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Theorem List for Metamath Proof Explorer - 24001-24100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembtwnswapid 24001 If you can swap the first two arguments of a betweenness statement, then those arguments are identical. Theorem 3.4 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.)

Theorembtwnswapid2 24002 If you can swap arguments one and three of a betweenness statement, then those arguments are identical. (Contributed by Scott Fenton, 7-Oct-2013.)

Theorembtwnintr 24003 Inner transitivity law for betweenness. Left hand side of Theorem 3.5 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.)

Theorembtwnexch3 24004 Exchange the first endpoint in betweenness. Left hand side of Theorem 3.6 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.)

Theorembtwnexch3and 24005 Deduction form of btwnexch3 24004. (Contributed by Scott Fenton, 13-Oct-2013.)

Theorembtwnouttr2 24006 Outer transitivity law for betweenness. Left hand side of Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.)

Theorembtwnexch2 24007 Exchange the outer point of two betweenness statements. Right hand side of Theorem 3.5 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 14-Jun-2013.)

Theorembtwnouttr 24008 Outer transitivity law for betweenness. Right hand side of Theorem 3.7 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 14-Jun-2013.)

Theorembtwnexch 24009 Outer transitivity law for betweenness. Right hand side of Theorem 3.6 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 24-Sep-2013.)

Theorembtwnexchand 24010 Deduction form of btwnexch 24009. (Contributed by Scott Fenton, 13-Oct-2013.)

Theorembtwndiff 24011* There is always a distinct from such that lies between and . Theorem 3.14 of [Schwabhauser] p. 32. (Contributed by Scott Fenton, 24-Sep-2013.)

Theoremtrisegint 24012* A line segment between two sides of a triange intersects a segment crossing from the remaining side to the opposite vertex. Theorem 3.17 of [Schwabhauser] p. 33. (Contributed by Scott Fenton, 24-Sep-2013.)

16.7.34  Segment Transportation

Syntaxctransport 24013 Declare the syntax for the segment transport function.
TransportTo

Definitiondf-transport 24014* Define the segment transport function. See fvtransport 24016 for an explanation of the function. (Contributed by Scott Fenton, 18-Oct-2013.)
TransportTo Cgr

Theoremfuntransport 24015 The TransportTo relationship is a function. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
TransportTo

Theoremfvtransport 24016* Calculate the value of the TransportTo function. This function takes four points, through , where and are distinct. It then returns the point that extends by the length of . (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
TransportTo Cgr

Theoremtransportcl 24017 Closure law for segment transport. (Contributed by Scott Fenton, 19-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
TransportTo

Theoremtransportprops 24018 Calculate the defining properties of the transport function (Contributed by Scott Fenton, 19-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
TransportTo TransportTo Cgr

16.7.35  Properties relating betweenness and congruence

Syntaxcifs 24019 Declare the syntax for the inner five segment predicate.

Syntaxccgr3 24020 Declare the syntax for the three place congruence predicate.
Cgr3

Syntaxccolin 24021 Declare the syntax for the colinearity predicate.

Syntaxcfs 24022 Declare the syntax for the five segment predicate.

Definitiondf-ifs 24023* The inner five segment configuration is an abbreviation for another congruence condition. See brifs 24027 and ifscgr 24028 for how it is used. Definition 4.1 of [Schwabhauser] p. 34. (Contributed by Scott Fenton, 26-Sep-2013.)
Cgr Cgr Cgr Cgr

Definitiondf-cgr3 24024* The three place congruence predicate. This is an abbreviation for saying that all three pair in a triple are congruent with each other. Three place form of Definition 4.4 of [Schwabhauser] p. 35. (Contributed by Scott Fenton, 4-Oct-2013.)
Cgr3 Cgr Cgr Cgr

Definitiondf-colinear 24025* The colinearity predicate states that the three points in its arguments sit on one line. Definition 4.10 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 25-Oct-2013.)

