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Theorem List for Metamath Proof Explorer - 26001-26100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxccgr3 26001 Declare the syntax for the three place congruence predicate.
Cgr3

Syntaxccolin 26002 Declare the syntax for the colinearity predicate.

Syntaxcfs 26003 Declare the syntax for the five segment predicate.

Definitiondf-ifs 26004* The inner five segment configuration is an abbreviation for another congruence condition. See brifs 26008 and ifscgr 26009 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 26005* 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 26006* 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 26007* The general five segment configuration is a generalization of the outer and inner five segment configurations. See brfs 26044 and fscgr 26045 for its use. Definition 4.15 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3 Cgr Cgr

Theorembrifs 26008 Binary relationship form of the inner five segment predicate. (Contributed by Scott Fenton, 26-Sep-2013.)
Cgr Cgr Cgr Cgr

Theoremifscgr 26009 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 26010 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 26011 Binary relationship form of the three-place congruence predicate. (Contributed by Scott Fenton, 4-Oct-2013.)
Cgr3 Cgr Cgr Cgr

Theoremcgr3permute3 26012 Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3 Cgr3

Theoremcgr3permute1 26013 Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3 Cgr3

Theoremcgr3permute2 26014 Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3 Cgr3

Theoremcgr3permute4 26015 Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3 Cgr3

Theoremcgr3permute5 26016 Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3 Cgr3

Theoremcgr3tr4 26017 Transitivity law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3 Cgr3 Cgr3

Theoremcgr3com 26018 Commutativity law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3 Cgr3

Theoremcgr3rflx 26019 Identity law for three-place congruence. (Contributed by Scott Fenton, 6-Oct-2013.)
Cgr3

Theoremcgrxfr 26020* 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 26021 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 26022 Colinearity is a relationship. (Contributed by Scott Fenton, 7-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorembrcolinear2 26023* Alternate colinearity binary relationship. (Contributed by Scott Fenton, 7-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorembrcolinear 26024 The binary relationship form of the colinearity predicate. (Contributed by Scott Fenton, 5-Oct-2013.)

Theoremcolinearex 26025 The colinear predicate exists. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremcolineardim1 26026 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 26027 Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.)

Theoremcolinearperm3 26028 Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.)

Theoremcolinearperm2 26029 Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.)

Theoremcolinearperm4 26030 Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.)

Theoremcolinearperm5 26031 Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.)

Theoremcolineartriv1 26032 Trivial case of colinearity. Theorem 4.12 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 5-Oct-2013.)

Theoremcolineartriv2 26033 Trivial case of colinearity. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorembtwncolinear1 26034 Betweenness implies colinearity. (Contributed by Scott Fenton, 7-Oct-2013.)

Theorembtwncolinear2 26035 Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorembtwncolinear3 26036 Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorembtwncolinear4 26037 Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorembtwncolinear5 26038 Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorembtwncolinear6 26039 Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremcolinearxfr 26040 Transfer law for colinearity. Theorem 4.13 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3

Theoremlineext 26041* Extend a line with a missing point. Theorem 4.14 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 6-Oct-2013.)
Cgr Cgr3

Theorembrofs2 26042 Change some conditions for outer five segment predicate. (Contributed by Scott Fenton, 6-Oct-2013.)
Cgr3 Cgr Cgr

Theorembrifs2 26043 Change some conditions for inner five segment predicate. (Contributed by Scott Fenton, 6-Oct-2013.)
Cgr3 Cgr Cgr

Theorembrfs 26044 Binary relationship form of the general five segment predicate. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3 Cgr Cgr

Theoremfscgr 26045 Congruence law for the general five segment configuration. Theorem 4.16 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr

Theoremlinecgr 26046 Congruence rule for lines. Theorem 4.17 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 6-Oct-2013.)
Cgr Cgr Cgr

Theoremlinecgrand 26047 Deduction form of linecgr 26046. (Contributed by Scott Fenton, 14-Oct-2013.)
Cgr        Cgr        Cgr

Theoremlineid 26048 Identity law for points on lines. Theorem 4.18 of [Schwabhauser] p. 38. (Contributed by Scott Fenton, 7-Oct-2013.)
Cgr Cgr

Theoremidinside 26049 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 26050 If , , and fall in order on a line, and and are congruent, then . (Contributed by Scott Fenton, 7-Oct-2013.)
Cgr

Theoremendofsegidand 26051 Deduction form of endofsegid 26050. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Cgr

19.7.46  Connectivity of betweenness

Theorembtwnconn1lem1 26052 Lemma for btwnconn1 26066. 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 26053 Lemma for btwnconn1 26066. 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 26054 Lemma for btwnconn1 26066. Establish the next congruence in the series. (Contributed by Scott Fenton, 8-Oct-2013.)
Cgr Cgr Cgr Cgr Cgr

Theorembtwnconn1lem4 26055 Lemma for btwnconn1 26066. 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 26056 Lemma for btwnconn1 26066. 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 26057 Lemma for btwnconn1 26066. Next, we show that is the midpoint of and (Contributed by Scott Fenton, 8-Oct-2013.)
Cgr Cgr Cgr Cgr Cgr

Theorembtwnconn1lem7 26058 Lemma for btwnconn1 26066. Under our assumptions, and are distinct. (Contributed by Scott Fenton, 8-Oct-2013.)
Cgr Cgr Cgr Cgr

Theorembtwnconn1lem8 26059 Lemma for btwnconn1 26066. 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 26060 Lemma for btwnconn1 26066. 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 26061 Lemma for btwnconn1 26066. 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 26062 Lemma for btwnconn1 26066. Now, we establish that and are equidistant from (Contributed by Scott Fenton, 8-Oct-2013.)
Cgr Cgr Cgr Cgr Cgr Cgr Cgr Cgr

