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Theorem List for Metamath Proof Explorer - 25901-26000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremaxsegcon 25901* Any segment can be extended to a point such that is congruent to . Axiom A4 of [Schwabhauser] p. 11. (Contributed by Scott Fenton, 4-Jun-2013.)
Cgr

Theoremax5seglem1 25902* Lemma for ax5seg 25912. Rexpress a one congruence sum given betweenness. (Contributed by Scott Fenton, 11-Jun-2013.)

Theoremax5seglem2 25903* Lemma for ax5seg 25912. Rexpress another congruence sum given betweenness. (Contributed by Scott Fenton, 11-Jun-2013.)

Theoremax5seglem3a 25904 Lemma for ax5seg 25912. (Contributed by Scott Fenton, 7-May-2015.)

Theoremax5seglem3 25905* Lemma for ax5seg 25912. Combine congruences for points on a line. (Contributed by Scott Fenton, 11-Jun-2013.)
Cgr Cgr

Theoremax5seglem4 25906* Lemma for ax5seg 25912. Given two distinct points, the scaling constant in a betweenness statement is non-zero. (Contributed by Scott Fenton, 11-Jun-2013.)

Theoremax5seglem5 25907* Lemma for ax5seg 25912. If is between and , and is distinct from , then is distinct from . (Contributed by Scott Fenton, 11-Jun-2013.)

Theoremax5seglem6 25908* Lemma for ax5seg 25912. Given two line segments that are divided into pieces, if the pieces are congruent, then the scaling constant is the same. (Contributed by Scott Fenton, 12-Jun-2013.)
Cgr Cgr

Theoremax5seglem7 25909 Lemma for ax5seg 25912. An algebraic calculation needed further down the line. (Contributed by Scott Fenton, 12-Jun-2013.)

Theoremax5seglem8 25910 Lemma for ax5seg 25912. Use the weak deduction theorem to eliminate the hypotheses from ax5seglem7 25909. (Contributed by Scott Fenton, 11-Jun-2013.)

Theoremax5seglem9 25911* Lemma for ax5seg 25912. Take the calculation in ax5seglem8 25910 and turn it into a series of measurements. (Contributed by Scott Fenton, 12-Jun-2013.) (Revised by Mario Carneiro, 22-May-2014.)

Theoremax5seg 25912 The five segment axiom. Take two triangles and , a point on , and a point on . If all corresponding line segments except for and are congruent, then so are and . Axiom A5 of [Schwabhauser] p. 11. (Contributed by Scott Fenton, 12-Jun-2013.)
Cgr Cgr Cgr Cgr Cgr

Theoremaxbtwnid 25913 Points are indivisible. That is, if lies between and , then . Axiom A6 of [Schwabhauser] p. 11. (Contributed by Scott Fenton, 3-Jun-2013.)

Theoremaxpaschlem 25914* Lemma for axpasch 25915. Set up coefficents used in the proof. (Contributed by Scott Fenton, 5-Jun-2013.)

Theoremaxpasch 25915* The inner Pasch axiom. Take a triangle , a point on , and a point extending . Then and intersect at some point . Axiom A7 of [Schwabhauser] p. 12. (Contributed by Scott Fenton, 3-Jun-2013.)

Theoremaxlowdimlem1 25916 Lemma for axlowdim 25935. Establish a particular constant function as a function. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremaxlowdimlem2 25917 Lemma for axlowdim 25935. Show that two sets are disjoint. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremaxlowdimlem3 25918 Lemma for axlowdim 25935. Set up a union property for an interval of integers. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremaxlowdimlem4 25919 Lemma for axlowdim 25935. Set up a particular constant function. (Contributed by Scott Fenton, 17-Apr-2013.)

