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Theorem List for Metamath Proof Explorer - 24601-24700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfveere 24601 The function value of a point is a real. (Contributed by Scott Fenton, 10-Jun-2013.)

Theoremfveecn 24602 The function value of a point is a complex. (Contributed by Scott Fenton, 10-Jun-2013.)

Theoremeqeefv 24603* Two points are equal iff they agree in all dimensions. (Contributed by Scott Fenton, 10-Jun-2013.)

Theoremeqeelen 24604* Two points are equal iff the square of the distance between them is zero. (Contributed by Scott Fenton, 10-Jun-2013.) (Revised by Mario Carneiro, 22-May-2014.)

Theorembrbtwn2 24605* Alternate characterization of betweenness, with no existential quantifiers. (Contributed by Scott Fenton, 24-Jun-2013.)

Theoremcolinearalglem1 24606 Lemma for colinearalg 24610. Expand out a multiplication. (Contributed by Scott Fenton, 24-Jun-2013.)

Theoremcolinearalglem2 24607* Lemma for colinearalg 24610. Translate between two forms of the colinearity condition. (Contributed by Scott Fenton, 24-Jun-2013.)

Theoremcolinearalglem3 24608* Lemma for colinearalg 24610. Translate between two forms of the colinearity condition. (Contributed by Scott Fenton, 24-Jun-2013.)

Theoremcolinearalglem4 24609* Lemma for colinearalg 24610. Prove a disjunction that will be needed in the final proof. (Contributed by Scott Fenton, 27-Jun-2013.)

Theoremcolinearalg 24610* An algebraic characterization of colinearity. Note the similarity to brbtwn2 24605. (Contributed by Scott Fenton, 24-Jun-2013.)

Theoremeleesub 24611* Membership of a subtraction mapping in a Euclidean space. (Contributed by Scott Fenton, 17-Jul-2013.)

Theoremeleesubd 24612* Membership of a subtraction mapping in a Euclidean space. Deduction form of eleesub 24611. (Contributed by Scott Fenton, 17-Jul-2013.)

18.7.34  Tarski's axioms for geometry

Theoremaxdimuniq 24613 The unique dimensional axiom. If a point is in dimensional space and in dimensional space, then . This axiom is not traditionally presented with Tarski's axioms, but we require it here as we are considering spaces in arbitrary dimensions. (Contributed by Scott Fenton, 24-Sep-2013.)

Theoremaxcgrrflx 24614 is as far from as is from . Axiom A1 of [Schwabhauser] p. 10. (Contributed by Scott Fenton, 3-Jun-2013.)
Cgr

Theoremaxcgrtr 24615 Congruence is transitive. Axiom A2 of [Schwabhauser] p. 10. (Contributed by Scott Fenton, 3-Jun-2013.)
Cgr Cgr Cgr

Theoremaxcgrid 24616 If there is no distance between and , then . Axiom A3 of [Schwabhauser] p. 10. (Contributed by Scott Fenton, 3-Jun-2013.)
Cgr

Theoremaxsegconlem1 24617* Lemma for axsegcon 24627. Handle the degenerate case. (Contributed by Scott Fenton, 7-Jun-2013.)

Theoremaxsegconlem2 24618* Lemma for axsegcon 24627. Show that the square of the distance between two points is a real number. (Contributed by Scott Fenton, 17-Sep-2013.)

Theoremaxsegconlem3 24619* Lemma for axsegcon 24627. Show that the square of the distance between two points is non-negative. (Contributed by Scott Fenton, 17-Sep-2013.)

Theoremaxsegconlem4 24620* Lemma for axsegcon 24627. Show that the distance between two points is a real number. (Contributed by Scott Fenton, 17-Sep-2013.)

Theoremaxsegconlem5 24621* Lemma for axsegcon 24627. Show that the distance between two points is non-negative. (Contributed by Scott Fenton, 17-Sep-2013.)

Theoremaxsegconlem6 24622* Lemma for axsegcon 24627. Show that the distance between two distinct points is positive. (Contributed by Scott Fenton, 17-Sep-2013.)

Theoremaxsegconlem7 24623* Lemma for axsegcon 24627. Show that a particular ratio of distances is in the closed unit interval. (Contributed by Scott Fenton, 18-Sep-2013.)

Theoremaxsegconlem8 24624* Lemma for axsegcon 24627. Show that a particular mapping generates a point. (Contributed by Scott Fenton, 18-Sep-2013.)

Theoremaxsegconlem9 24625* Lemma for axsegcon 24627. Show that is congruent to . (Contributed by Scott Fenton, 19-Sep-2013.)

Theoremaxsegconlem10 24626* Lemma for axsegcon 24627. Show that the scaling constant from axsegconlem7 24623 produces the betweenness condition for , and . (Contributed by Scott Fenton, 21-Sep-2013.)

Theoremaxsegcon 24627* 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 24628* Lemma for ax5seg 24638. Rexpress a one congruence sum given betweenness. (Contributed by Scott Fenton, 11-Jun-2013.)

