gβπ€)π₯))}) / πβ¦((π€ sSet β¨(Itvβndx), πβ©) sSet β¨(LineGβndx), (π₯ β (Baseβπ€), π¦ β (Baseβπ€) β¦ {π§ β (Baseβπ€) β£ (π§ β (π₯ππ¦) β¨ π₯ β (π§ππ¦) β¨ π¦ β (π₯ππ§))})β©)) | ||
Theorem | ttgval 28393* | Define a function to augment a subcomplex Hilbert space with betweenness and a line definition. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Proof shortened by AV, 9-Nov-2024.) |
β’ πΊ = (toTGβπ») & β’ π΅ = (Baseβπ») & β’ β = (-gβπ») & β’ Β· = ( Β·π βπ») & β’ πΌ = (ItvβπΊ) β β’ (π» β π β (πΊ = ((π» sSet β¨(Itvβndx), (π₯ β π΅, π¦ β π΅ β¦ {π§ β π΅ β£ βπ β (0[,]1)(π§ β π₯) = (π Β· (π¦ β π₯))})β©) sSet β¨(LineGβndx), (π₯ β π΅, π¦ β π΅ β¦ {π§ β π΅ β£ (π§ β (π₯πΌπ¦) β¨ π₯ β (π§πΌπ¦) β¨ π¦ β (π₯πΌπ§))})β©) β§ πΌ = (π₯ β π΅, π¦ β π΅ β¦ {π§ β π΅ β£ βπ β (0[,]1)(π§ β π₯) = (π Β· (π¦ β π₯))}))) | ||
Theorem | ttgvalOLD 28394* | Obsolete proof of ttgval 28393 as of 9-Nov-2024. Define a function to augment a subcomplex Hilbert space with betweenness and a line definition. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ πΊ = (toTGβπ») & β’ π΅ = (Baseβπ») & β’ β = (-gβπ») & β’ Β· = ( Β·π βπ») & β’ πΌ = (ItvβπΊ) β β’ (π» β π β (πΊ = ((π» sSet β¨(Itvβndx), (π₯ β π΅, π¦ β π΅ β¦ {π§ β π΅ β£ βπ β (0[,]1)(π§ β π₯) = (π Β· (π¦ β π₯))})β©) sSet β¨(LineGβndx), (π₯ β π΅, π¦ β π΅ β¦ {π§ β π΅ β£ (π§ β (π₯πΌπ¦) β¨ π₯ β (π§πΌπ¦) β¨ π¦ β (π₯πΌπ§))})β©) β§ πΌ = (π₯ β π΅, π¦ β π΅ β¦ {π§ β π΅ β£ βπ β (0[,]1)(π§ β π₯) = (π Β· (π¦ β π₯))}))) | ||
Theorem | ttglem 28395 | Lemma for ttgbas 28397, ttgvsca 28402 etc. (Contributed by Thierry Arnoux, 15-Apr-2019.) (Revised by AV, 29-Oct-2024.) |
β’ πΊ = (toTGβπ») & β’ πΈ = Slot (πΈβndx) & β’ (πΈβndx) β (LineGβndx) & β’ (πΈβndx) β (Itvβndx) β β’ (πΈβπ») = (πΈβπΊ) | ||
Theorem | ttglemOLD 28396 | Obsolete version of ttglem 28395 as of 29-Oct-2024. Lemma for ttgbas 28397 and ttgvsca 28402. (Contributed by Thierry Arnoux, 15-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ πΊ = (toTGβπ») & β’ πΈ = Slot π & β’ π β β & β’ π < ;16 β β’ (πΈβπ») = (πΈβπΊ) | ||
Theorem | ttgbas 28397 | The base set of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Revised by AV, 29-Oct-2024.) |
β’ πΊ = (toTGβπ») & β’ π΅ = (Baseβπ») β β’ π΅ = (BaseβπΊ) | ||
Theorem | ttgbasOLD 28398 | Obsolete proof of ttgbas 28397 as of 29-Oct-2024. The base set of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ πΊ = (toTGβπ») & β’ π΅ = (Baseβπ») β β’ π΅ = (BaseβπΊ) | ||
Theorem | ttgplusg 28399 | The addition operation of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Revised by AV, 29-Oct-2024.) |
β’ πΊ = (toTGβπ») & β’ + = (+gβπ») β β’ + = (+gβπΊ) | ||
Theorem | ttgplusgOLD 28400 | Obsolete proof of ttgplusg 28399 as of 29-Oct-2024. The addition operation of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ πΊ = (toTGβπ») & β’ + = (+gβπ») β β’ + = (+gβπΊ) |
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