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Theorem List for Metamath Proof Explorer - 23801-23900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcgrdegen 23801 Two congruent segments are either both degenrate or both non-degenerate. (Contributed by Scott Fenton, 12-Jun-2013.)
Cgr

Theorembrofs 23802 Binary relationship form of the outer five segment predicate. (Contributed by Scott Fenton, 21-Sep-2013.)
Cgr Cgr Cgr Cgr

Theorem5segofs 23803 Rephrase ax5seg 23740 using the outer five segment predicate. Theorem 2.10 of [Schwabhauser] p. 28. (Contributed by Scott Fenton, 21-Sep-2013.)
Cgr

Theoremofscom 23804 The outer five segment predicate commutes. (Contributed by Scott Fenton, 26-Sep-2013.)

Theoremcgrextend 23805 Link congruence over a pair of line segments. Theorem 2.11 of [Schwabhauser] p. 29. (Contributed by Scott Fenton, 12-Jun-2013.)
Cgr Cgr Cgr

Theoremcgrextendand 23806 Deduction form of cgrextend 23805. (Contributed by Scott Fenton, 14-Oct-2013.)
Cgr        Cgr        Cgr

Theoremsegconeq 23807 Two points that satsify the conclusion of axsegcon 23729 are identical. Uniqueness portion of Theorem 2.12 of [Schwabhauser] p. 29. (Contributed by Scott Fenton, 12-Jun-2013.)
Cgr Cgr

Theoremsegconeu 23808* Existential uniqueness version of segconeq 23807. (Contributed by Scott Fenton, 19-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Cgr

16.6.33  Betweenness properties

Theorembtwntriv2 23809 Betweeness always holds for the second endpoint. Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.)

Theorembtwncomim 23810 Betweeness commutes. Implication version. Theorem 3.2 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.)

Theorembtwncom 23811 Betweeness commutes. (Contributed by Scott Fenton, 12-Jun-2013.)

Theorembtwncomand 23812 Deduction form of btwncom 23811. (Contributed by Scott Fenton, 14-Oct-2013.)

Theorembtwntriv1 23813 Betweeness always holds for the first endpoint. Theorem 3.3 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.)

Theorembtwnswapid 23814 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 23815 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 23816 Inner transitivity law for betweenness. Left hand side of Theorem 3.5 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.)

Theorembtwnexch3 23817 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 23818 Deduction form of btwnexch3 23817. (Contributed by Scott Fenton, 13-Oct-2013.)

Theorembtwnouttr2 23819 Outer transitivity law for betweenness. Left hand side of Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.)

Theorembtwnexch2 23820 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 23821 Outer transitivity law for betweenness. Right hand side of Theorem 3.7 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 14-Jun-2013.)

Theorembtwnexch 23822 Outer transitivity law for betweenness. Right hand side of Theorem 3.6 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 24-Sep-2013.)

Theorembtwnexchand 23823 Deduction form of btwnexch 23822. (Contributed by Scott Fenton, 13-Oct-2013.)

Theorembtwndiff 23824* 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 23825* 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.)

16.6.34  Segment Transportation

Syntaxctransport 23826 Declare the syntax for the segment transport function.
TransportTo

Definitiondf-transport 23827* Define the segment transport function. See fvtransport 23829 for an explanation of the function. (Contributed by Scott Fenton, 18-Oct-2013.)
TransportTo Cgr

Theoremfuntransport 23828 The TransportTo relationship is a function. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
TransportTo

Theoremfvtransport 23829* 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 23830 Closure law for segment transport. (Contributed by Scott Fenton, 19-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
TransportTo

Theoremtransportprops 23831 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

16.6.35  Properties relating betweenness and congruence

Syntaxcifs 23832 Declare the syntax for the inner five segment predicate.

Syntaxccgr3 23833 Declare the syntax for the three place congruence predicate.
Cgr3

Syntaxccolin 23834 Declare the syntax for the colinearity predicate.

Syntaxcfs 23835 Declare the syntax for the five segment predicate.

Definitiondf-ifs 23836* The inner five segment configuration is an abbreviation for another congruence condition. See brifs 23840 and ifscgr 23841 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 23837* 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 23838* 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 23839* The general five segment configuration is a generalization of the outer and inner five segment configurations. See brfs 23876 and fscgr 23877 for its use. Definition 4.15 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3 Cgr Cgr

Theorembrifs 23840 Binary relationship form of the inner five segment predicate. (Contributed by Scott Fenton, 26-Sep-2013.)
Cgr Cgr Cgr Cgr

Theoremifscgr 23841 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 23842 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 23843 Binary relationship form of the three-place congruence predicate. (Contributed by Scott Fenton, 4-Oct-2013.)
Cgr3 Cgr Cgr Cgr

Theoremcgr3permute3 23844 Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3 Cgr3

Theoremcgr3permute1 23845 Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3 Cgr3

Theoremcgr3permute2 23846 Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3 Cgr3

Theoremcgr3permute4 23847 Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3 Cgr3

Theoremcgr3permute5 23848 Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3 Cgr3

Theoremcgr3tr4 23849 Transitivity law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3 Cgr3 Cgr3

Theoremcgr3com 23850 Commutativity law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3 Cgr3

Theoremcgr3rflx 23851 Identity law for three-place congruence. (Contributed by Scott Fenton, 6-Oct-2013.)
Cgr3

Theoremcgrxfr 23852* 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 23853 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 23854 Colinearity is a relationship. (Contributed by Scott Fenton, 7-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorembrcolinear2 23855* Alternate colinearity binary relationship. (Contributed by Scott Fenton, 7-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorembrcolinear 23856 The binary relationship form of the colinearity predicate. (Contributed by Scott Fenton, 5-Oct-2013.)