Definitiondf-fs 24026* The general five segment configuration is a generalization of the outer and inner five segment configurations. See brfs 24063 and fscgr 24064 for its use. Definition 4.15 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3 Cgr Cgr

Theorembrifs 24027 Binary relationship form of the inner five segment predicate. (Contributed by Scott Fenton, 26-Sep-2013.)
Cgr Cgr Cgr Cgr

Theoremifscgr 24028 Inner five segment congruence. Take two triangles, and , with between and and between and . If the other components of the triangles are congruent, then so are and . Theorem 4.2 of [Schwabhauser] p. 34. (Contributed by Scott Fenton, 27-Sep-2013.)
Cgr

Theoremcgrsub 24029 Removing identical parts from the end of a line segment preserves congruence. Theorem 4.3 of [Schwabhauser] p. 35. (Contributed by Scott Fenton, 4-Oct-2013.)
Cgr Cgr Cgr

Theorembrcgr3 24030 Binary relationship form of the three-place congruence predicate. (Contributed by Scott Fenton, 4-Oct-2013.)
Cgr3 Cgr Cgr Cgr

Theoremcgr3permute3 24031 Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3 Cgr3

Theoremcgr3permute1 24032 Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3 Cgr3

Theoremcgr3permute2 24033 Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3 Cgr3

Theoremcgr3permute4 24034 Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3 Cgr3

Theoremcgr3permute5 24035 Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3 Cgr3

Theoremcgr3tr4 24036 Transitivity law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3 Cgr3 Cgr3

Theoremcgr3com 24037 Commutativity law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3 Cgr3

Theoremcgr3rflx 24038 Identity law for three-place congruence. (Contributed by Scott Fenton, 6-Oct-2013.)
Cgr3

Theoremcgrxfr 24039* A line segment can be divided at the same place as a congruent line segment is divided. Theorem 4.5 of [Schwabhauser] p. 35. (Contributed by Scott Fenton, 4-Oct-2013.)
Cgr Cgr3

Theorembtwnxfr 24040 A condition for extending betweenness to a new set of points based on congruence with another set of points. Theorem 4.6 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 4-Oct-2013.)
Cgr3

Theoremcolinrel 24041 Colinearity is a relationship. (Contributed by Scott Fenton, 7-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorembrcolinear2 24042* Alternate colinearity binary relationship. (Contributed by Scott Fenton, 7-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorembrcolinear 24043 The binary relationship form of the colinearity predicate. (Contributed by Scott Fenton, 5-Oct-2013.)

Theoremcolinearex 24044 The colinear predicate exists. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremcolineardim1 24045 If is colinear with and , then is in the same space as . (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremcolinearperm1 24046 Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.)

Theoremcolinearperm3 24047 Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.)

Theoremcolinearperm2 24048 Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.)

Theoremcolinearperm4 24049 Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.)

Theoremcolinearperm5 24050 Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.)

Theoremcolineartriv1 24051 Trivial case of colinearity. Theorem 4.12 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 5-Oct-2013.)

Theoremcolineartriv2 24052 Trivial case of colinearity. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorembtwncolinear1 24053 Betweenness implies colinearity. (Contributed by Scott Fenton, 7-Oct-2013.)

Theorembtwncolinear2 24054 Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorembtwncolinear3 24055 Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorembtwncolinear4 24056 Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorembtwncolinear5 24057 Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorembtwncolinear6 24058 Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremcolinearxfr 24059 Transfer law for colinearity. Theorem 4.13 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3

Theoremlineext 24060* Extend a line with a missing point. Theorem 4.14 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 6-Oct-2013.)
Cgr Cgr3

Theorembrofs2 24061 Change some conditions for outer five segment predicate. (Contributed by Scott Fenton, 6-Oct-2013.)
Cgr3 Cgr Cgr

Theorembrifs2 24062 Change some conditions for inner five segment predicate. (Contributed by Scott Fenton, 6-Oct-2013.)
Cgr3 Cgr Cgr