Theoremaxlowdimlem5 25920 Lemma for axlowdim 25935. Show that a particular union is a point in Euclidean space. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremaxlowdimlem6 25921 Lemma for axlowdim 25935. Show that three points are non-colinear. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremaxlowdimlem7 25922 Lemma for axlowdim 25935. Set up a point in Euclidean space. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremaxlowdimlem8 25923 Lemma for axlowdim 25935. Calulate the value of at three. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem9 25924 Lemma for axlowdim 25935. Calulate the value of away from three. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem10 25925 Lemma for axlowdim 25935. Set up a family of points in Euclidean space. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem11 25926 Lemma for axlowdim 25935. Calculate the value of at its distinguished point. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem12 25927 Lemma for axlowdim 25935. Calculate the value of away from its distunguished point. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem13 25928 Lemma for axlowdim 25935. Establish that and are different points. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem14 25929 Lemma for axlowdim 25935. Take two possible from axlowdimlem10 25925. They are the same iff their distunguished values are the same. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem15 25930* Lemma for axlowdim 25935. Set up a one-to-one function of points. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem16 25931* Lemma for axlowdim 25935. Set up a summation that will help establish distance. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem17 25932 Lemma for axlowdim 25935. Establish a congruence result. (Contributed by Scott Fenton, 22-Apr-2013.) (Proof shortened by Mario Carneiro, 22-May-2014.)
Cgr

Theoremaxlowdim1 25933* The lower dimensional axiom for one dimension. In any dimension, there are at least two distinct points. Theorem 3.13 of [Schwabhauser] p. 32, where it is derived from axlowdim2 25934. (Contributed by Scott Fenton, 22-Apr-2013.)

Theoremaxlowdim2 25934* The lower two-dimensional axiom. In any space where the dimension is greater than one, there are three non-colinear points. Axiom A8 of [Schwabhauser] p. 12. (Contributed by Scott Fenton, 15-Apr-2013.)

Theoremaxlowdim 25935* The general lower dimensional axiom. Take a dimension greater than or equal to three. Then, there are three non-colinear points in dimensional space that are equidistant from distinct points. Derived from remarks in "Tarski's System of Geometry", by Alfred Tarski and Steven Givant, Bull. Symbolic Logic Volume 5, Number 2 (1999), 175-214. (Contributed by Scott Fenton, 22-Apr-2013.)
Cgr Cgr Cgr

Theoremaxeuclidlem 25936* Lemma for axeuclid 25937. Handle the algebraic aspects of the theorem. (Contributed by Scott Fenton, 9-Sep-2013.)

Theoremaxeuclid 25937* Euclid's axiom. Take an angle and a point between and . Now, if you extend the segment to a point , then lies between two points and that lie on the angle. Axiom A10 of [Schwabhauser] p. 13. (Contributed by Scott Fenton, 9-Sep-2013.)

Theoremaxcontlem1 25938* Lemma for axcont 25950. Change bound variables for later use. (Contributed by Scott Fenton, 20-Jun-2013.)

Theoremaxcontlem2 25939* Lemma for axcont 25950. The idea here is to set up a mapping that will allow us to transfer dedekind 25222 to two sets of points. Here, we set up and show its domain and range. (Contributed by Scott Fenton, 17-Jun-2013.)

Theoremaxcontlem3 25940* Lemma for axcont 25950. Given the separation assumption, is a subset of . (Contributed by Scott Fenton, 18-Jun-2013.)

Theoremaxcontlem4 25941* Lemma for axcont 25950. Given the separation assumption, is a subset of . (Contributed by Scott Fenton, 18-Jun-2013.)

Theoremaxcontlem5 25942* Lemma for axcont 25950. Compute the value of . (Contributed by Scott Fenton, 18-Jun-2013.)

Theoremaxcontlem6 25943* Lemma for axcont 25950. State the defining properties of the value of (Contributed by Scott Fenton, 19-Jun-2013.)

Theoremaxcontlem7 25944* Lemma for axcont 25950. Given two points in , one preceeds the other iff its scaling constant is less than the other point's. (Contributed by Scott Fenton, 18-Jun-2013.)

Theoremaxcontlem8 25945* Lemma for axcont 25950. A point in is between two others if its function value falls in the middle. (Contributed by Scott Fenton, 18-Jun-2013.)

Theoremaxcontlem9 25946* Lemma for axcont 25950. Given the separation assumption, all values of over are less than or equal to all values of over . (Contributed by Scott Fenton, 20-Jun-2013.)