Theoremax5seglem2 24629* Lemma for ax5seg 24638. Rexpress another congruence sum given betweenness. (Contributed by Scott Fenton, 11-Jun-2013.)

Theoremax5seglem3a 24630 Lemma for ax5seg 24638. (Contributed by Scott Fenton, 7-May-2015.)

Theoremax5seglem3 24631* Lemma for ax5seg 24638. Combine congruences for points on a line. (Contributed by Scott Fenton, 11-Jun-2013.)
Cgr Cgr

Theoremax5seglem4 24632* Lemma for ax5seg 24638. Given two distinct points, the scaling constant in a betweenness statement is non-zero. (Contributed by Scott Fenton, 11-Jun-2013.)

Theoremax5seglem5 24633* Lemma for ax5seg 24638. If is between and , and is distinct from , then is distinct from . (Contributed by Scott Fenton, 11-Jun-2013.)

Theoremax5seglem6 24634* Lemma for ax5seg 24638. 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 24635 Lemma for ax5seg 24638. An algebraic calculation needed further down the line. (Contributed by Scott Fenton, 12-Jun-2013.)

Theoremax5seglem8 24636 Lemma for ax5seg 24638. Use the weak deduction theorem to eliminate the hypotheses from ax5seglem7 24635. (Contributed by Scott Fenton, 11-Jun-2013.)

Theoremax5seglem9 24637* Lemma for ax5seg 24638. Take the calculation in ax5seglem8 24636 and turn it into a series of measurements. (Contributed by Scott Fenton, 12-Jun-2013.) (Revised by Mario Carneiro, 22-May-2014.)

Theoremax5seg 24638 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 24639 Points are indivisible. That is, if lies between and , then . Axiom A6 of [Schwabhauser] p. 11. (Contributed by Scott Fenton, 3-Jun-2013.)

Theoremaxpaschlem 24640* Lemma for axpasch 24641. Set up coefficents used in the proof. (Contributed by Scott Fenton, 5-Jun-2013.)

Theoremaxpasch 24641* 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 24642 Lemma for axlowdim 24661. Establish a particular constant function as a function. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremaxlowdimlem2 24643 Lemma for axlowdim 24661. Show that two sets are disjoint. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremaxlowdimlem3 24644 Lemma for axlowdim 24661. Set up a union property for an interval of integers. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremaxlowdimlem4 24645 Lemma for axlowdim 24661. Set up a particular constant function. (Contributed by Scott Fenton, 17-Apr-2013.)

Theoremaxlowdimlem5 24646 Lemma for axlowdim 24661. Show that a particular union is a point in Euclidean space. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremaxlowdimlem6 24647 Lemma for axlowdim 24661. Show that three points are non-colinear. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremaxlowdimlem7 24648 Lemma for axlowdim 24661. Set up a point in Euclidean space. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremaxlowdimlem8 24649 Lemma for axlowdim 24661. Calulate the value of at three. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem9 24650 Lemma for axlowdim 24661. Calulate the value of away from three. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem10 24651 Lemma for axlowdim 24661. Set up a family of points in Euclidean space. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem11 24652 Lemma for axlowdim 24661. Calculate the value of at its distinguished point. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem12 24653 Lemma for axlowdim 24661. Calculate the value of away from its distunguished point. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem13 24654 Lemma for axlowdim 24661. Establish that and are different points. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem14 24655 Lemma for axlowdim 24661. Take two possible from axlowdimlem10 24651. They are the same iff their distunguished values are the same. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem15 24656* Lemma for axlowdim 24661. Set up a one-to-one function of points. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem16 24657* Lemma for axlowdim 24661. Set up a summation that will help establish distance. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem17 24658 Lemma for axlowdim 24661. Establish a congruence result. (Contributed by Scott Fenton, 22-Apr-2013.) (Proof shortened by Mario Carneiro, 22-May-2014.)
Cgr

Theoremaxlowdim1 24659* 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 24660. (Contributed by Scott Fenton, 22-Apr-2013.)

Theoremaxlowdim2 24660* 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 24661* 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 24662* Lemma for axeuclid 24663. Handle the algebraic aspects of the theorem. (Contributed by Scott Fenton, 9-Sep-2013.)

Theoremaxeuclid 24663* 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 24664* Lemma for axcont 24676. Change bound variables for later use. (Contributed by Scott Fenton, 20-Jun-2013.)

Theoremaxcontlem2 24665* Lemma for axcont 24676. The idea here is to set up a mapping that will allow us to transfer dedekind 24097 to two sets of points. Here, we set up and show its domain and range. (Contributed by Scott Fenton, 17-Jun-2013.)

Theoremaxcontlem3 24666* Lemma for axcont 24676. Given the separation assumption, is a subset of . (Contributed by Scott Fenton, 18-Jun-2013.)

Theoremaxcontlem4 24667* Lemma for axcont 24676. Given the separation assumption, is a subset of . (Contributed by Scott Fenton, 18-Jun-2013.)

