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Theorem List for Metamath Proof Explorer - 25701-25800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxcbtwn 25701 Declare the syntax for the Euclidean betweenness predicate.

Syntaxccgr 25702 Declare the syntax for the Euclidean congruence predicate.
Cgr

Definitiondf-ee 25703 Define the Euclidean space generator. For details, see elee 25706. (Contributed by Scott Fenton, 3-Jun-2013.)

Definitiondf-btwn 25704* Define the Euclidean betweenness predicate. For details, see brbtwn 25711. (Contributed by Scott Fenton, 3-Jun-2013.)

Definitiondf-cgr 25705* Define the Euclidean congruence predicate. For details, see brcgr 25712. (Contributed by Scott Fenton, 3-Jun-2013.)
Cgr

Theoremelee 25706 Membership in a Euclidean space. We define Euclidean space here using Cartesian coordinates over space. We later abstract away from this using Tarski's geometry axioms, so this exact definition is unimportant. (Contributed by Scott Fenton, 3-Jun-2013.)

Theoremmptelee 25707* A condition for a mapping to be an element of a Euclidean space. (Contributed by Scott Fenton, 7-Jun-2013.)

Theoremeleenn 25708 If is in , then is a natural. (Contributed by Scott Fenton, 1-Jul-2013.)

Theoremeleei 25709 The forward direction of elee 25706. (Contributed by Scott Fenton, 1-Jul-2013.)

Theoremeedimeq 25710 A point belongs to at most one Euclidean space. (Contributed by Scott Fenton, 1-Jul-2013.)

Theorembrbtwn 25711* The binary relationship form of the betweenness predicate. The statement should be informally read as " lies on a line segment between and . This exact definition is abstracted away by Tarski's geometry axioms later on. (Contributed by Scott Fenton, 3-Jun-2013.)

Theorembrcgr 25712* The binary relationship form of the congruence predicate. The statement Cgr should be read informally as "the dimensional point is as far from as is from , or "the line segment is congruent to the line segment . This particular definition is encapsulated by Tarski's axioms later on. (Contributed by Scott Fenton, 3-Jun-2013.)
Cgr

Theoremfveere 25713 The function value of a point is a real. (Contributed by Scott Fenton, 10-Jun-2013.)

Theoremfveecn 25714 The function value of a point is a complex. (Contributed by Scott Fenton, 10-Jun-2013.)

Theoremeqeefv 25715* Two points are equal iff they agree in all dimensions. (Contributed by Scott Fenton, 10-Jun-2013.)

Theoremeqeelen 25716* 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 25717* Alternate characterization of betweenness, with no existential quantifiers. (Contributed by Scott Fenton, 24-Jun-2013.)

Theoremcolinearalglem1 25718 Lemma for colinearalg 25722. Expand out a multiplication. (Contributed by Scott Fenton, 24-Jun-2013.)

Theoremcolinearalglem2 25719* Lemma for colinearalg 25722. Translate between two forms of the colinearity condition. (Contributed by Scott Fenton, 24-Jun-2013.)

Theoremcolinearalglem3 25720* Lemma for colinearalg 25722. Translate between two forms of the colinearity condition. (Contributed by Scott Fenton, 24-Jun-2013.)

Theoremcolinearalglem4 25721* Lemma for colinearalg 25722. Prove a disjunction that will be needed in the final proof. (Contributed by Scott Fenton, 27-Jun-2013.)

Theoremcolinearalg 25722* An algebraic characterization of colinearity. Note the similarity to brbtwn2 25717. (Contributed by Scott Fenton, 24-Jun-2013.)

Theoremeleesub 25723* Membership of a subtraction mapping in a Euclidean space. (Contributed by Scott Fenton, 17-Jul-2013.)

Theoremeleesubd 25724* Membership of a subtraction mapping in a Euclidean space. Deduction form of eleesub 25723. (Contributed by Scott Fenton, 17-Jul-2013.)

19.7.40  Tarski's axioms for geometry

Theoremaxdimuniq 25725 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 25726 is as far from as is from . Axiom A1 of [Schwabhauser] p. 10. (Contributed by Scott Fenton, 3-Jun-2013.)
Cgr

Theoremaxcgrtr 25727 Congruence is transitive. Axiom A2 of [Schwabhauser] p. 10. (Contributed by Scott Fenton, 3-Jun-2013.)
Cgr Cgr Cgr

Theoremaxcgrid 25728 If there is no distance between and , then . Axiom A3 of [Schwabhauser] p. 10. (Contributed by Scott Fenton, 3-Jun-2013.)
Cgr

Theoremaxsegconlem1 25729* Lemma for axsegcon 25739. Handle the degenerate case. (Contributed by Scott Fenton, 7-Jun-2013.)

Theoremaxsegconlem2 25730* Lemma for axsegcon 25739. Show that the square of the distance between two points is a real number. (Contributed by Scott Fenton, 17-Sep-2013.)