Theoremaxcontlem10 25947* Lemma for axcont 25950. Given a handful of assumptions, derive the conclusion of the final theorem. (Contributed by Scott Fenton, 20-Jun-2013.)

Theoremaxcontlem11 25948* Lemma for axcont 25950. Eliminate the hypotheses from axcontlem10 25947. (Contributed by Scott Fenton, 20-Jun-2013.)

Theoremaxcontlem12 25949* Lemma for axcont 25950. Eliminate the trivial cases from the previous lemmas. (Contributed by Scott Fenton, 20-Jun-2013.)

Theoremaxcont 25950* The axiom of continuity. Take two sets of points and . If all the points in come before the points of on a line, then there is a point separating the two. Axiom A11 of [Schwabhauser] p. 13. (Contributed by Scott Fenton, 20-Jun-2013.)

19.7.42  Congruence properties

Syntaxcofs 25951 Declare the syntax for the outer five segment configuration.

Definitiondf-ofs 25952* The outer five segment configuration is an abbreviation for the conditions of the Five Segment Axiom (ax5seg 25912). See brofs 25974 and 5segofs 25975 for how it is used. Definition 2.10 of [Schwabhauser] p. 28. (Contributed by Scott Fenton, 21-Sep-2013.)
Cgr Cgr Cgr Cgr

Theoremcgrrflx2d 25953 Deduction form of axcgrrflx 25888. (Contributed by Scott Fenton, 13-Oct-2013.)
Cgr

Theoremcgrtr4d 25954 Deduction form of axcgrtr 25889. (Contributed by Scott Fenton, 13-Oct-2013.)
Cgr        Cgr        Cgr

Theoremcgrtr4and 25955 Deduction form of axcgrtr 25889. (Contributed by Scott Fenton, 13-Oct-2013.)
Cgr        Cgr        Cgr

Theoremcgrrflx 25956 Reflexivity law for congruence. Theorem 2.1 of [Schwabhauser] p. 27. (Contributed by Scott Fenton, 12-Jun-2013.)
Cgr

Theoremcgrrflxd 25957 Deduction form of cgrrflx 25956. (Contributed by Scott Fenton, 13-Oct-2013.)
Cgr

Theoremcgrcomim 25958 Congruence commutes on the two sides. Implication version. Theorem 2.2 of [Schwabhauser] p. 27. (Contributed by Scott Fenton, 12-Jun-2013.)
Cgr Cgr

Theoremcgrcom 25959 Congruence commutes between the two sides. (Contributed by Scott Fenton, 12-Jun-2013.)
Cgr Cgr

Theoremcgrcomand 25960 Deduction form of cgrcom 25959. (Contributed by Scott Fenton, 13-Oct-2013.)
Cgr        Cgr

Theoremcgrtr 25961 Transitivity law for congruence. Theorem 2.3 of [Schwabhauser] p. 27. (Contributed by Scott Fenton, 24-Sep-2013.)
Cgr Cgr Cgr

Theoremcgrtrand 25962 Deduction form of cgrtr 25961. (Contributed by Scott Fenton, 13-Oct-2013.)
Cgr        Cgr        Cgr

Theoremcgrtr3 25963 Transitivity law for congruence. (Contributed by Scott Fenton, 7-Oct-2013.)
Cgr Cgr Cgr

Theoremcgrtr3and 25964 Deduction form of cgrtr3 25963. (Contributed by Scott Fenton, 13-Oct-2013.)
Cgr        Cgr        Cgr

Theoremcgrcoml 25965 Congruence commutes on the left. Biconditional version of Theorem 2.4 of [Schwabhauser] p. 27. (Contributed by Scott Fenton, 12-Jun-2013.)
Cgr Cgr

Theoremcgrcomr 25966 Congruence commutes on the right. Biconditional version of Theorem 2.5 of [Schwabhauser] p. 27. (Contributed by Scott Fenton, 12-Jun-2013.)
Cgr Cgr

Theoremcgrcomlr 25967 Congruence commutes on both sides. (Contributed by Scott Fenton, 12-Jun-2013.)
Cgr Cgr