Theorembrfs 24063 Binary relationship form of the general five segment predicate. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3 Cgr Cgr

Theoremfscgr 24064 Congruence law for the general five segment configuration. Theorem 4.16 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr

Theoremlinecgr 24065 Congruence rule for lines. Theorem 4.17 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 6-Oct-2013.)
Cgr Cgr Cgr

Theoremlinecgrand 24066 Deduction form of linecgr 24065. (Contributed by Scott Fenton, 14-Oct-2013.)
Cgr        Cgr        Cgr

Theoremlineid 24067 Identity law for points on lines. Theorem 4.18 of [Schwabhauser] p. 38. (Contributed by Scott Fenton, 7-Oct-2013.)
Cgr Cgr

Theoremidinside 24068 Law for finding a point inside a segment. Theorem 4.19 of [Schwabhauser] p. 38. (Contributed by Scott Fenton, 7-Oct-2013.)
Cgr Cgr

Theoremendofsegid 24069 If , , and fall in order on a line, and and are congruent, then . (Contributed by Scott Fenton, 7-Oct-2013.)
Cgr

Theoremendofsegidand 24070 Deduction form of endofsegid 24069. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Cgr

16.7.36  Connectivity of betweenness

Theorembtwnconn1lem1 24071 Lemma for btwnconn1 24085. The next several lemmas introduce various properties of hypothetical points that end up eliminating alternatives to connectivity. We begin by showing a congruence property of those hypothetical points. (Contributed by Scott Fenton, 8-Oct-2013.)
Cgr Cgr Cgr Cgr Cgr

Theorembtwnconn1lem2 24072 Lemma for btwnconn1 24085. Now, we show that two of the hypotheticals we introduced in the first lemma are identical. (Contributed by Scott Fenton, 8-Oct-2013.)
Cgr Cgr Cgr Cgr

Theorembtwnconn1lem3 24073 Lemma for btwnconn1 24085. Establish the next congruence in the series. (Contributed by Scott Fenton, 8-Oct-2013.)
Cgr Cgr Cgr Cgr Cgr

Theorembtwnconn1lem4 24074 Lemma for btwnconn1 24085. Assuming , we now attempt to force from here out via a series of congruences. (Contributed by Scott Fenton, 8-Oct-2013.)
Cgr Cgr Cgr Cgr Cgr

Theorembtwnconn1lem5 24075 Lemma for btwnconn1 24085. Now, we introduce , the intersection of and . We begin by showing that it is the midpoint of and (Contributed by Scott Fenton, 8-Oct-2013.)
Cgr Cgr Cgr Cgr Cgr

Theorembtwnconn1lem6 24076 Lemma for btwnconn1 24085. Next, we show that is the midpoint of and (Contributed by Scott Fenton, 8-Oct-2013.)
Cgr Cgr Cgr Cgr Cgr

Theorembtwnconn1lem7 24077 Lemma for btwnconn1 24085. Under our assumptions, and are distinct. (Contributed by Scott Fenton, 8-Oct-2013.)
Cgr Cgr Cgr Cgr

Theorembtwnconn1lem8 24078 Lemma for btwnconn1 24085. Now, we introduce the last three points used in the construction: , , and will turn out to be equal further down, and will provide us with the key to the final statement. We begin by establishing congruence of and (Contributed by Scott Fenton, 8-Oct-2013.)
Cgr Cgr Cgr Cgr Cgr Cgr Cgr Cgr

Theorembtwnconn1lem9 24079 Lemma for btwnconn1 24085. Now, a quick use of transitivity to establish congruence on and (Contributed by Scott Fenton, 8-Oct-2013.)
Cgr Cgr Cgr Cgr Cgr Cgr Cgr Cgr

Theorembtwnconn1lem10 24080 Lemma for btwnconn1 24085. Now we establish a congruence that will give us when we compute later on. (Contributed by Scott Fenton, 8-Oct-2013.)
Cgr Cgr Cgr Cgr Cgr Cgr Cgr Cgr