Theoremaxsegconlem3 25731* Lemma for axsegcon 25739. Show that the square of the distance between two points is non-negative. (Contributed by Scott Fenton, 17-Sep-2013.)

Theoremaxsegconlem4 25732* Lemma for axsegcon 25739. Show that the distance between two points is a real number. (Contributed by Scott Fenton, 17-Sep-2013.)

Theoremaxsegconlem5 25733* Lemma for axsegcon 25739. Show that the distance between two points is non-negative. (Contributed by Scott Fenton, 17-Sep-2013.)

Theoremaxsegconlem6 25734* Lemma for axsegcon 25739. Show that the distance between two distinct points is positive. (Contributed by Scott Fenton, 17-Sep-2013.)

Theoremaxsegconlem7 25735* Lemma for axsegcon 25739. Show that a particular ratio of distances is in the closed unit interval. (Contributed by Scott Fenton, 18-Sep-2013.)

Theoremaxsegconlem8 25736* Lemma for axsegcon 25739. Show that a particular mapping generates a point. (Contributed by Scott Fenton, 18-Sep-2013.)

Theoremaxsegconlem9 25737* Lemma for axsegcon 25739. Show that is congruent to . (Contributed by Scott Fenton, 19-Sep-2013.)

Theoremaxsegconlem10 25738* Lemma for axsegcon 25739. Show that the scaling constant from axsegconlem7 25735 produces the betweenness condition for , and . (Contributed by Scott Fenton, 21-Sep-2013.)

Theoremaxsegcon 25739* 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 25740* Lemma for ax5seg 25750. Rexpress a one congruence sum given betweenness. (Contributed by Scott Fenton, 11-Jun-2013.)

Theoremax5seglem2 25741* Lemma for ax5seg 25750. Rexpress another congruence sum given betweenness. (Contributed by Scott Fenton, 11-Jun-2013.)

Theoremax5seglem3a 25742 Lemma for ax5seg 25750. (Contributed by Scott Fenton, 7-May-2015.)

Theoremax5seglem3 25743* Lemma for ax5seg 25750. Combine congruences for points on a line. (Contributed by Scott Fenton, 11-Jun-2013.)
Cgr Cgr

Theoremax5seglem4 25744* Lemma for ax5seg 25750. Given two distinct points, the scaling constant in a betweenness statement is non-zero. (Contributed by Scott Fenton, 11-Jun-2013.)

Theoremax5seglem5 25745* Lemma for ax5seg 25750. If is between and , and is distinct from , then is distinct from . (Contributed by Scott Fenton, 11-Jun-2013.)

Theoremax5seglem6 25746* Lemma for ax5seg 25750. 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 25747 Lemma for ax5seg 25750. An algebraic calculation needed further down the line. (Contributed by Scott Fenton, 12-Jun-2013.)

Theoremax5seglem8 25748 Lemma for ax5seg 25750. Use the weak deduction theorem to eliminate the hypotheses from ax5seglem7 25747. (Contributed by Scott Fenton, 11-Jun-2013.)

Theoremax5seglem9 25749* Lemma for ax5seg 25750. Take the calculation in ax5seglem8 25748 and turn it into a series of measurements. (Contributed by Scott Fenton, 12-Jun-2013.) (Revised by Mario Carneiro, 22-May-2014.)

Theoremax5seg 25750 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 25751 Points are indivisible. That is, if lies between and , then . Axiom A6 of [Schwabhauser] p. 11. (Contributed by Scott Fenton, 3-Jun-2013.)

Theoremaxpaschlem 25752* Lemma for axpasch 25753. Set up coefficents used in the proof. (Contributed by Scott Fenton, 5-Jun-2013.)

Theoremaxpasch 25753* 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 25754 Lemma for axlowdim 25773. Establish a particular constant function as a function. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremaxlowdimlem2 25755 Lemma for axlowdim 25773. Show that two sets are disjoint. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremaxlowdimlem3 25756 Lemma for axlowdim 25773. Set up a union property for an interval of integers. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremaxlowdimlem4 25757 Lemma for axlowdim 25773. Set up a particular constant function. (Contributed by Scott Fenton, 17-Apr-2013.)

Theoremaxlowdimlem5 25758 Lemma for axlowdim 25773. Show that a particular union is a point in Euclidean space. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremaxlowdimlem6 25759 Lemma for axlowdim 25773. Show that three points are non-colinear. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremaxlowdimlem7 25760 Lemma for axlowdim 25773. Set up a point in Euclidean space. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremaxlowdimlem8 25761 Lemma for axlowdim 25773. Calulate the value of at three. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem9 25762 Lemma for axlowdim 25773. Calulate the value of away from three. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem10 25763 Lemma for axlowdim 25773. Set up a family of points in Euclidean space. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem11 25764 Lemma for axlowdim 25773. Calculate the value of at its distinguished point. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem12 25765 Lemma for axlowdim 25773. Calculate the value of away from its distunguished point. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem13 25766 Lemma for axlowdim 25773. Establish that and are different points. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem14 25767 Lemma for axlowdim 25773. Take two possible from axlowdimlem10 25763. They are the same iff their distunguished values are the same. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem15 25768* Lemma for axlowdim 25773. Set up a one-to-one function of points. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem16 25769* Lemma for axlowdim 25773. Set up a summation that will help establish distance. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem17 25770 Lemma for axlowdim 25773. Establish a congruence result. (Contributed by Scott Fenton, 22-Apr-2013.) (Proof shortened by Mario Carneiro, 22-May-2014.)
Cgr

Theoremaxlowdim1 25771* 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 25772. (Contributed by Scott Fenton, 22-Apr-2013.)