Theoremaxcontlem5 24668* Lemma for axcont 24676. Compute the value of . (Contributed by Scott Fenton, 18-Jun-2013.)

Theoremaxcontlem6 24669* Lemma for axcont 24676. State the defining properties of the value of (Contributed by Scott Fenton, 19-Jun-2013.)

Theoremaxcontlem7 24670* Lemma for axcont 24676. 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 24671* Lemma for axcont 24676. A point in is between two others if its function value falls in the middle. (Contributed by Scott Fenton, 18-Jun-2013.)

Theoremaxcontlem9 24672* Lemma for axcont 24676. 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 24673* Lemma for axcont 24676. Given a handful of assumptions, derive the conclusion of the final theorem. (Contributed by Scott Fenton, 20-Jun-2013.)

Theoremaxcontlem11 24674* Lemma for axcont 24676. Eliminate the hypotheses from axcontlem10 24673. (Contributed by Scott Fenton, 20-Jun-2013.)

Theoremaxcontlem12 24675* Lemma for axcont 24676. Eliminate the trivial cases from the previous lemmas. (Contributed by Scott Fenton, 20-Jun-2013.)

Theoremaxcont 24676* 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.)

18.7.35  Congruence properties

Syntaxcofs 24677 Declare the syntax for the outer five segment configuration.

Definitiondf-ofs 24678* The outer five segment configuration is an abbreviation for the conditions of the Five Segment Axiom (ax5seg 24638). See brofs 24700 and 5segofs 24701 for how it is used. Definition 2.10 of [Schwabhauser] p. 28. (Contributed by Scott Fenton, 21-Sep-2013.)
Cgr Cgr Cgr Cgr

Theoremcgrrflx2d 24679 Deduction form of axcgrrflx 24614. (Contributed by Scott Fenton, 13-Oct-2013.)
Cgr

Theoremcgrtr4d 24680 Deduction form of axcgrtr 24615. (Contributed by Scott Fenton, 13-Oct-2013.)
Cgr        Cgr        Cgr

Theoremcgrtr4and 24681 Deduction form of axcgrtr 24615. (Contributed by Scott Fenton, 13-Oct-2013.)
Cgr        Cgr        Cgr

Theoremcgrrflx 24682 Reflexivity law for congruence. Theorem 2.1 of [Schwabhauser] p. 27. (Contributed by Scott Fenton, 12-Jun-2013.)
Cgr

Theoremcgrrflxd 24683 Deduction form of cgrrflx 24682. (Contributed by Scott Fenton, 13-Oct-2013.)
Cgr

Theoremcgrcomim 24684 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 24685 Congruence commutes between the two sides. (Contributed by Scott Fenton, 12-Jun-2013.)
Cgr Cgr

Theoremcgrcomand 24686 Deduction form of cgrcom 24685. (Contributed by Scott Fenton, 13-Oct-2013.)
Cgr        Cgr

Theoremcgrtr 24687 Transitivity law for congruence. Theorem 2.3 of [Schwabhauser] p. 27. (Contributed by Scott Fenton, 24-Sep-2013.)
Cgr Cgr Cgr

Theoremcgrtrand 24688 Deduction form of cgrtr 24687. (Contributed by Scott Fenton, 13-Oct-2013.)
Cgr        Cgr        Cgr

Theoremcgrtr3 24689 Transitivity law for congruence. (Contributed by Scott Fenton, 7-Oct-2013.)
Cgr Cgr Cgr

Theoremcgrtr3and 24690 Deduction form of cgrtr3 24689. (Contributed by Scott Fenton, 13-Oct-2013.)
Cgr        Cgr        Cgr

Theoremcgrcoml 24691 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 24692 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 24693 Congruence commutes on both sides. (Contributed by Scott Fenton, 12-Jun-2013.)
Cgr Cgr

Theoremcgrcomland 24694 Deduction form of cgrcoml 24691. (Contributed by Scott Fenton, 14-Oct-2013.)
Cgr        Cgr

Theoremcgrcomrand 24695 Deduction form of cgrcoml 24691. (Contributed by Scott Fenton, 14-Oct-2013.)
Cgr        Cgr

Theoremcgrcomlrand 24696 Deduction form of cgrcomlr 24693. (Contributed by Scott Fenton, 14-Oct-2013.)
Cgr        Cgr

Theoremcgrtriv 24697 Degenerate segments are congruent. Theorem 2.8 of [Schwabhauser] p. 28. (Contributed by Scott Fenton, 12-Jun-2013.)
Cgr

Theoremcgrid2 24698 Identity law for congruence. (Contributed by Scott Fenton, 12-Jun-2013.)
Cgr

Theoremcgrdegen 24699 Two congruent segments are either both degenrate or both non-degenerate. (Contributed by Scott Fenton, 12-Jun-2013.)
Cgr

Theorembrofs 24700 Binary relationship form of the outer five segment predicate. (Contributed by Scott Fenton, 21-Sep-2013.)
Cgr Cgr Cgr Cgr

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