Theorembtwnconn1lem11 24081 Lemma for btwnconn1 24085. Now, we establish that and are equidistant from (Contributed by Scott Fenton, 8-Oct-2013.)
Cgr Cgr Cgr Cgr Cgr Cgr Cgr Cgr

Theorembtwnconn1lem12 24082 Lemma for btwnconn1 24085. Using a long string of invocations of linecgr 24065, we show that . (Contributed by Scott Fenton, 9-Oct-2013.)
Cgr Cgr Cgr Cgr Cgr Cgr Cgr

Theorembtwnconn1lem13 24083 Lemma for btwnconn1 24085. Begin back-filling and eliminating hypotheses. (Contributed by Scott Fenton, 9-Oct-2013.)
Cgr Cgr Cgr Cgr

Theorembtwnconn1lem14 24084 Lemma for btwnconn1 24085. Final statement of the theorem when . (Contributed by Scott Fenton, 9-Oct-2013.)

Theorembtwnconn1 24085 Connectitivy law for betweenness. Theorem 5.1 of [Schwabhauser] p. 39-41. (Contributed by Scott Fenton, 9-Oct-2013.)

Theorembtwnconn2 24086 Another connectivity law for betweenness. Theorem 5.2 of [Schwabhauser] p. 41. (Contributed by Scott Fenton, 9-Oct-2013.)

Theorembtwnconn3 24087 Inner connectivity law for betweenness. Theorem 5.3 of [Schwabhauser] p. 41. (Contributed by Scott Fenton, 9-Oct-2013.)

Theoremmidofsegid 24088 If two points fall in the same place in the middle of a segment, then they are identical. (Contributed by Scott Fenton, 16-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Cgr

Theoremsegcon2 24089* Generalization of axsegcon 23916. This time, we generate an endpoint for a segment on the ray congruent to and starting at , as opposed to axsegcon 23916, where the segment starts at (Contributed by Scott Fenton, 14-Oct-2013.) (Removed unneeded inequality, 15-Oct-2013.)
Cgr

16.7.37  Segment less than or equal to

Syntaxcsegle 24090 Declare the constant for the segment less than or equal to relationship.

Definitiondf-segle 24091* Define the segment length comparison relationship. This relationship expresses that the segment is no longer than . In this section, we establish various properties of this relationship showing that it is a transitive, reflexive relationship on pairs of points that is substitutive under congruence. Definition 5.4 of [Schwabhauser] p. 41. (Contributed by Scott Fenton, 11-Oct-2013.)
Cgr

Theorembrsegle 24092* Binary relationship form of the segment comparison relationship. (Contributed by Scott Fenton, 11-Oct-2013.)
Cgr

Theorembrsegle2 24093* Alternate characterization of segment comparison. Theorem 5.5 of [Schwabhauser] p. 41-42. (Contributed by Scott Fenton, 11-Oct-2013.)
Cgr

Theoremseglecgr12im 24094 Substitution law for segment comparison under congruence. Theorem 5.6 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 11-Oct-2013.)
Cgr Cgr

Theoremseglecgr12 24095 Substitution law for segment comparison under congruence. Biconditional version. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Cgr Cgr

Theoremseglerflx 24096 Segment comparison is reflexive. Theorem 5.7 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 11-Oct-2013.)

Theoremseglemin 24097 Any segment is at least as long as a degenerate segment. Theorem 5.11 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 11-Oct-2013.)

Theoremsegletr 24098 Segment less than is transitive. Theorem 5.8 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 11-Oct-2013.)

Theoremsegleantisym 24099 Antisymmetry law for segment comparison. Theorem 5.9 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 14-Oct-2013.)
Cgr

Theoremseglelin 24100 Linearity law for segment comparison. Theorem 5.10 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 14-Oct-2013.)

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