Theoremcolinearex 23857 The colinear predicate exists. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremcolineardim1 23858 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 23859 Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.)

Theoremcolinearperm3 23860 Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.)

Theoremcolinearperm2 23861 Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.)

Theoremcolinearperm4 23862 Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.)

Theoremcolinearperm5 23863 Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.)

Theoremcolineartriv1 23864 Trivial case of colinearity. Theorem 4.12 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 5-Oct-2013.)

Theoremcolineartriv2 23865 Trivial case of colinearity. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorembtwncolinear1 23866 Betweenness implies colinearity. (Contributed by Scott Fenton, 7-Oct-2013.)

Theorembtwncolinear2 23867 Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorembtwncolinear3 23868 Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorembtwncolinear4 23869 Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorembtwncolinear5 23870 Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorembtwncolinear6 23871 Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremcolinearxfr 23872 Transfer law for colinearity. Theorem 4.13 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3

Theoremlineext 23873* Extend a line with a missing point. Theorem 4.14 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 6-Oct-2013.)
Cgr Cgr3

Theorembrofs2 23874 Change some conditions for outer five segment predicate. (Contributed by Scott Fenton, 6-Oct-2013.)
Cgr3 Cgr Cgr

Theorembrifs2 23875 Change some conditions for inner five segment predicate. (Contributed by Scott Fenton, 6-Oct-2013.)
Cgr3 Cgr Cgr

Theorembrfs 23876 Binary relationship form of the general five segment predicate. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr3 Cgr Cgr

Theoremfscgr 23877 Congruence law for the general five segment configuration. Theorem 4.16 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 5-Oct-2013.)
Cgr

Theoremlinecgr 23878 Congruence rule for lines. Theorem 4.17 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 6-Oct-2013.)
Cgr Cgr Cgr

Theoremlinecgrand 23879 Deduction form of linecgr 23878. (Contributed by Scott Fenton, 14-Oct-2013.)
Cgr        Cgr        Cgr

Theoremlineid 23880 Identity law for points on lines. Theorem 4.18 of [Schwabhauser] p. 38. (Contributed by Scott Fenton, 7-Oct-2013.)
Cgr Cgr

Theoremidinside 23881 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 23882 If , , and fall in order on a line, and and are congruent, then . (Contributed by Scott Fenton, 7-Oct-2013.)
Cgr

Theoremendofsegidand 23883 Deduction form of endofsegid 23882. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Cgr

16.6.36  Connectivity of betweenness

Theorembtwnconn1lem1 23884 Lemma for btwnconn1 23898. 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 23885 Lemma for btwnconn1 23898. 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 23886 Lemma for btwnconn1 23898. Establish the next congruence in the series. (Contributed by Scott Fenton, 8-Oct-2013.)
Cgr Cgr Cgr Cgr Cgr

Theorembtwnconn1lem4 23887 Lemma for btwnconn1 23898. 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 23888 Lemma for btwnconn1 23898. 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 23889 Lemma for btwnconn1 23898. Next, we show that is the midpoint of and (Contributed by Scott Fenton, 8-Oct-2013.)
Cgr Cgr Cgr Cgr Cgr

Theorembtwnconn1lem7 23890 Lemma for btwnconn1 23898. Under our assumptions, and are distinct. (Contributed by Scott Fenton, 8-Oct-2013.)
Cgr Cgr Cgr Cgr

Theorembtwnconn1lem8 23891 Lemma for btwnconn1 23898. 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 23892 Lemma for btwnconn1 23898. 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 23893 Lemma for btwnconn1 23898. 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 23894 Lemma for btwnconn1 23898. Now, we establish that and are equidistant from (Contributed by Scott Fenton, 8-Oct-2013.)
Cgr Cgr Cgr Cgr Cgr Cgr Cgr Cgr

Theorembtwnconn1lem12 23895 Lemma for btwnconn1 23898. Using a long string of invocations of linecgr 23878, we show that . (Contributed by Scott Fenton, 9-Oct-2013.)
Cgr Cgr Cgr Cgr Cgr Cgr Cgr

Theorembtwnconn1lem13 23896 Lemma for btwnconn1 23898. Begin back-filling and eliminating hypotheses. (Contributed by Scott Fenton, 9-Oct-2013.)
Cgr Cgr Cgr Cgr

Theorembtwnconn1lem14 23897 Lemma for btwnconn1 23898. Final statement of the theorem when . (Contributed by Scott Fenton, 9-Oct-2013.)

Theorembtwnconn1 23898 Connectitivy law for betweenness. Theorem 5.1 of [Schwabhauser] p. 39-41. (Contributed by Scott Fenton, 9-Oct-2013.)

Theorembtwnconn2 23899 Another connectivity law for betweenness. Theorem 5.2 of [Schwabhauser] p. 41. (Contributed by Scott Fenton, 9-Oct-2013.)

Theorembtwnconn3 23900 Inner connectivity law for betweenness. Theorem 5.3 of [Schwabhauser] p. 41. (Contributed by Scott Fenton, 9-Oct-2013.)

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