Theoremcgrcomland 25968 Deduction form of cgrcoml 25965. (Contributed by Scott Fenton, 14-Oct-2013.)
Cgr        Cgr

Theoremcgrcomrand 25969 Deduction form of cgrcoml 25965. (Contributed by Scott Fenton, 14-Oct-2013.)
Cgr        Cgr

Theoremcgrcomlrand 25970 Deduction form of cgrcomlr 25967. (Contributed by Scott Fenton, 14-Oct-2013.)
Cgr        Cgr

Theoremcgrtriv 25971 Degenerate segments are congruent. Theorem 2.8 of [Schwabhauser] p. 28. (Contributed by Scott Fenton, 12-Jun-2013.)
Cgr

Theoremcgrid2 25972 Identity law for congruence. (Contributed by Scott Fenton, 12-Jun-2013.)
Cgr

Theoremcgrdegen 25973 Two congruent segments are either both degenrate or both non-degenerate. (Contributed by Scott Fenton, 12-Jun-2013.)
Cgr

Theorembrofs 25974 Binary relationship form of the outer five segment predicate. (Contributed by Scott Fenton, 21-Sep-2013.)
Cgr Cgr Cgr Cgr

Theorem5segofs 25975 Rephrase ax5seg 25912 using the outer five segment predicate. Theorem 2.10 of [Schwabhauser] p. 28. (Contributed by Scott Fenton, 21-Sep-2013.)
Cgr

Theoremofscom 25976 The outer five segment predicate commutes. (Contributed by Scott Fenton, 26-Sep-2013.)

Theoremcgrextend 25977 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 25978 Deduction form of cgrextend 25977. (Contributed by Scott Fenton, 14-Oct-2013.)
Cgr        Cgr        Cgr

Theoremsegconeq 25979 Two points that satsify the conclusion of axsegcon 25901 are identical. Uniqueness portion of Theorem 2.12 of [Schwabhauser] p. 29. (Contributed by Scott Fenton, 12-Jun-2013.)
Cgr Cgr

Theoremsegconeu 25980* Existential uniqueness version of segconeq 25979. (Contributed by Scott Fenton, 19-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Cgr

19.7.43  Betweenness properties

Theorembtwntriv2 25981 Betweeness always holds for the second endpoint. Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.)

Theorembtwncomim 25982 Betweeness commutes. Implication version. Theorem 3.2 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.)

Theorembtwncom 25983 Betweeness commutes. (Contributed by Scott Fenton, 12-Jun-2013.)

Theorembtwncomand 25984 Deduction form of btwncom 25983. (Contributed by Scott Fenton, 14-Oct-2013.)

Theorembtwntriv1 25985 Betweeness always holds for the first endpoint. Theorem 3.3 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.)

Theorembtwnswapid 25986 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 25987 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 25988 Inner transitivity law for betweenness. Left-hand side of Theorem 3.5 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.)

Theorembtwnexch3 25989 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 25990 Deduction form of btwnexch3 25989. (Contributed by Scott Fenton, 13-Oct-2013.)

Theorembtwnouttr2 25991 Outer transitivity law for betweenness. Left-hand side of Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.)

Theorembtwnexch2 25992 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 25993 Outer transitivity law for betweenness. Right-hand side of Theorem 3.7 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 14-Jun-2013.)

Theorembtwnexch 25994 Outer transitivity law for betweenness. Right-hand side of Theorem 3.6 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 24-Sep-2013.)

Theorembtwnexchand 25995 Deduction form of btwnexch 25994. (Contributed by Scott Fenton, 13-Oct-2013.)

Theorembtwndiff 25996* 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 25997* 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.)

19.7.44  Segment Transportation

Syntaxctransport 25998 Declare the syntax for the segment transport function.
TransportTo

Definitiondf-transport 25999* Define the segment transport function. See fvtransport 26001 for an explanation of the function. (Contributed by Scott Fenton, 18-Oct-2013.)
TransportTo Cgr

Theoremfuntransport 26000 The TransportTo relationship is a function. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
TransportTo

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