Theorembtwnconn1lem12 26063 Lemma for btwnconn1 26066. Using a long string of invocations of linecgr 26046, we show that . (Contributed by Scott Fenton, 9-Oct-2013.)
Cgr Cgr Cgr Cgr Cgr Cgr Cgr

Theorembtwnconn1lem13 26064 Lemma for btwnconn1 26066. Begin back-filling and eliminating hypotheses. (Contributed by Scott Fenton, 9-Oct-2013.)
Cgr Cgr Cgr Cgr

Theorembtwnconn1lem14 26065 Lemma for btwnconn1 26066. Final statement of the theorem when . (Contributed by Scott Fenton, 9-Oct-2013.)

Theorembtwnconn1 26066 Connectitivy law for betweenness. Theorem 5.1 of [Schwabhauser] p. 39-41. (Contributed by Scott Fenton, 9-Oct-2013.)

Theorembtwnconn2 26067 Another connectivity law for betweenness. Theorem 5.2 of [Schwabhauser] p. 41. (Contributed by Scott Fenton, 9-Oct-2013.)

Theorembtwnconn3 26068 Inner connectivity law for betweenness. Theorem 5.3 of [Schwabhauser] p. 41. (Contributed by Scott Fenton, 9-Oct-2013.)

Theoremmidofsegid 26069 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 26070* Generalization of axsegcon 25897. This time, we generate an endpoint for a segment on the ray congruent to and starting at , as opposed to axsegcon 25897, where the segment starts at (Contributed by Scott Fenton, 14-Oct-2013.) (Removed unneeded inequality, 15-Oct-2013.)
Cgr

19.7.47  Segment less than or equal to

Syntaxcsegle 26071 Declare the constant for the segment less than or equal to relationship.

Definitiondf-segle 26072* 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 26073* Binary relationship form of the segment comparison relationship. (Contributed by Scott Fenton, 11-Oct-2013.)
Cgr

Theorembrsegle2 26074* Alternate characterization of segment comparison. Theorem 5.5 of [Schwabhauser] p. 41-42. (Contributed by Scott Fenton, 11-Oct-2013.)
Cgr

Theoremseglecgr12im 26075 Substitution law for segment comparison under congruence. Theorem 5.6 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 11-Oct-2013.)
Cgr Cgr

Theoremseglecgr12 26076 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 26077 Segment comparison is reflexive. Theorem 5.7 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 11-Oct-2013.)

Theoremseglemin 26078 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 26079 Segment less than is transitive. Theorem 5.8 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 11-Oct-2013.)

Theoremsegleantisym 26080 Antisymmetry law for segment comparison. Theorem 5.9 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 14-Oct-2013.)
Cgr

Theoremseglelin 26081 Linearity law for segment comparison. Theorem 5.10 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 14-Oct-2013.)

Theorembtwnsegle 26082 If falls between and , then is no longer than . (Contributed by Scott Fenton, 16-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremcolinbtwnle 26083 Given three colinear points , , and , falls in the middle iff the two segments to are no longer than . Theorem 5.12 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

19.7.48  Outside of relationship

Syntaxcoutsideof 26084 Declare the syntax for the outside of constant.
OutsideOf

Definitiondf-outsideof 26085 The outside of relationship. This relationship expresses that , , and fall on a line, but is not on the segment . This definition is taken from theorem 6.4 of [Schwabhauser] p. 43, since it requires no dummy variables. (Contributed by Scott Fenton, 17-Oct-2013.)
OutsideOf

Theorembroutsideof 26086 Binary relationship form of OutsideOf. Theorem 6.4 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 17-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
OutsideOf

Theorembroutsideof2 26087 Alternate form of OutsideOf. Definition 6.1 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 17-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
OutsideOf

Theoremoutsidene1 26088 Outsideness implies inequality. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
OutsideOf

Theoremoutsidene2 26089 Outsideness implies inequality. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
OutsideOf

Theorembtwnoutside 26090 A principle linking outsideness to betweenness. Theorem 6.2 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
OutsideOf

Theorembroutsideof3 26091* Characterization of outsideness in terms of relationship to a fourth point. Theorem 6.3 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
OutsideOf

Theoremoutsideofrflx 26092 Reflexitivity of outsideness. Theorem 6.5 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
OutsideOf

Theoremoutsideofcom 26093 Commutitivity law for outsideness. Theorem 6.6 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
OutsideOf OutsideOf

Theoremoutsideoftr 26094 Transitivity law for outsideness. Theorem 6.7 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
OutsideOf OutsideOf OutsideOf

Theoremoutsideofeq 26095 Uniqueness law for OutsideOf. Analog of segconeq 25975. (Contributed by Scott Fenton, 24-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
OutsideOf Cgr OutsideOf Cgr

Theoremoutsideofeu 26096* Given a non-degenerate ray, there is a unique point congruent to the segment lying on the ray . Theorem 6.11 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 23-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
OutsideOf Cgr

Theoremoutsidele 26097 Relate OutsideOf to . Theorem 6.13 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 24-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
OutsideOf

Theoremoutsideofcol 26098 Outside of implies colinearity. (Contributed by Scott Fenton, 26-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
OutsideOf

19.7.49  Lines and Rays

Syntaxcline2 26099 Declare the constant for the line function.
Line

Syntaxcray 26100 Declare the constant for the ray function.
Ray

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