Theoremaxlowdim2 25772* 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 25773* 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 25774* Lemma for axeuclid 25775. Handle the algebraic aspects of the theorem. (Contributed by Scott Fenton, 9-Sep-2013.)

Theoremaxeuclid 25775* 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 25776* Lemma for axcont 25788. Change bound variables for later use. (Contributed by Scott Fenton, 20-Jun-2013.)

Theoremaxcontlem2 25777* Lemma for axcont 25788. The idea here is to set up a mapping that will allow us to transfer dedekind 25109 to two sets of points. Here, we set up and show its domain and range. (Contributed by Scott Fenton, 17-Jun-2013.)

Theoremaxcontlem3 25778* Lemma for axcont 25788. Given the separation assumption, is a subset of . (Contributed by Scott Fenton, 18-Jun-2013.)

Theoremaxcontlem4 25779* Lemma for axcont 25788. Given the separation assumption, is a subset of . (Contributed by Scott Fenton, 18-Jun-2013.)

Theoremaxcontlem5 25780* Lemma for axcont 25788. Compute the value of . (Contributed by Scott Fenton, 18-Jun-2013.)

Theoremaxcontlem6 25781* Lemma for axcont 25788. State the defining properties of the value of (Contributed by Scott Fenton, 19-Jun-2013.)

Theoremaxcontlem7 25782* Lemma for axcont 25788. 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 25783* Lemma for axcont 25788. A point in is between two others if its function value falls in the middle. (Contributed by Scott Fenton, 18-Jun-2013.)

Theoremaxcontlem9 25784* Lemma for axcont 25788. 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 25785* Lemma for axcont 25788. Given a handful of assumptions, derive the conclusion of the final theorem. (Contributed by Scott Fenton, 20-Jun-2013.)

Theoremaxcontlem11 25786* Lemma for axcont 25788. Eliminate the hypotheses from axcontlem10 25785. (Contributed by Scott Fenton, 20-Jun-2013.)

Theoremaxcontlem12 25787* Lemma for axcont 25788. Eliminate the trivial cases from the previous lemmas. (Contributed by Scott Fenton, 20-Jun-2013.)

Theoremaxcont 25788* 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.)

19.7.41  Congruence properties

Syntaxcofs 25789 Declare the syntax for the outer five segment configuration.

Definitiondf-ofs 25790* The outer five segment configuration is an abbreviation for the conditions of the Five Segment Axiom (ax5seg 25750). See brofs 25812 and 5segofs 25813 for how it is used. Definition 2.10 of [Schwabhauser] p. 28. (Contributed by Scott Fenton, 21-Sep-2013.)
Cgr Cgr Cgr Cgr

Theoremcgrrflx2d 25791 Deduction form of axcgrrflx 25726. (Contributed by Scott Fenton, 13-Oct-2013.)
Cgr

Theoremcgrtr4d 25792 Deduction form of axcgrtr 25727. (Contributed by Scott Fenton, 13-Oct-2013.)
Cgr        Cgr        Cgr

Theoremcgrtr4and 25793 Deduction form of axcgrtr 25727. (Contributed by Scott Fenton, 13-Oct-2013.)
Cgr        Cgr        Cgr

Theoremcgrrflx 25794 Reflexivity law for congruence. Theorem 2.1 of [Schwabhauser] p. 27. (Contributed by Scott Fenton, 12-Jun-2013.)
Cgr

Theoremcgrrflxd 25795 Deduction form of cgrrflx 25794. (Contributed by Scott Fenton, 13-Oct-2013.)
Cgr

Theoremcgrcomim 25796 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 25797 Congruence commutes between the two sides. (Contributed by Scott Fenton, 12-Jun-2013.)
Cgr Cgr

Theoremcgrcomand 25798 Deduction form of cgrcom 25797. (Contributed by Scott Fenton, 13-Oct-2013.)
Cgr        Cgr

Theoremcgrtr 25799 Transitivity law for congruence. Theorem 2.3 of [Schwabhauser] p. 27. (Contributed by Scott Fenton, 24-Sep-2013.)
Cgr Cgr Cgr

Theoremcgrtrand 25800 Deduction form of cgrtr 25799. (Contributed by Scott Fenton, 13-Oct-2013.)
Cgr        Cgr        Cgr

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