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Theorem List for Metamath Proof Explorer - 25801-25900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtfrqfree 25801* Calculate a quantifier-free version of the function from tfr1 6661 through tfr3 6663. (Contributed by Scott Fenton, 29-Apr-2014.)
Domain Domain Apply FullFun Restrict

Theoremdfint3 25802 Quantifier-free definition of class intersection. (Contributed by Scott Fenton, 13-Apr-2018.)

Theoremimagesset 25803 The Image functor applied to the converse of the subset relationship yields a subset of the subset relationship. (Contributed by Scott Fenton, 14-Apr-2018.)
Image

Theorembrub 25804* Binary relationship form of the upper bound functor. (Contributed by Scott Fenton, 3-May-2018.)
UB

Theorembrlb 25805* Binary relationship form of the lower bound functor. (Contributed by Scott Fenton, 3-May-2018.)
LB

19.7.39  Alternate ordered pairs

Syntaxcaltop 25806 Declare the syntax for an alternate ordered pair.

Syntaxcaltxp 25807 Declare the syntax for an alternate cross product.

Definitiondf-altop 25808 An alternative definition of ordered pairs. This definition removes a hypothesis from its defining theorem (see altopth 25819), making it more convenient in some circumstances. (Contributed by Scott Fenton, 22-Mar-2012.)

Definitiondf-altxp 25809* Define cross products of alternative ordered pairs. (Contributed by Scott Fenton, 23-Mar-2012.)

Theoremaltopex 25810 Alternative ordered pairs always exist. (Contributed by Scott Fenton, 22-Mar-2012.)

Theoremaltopthsn 25811 Two alternate ordered pairs are equal iff the singletons of their respective elements are equal. Note that this holds regardless of sethood of any of the elements. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremaltopeq12 25812 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)

Theoremaltopeq1 25813 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)

Theoremaltopeq2 25814 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)

Theoremaltopth1 25815 Equality of the first members of equal alternate ordered pairs, which holds regardless of the second members' sethood. (Contributed by Scott Fenton, 22-Mar-2012.)

Theoremaltopth2 25816 Equality of the second members of equal alternate ordered pairs, which holds regardless of the first members' sethood. (Contributed by Scott Fenton, 22-Mar-2012.)

Theoremaltopthg 25817 Alternate ordered pair theorem. (Contributed by Scott Fenton, 22-Mar-2012.)

Theoremaltopthbg 25818 Alternate ordered pair theorem. (Contributed by Scott Fenton, 14-Apr-2012.)

Theoremaltopth 25819 The alternate ordered pair theorem. If two alternate ordered pairs are equal, their first elements are equal and their second elements are equal. Note that and are not required to be a set due to a peculiarity of our specific ordered pair definition, as opposed to the regular ordered pairs used here, which (as in opth 4438), requires to be a set. (Contributed by Scott Fenton, 23-Mar-2012.)

Theoremaltopthb 25820 Alternate ordered pair theorem with different sethood requirements. See altopth 25819 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)

Theoremaltopthc 25821 Alternate ordered pair theorem with different sethood requirements. See altopth 25819 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)

Theoremaltopthd 25822 Alternate ordered pair theorem with different sethood requirements. See altopth 25819 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)

Theoremaltxpeq1 25823 Equality for alternate cross products. (Contributed by Scott Fenton, 24-Mar-2012.)

Theoremaltxpeq2 25824 Equality for alternate cross products. (Contributed by Scott Fenton, 24-Mar-2012.)

Theoremelaltxp 25825* Membership in alternate cross products. (Contributed by Scott Fenton, 23-Mar-2012.)

Theoremaltopelaltxp 25826 Alternate ordered pair membership in a cross product. Note that, unlike opelxp 4911, there is no sethood requirement here. (Contributed by Scott Fenton, 22-Mar-2012.)

Theoremaltxpsspw 25827 An inclusion rule for alternate cross products. (Contributed by Scott Fenton, 24-Mar-2012.)

Theoremaltxpexg 25828 The alternate cross product of two sets is a set. (Contributed by Scott Fenton, 24-Mar-2012.)

Theoremrankaltopb 25829 Compute the rank of an alternate ordered pair. (Contributed by Scott Fenton, 18-Dec-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremnfaltop 25830 Bound-variable hypothesis builder for alternate ordered pairs. (Contributed by Scott Fenton, 25-Sep-2015.)

Theoremsbcaltop 25831* Distribution of class substitution over alternate ordered pairs. (Contributed by Scott Fenton, 25-Sep-2015.)

19.7.40  Tarskian geometry

Syntaxcee 25832 Declare the syntax for the Euclidean space generator.

Syntaxcbtwn 25833 Declare the syntax for the Euclidean betweenness predicate.

Syntaxccgr 25834 Declare the syntax for the Euclidean congruence predicate.
Cgr

Definitiondf-ee 25835 Define the Euclidean space generator. For details, see elee 25838. (Contributed by Scott Fenton, 3-Jun-2013.)

Definitiondf-btwn 25836* Define the Euclidean betweenness predicate. For details, see brbtwn 25843. (Contributed by Scott Fenton, 3-Jun-2013.)

Definitiondf-cgr 25837* Define the Euclidean congruence predicate. For details, see brcgr 25844. (Contributed by Scott Fenton, 3-Jun-2013.)
Cgr

Theoremelee 25838 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 25839* A condition for a mapping to be an element of a Euclidean space. (Contributed by Scott Fenton, 7-Jun-2013.)

Theoremeleenn 25840 If is in , then is a natural. (Contributed by Scott Fenton, 1-Jul-2013.)

Theoremeleei 25841 The forward direction of elee 25838. (Contributed by Scott Fenton, 1-Jul-2013.)

Theoremeedimeq 25842 A point belongs to at most one Euclidean space. (Contributed by Scott Fenton, 1-Jul-2013.)

Theorembrbtwn 25843* 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 25844* 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 25845 The function value of a point is a real. (Contributed by Scott Fenton, 10-Jun-2013.)

Theoremfveecn 25846 The function value of a point is a complex. (Contributed by Scott Fenton, 10-Jun-2013.)

Theoremeqeefv 25847* Two points are equal iff they agree in all dimensions. (Contributed by Scott Fenton, 10-Jun-2013.)

Theoremeqeelen 25848* 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 25849* Alternate characterization of betweenness, with no existential quantifiers. (Contributed by Scott Fenton, 24-Jun-2013.)

Theoremcolinearalglem1 25850 Lemma for colinearalg 25854. Expand out a multiplication. (Contributed by Scott Fenton, 24-Jun-2013.)

Theoremcolinearalglem2 25851* Lemma for colinearalg 25854. Translate between two forms of the colinearity condition. (Contributed by Scott Fenton, 24-Jun-2013.)

Theoremcolinearalglem3 25852* Lemma for colinearalg 25854. Translate between two forms of the colinearity condition. (Contributed by Scott Fenton, 24-Jun-2013.)

Theoremcolinearalglem4 25853* Lemma for colinearalg 25854. Prove a disjunction that will be needed in the final proof. (Contributed by Scott Fenton, 27-Jun-2013.)

Theoremcolinearalg 25854* An algebraic characterization of colinearity. Note the similarity to brbtwn2 25849. (Contributed by Scott Fenton, 24-Jun-2013.)

Theoremeleesub 25855* Membership of a subtraction mapping in a Euclidean space. (Contributed by Scott Fenton, 17-Jul-2013.)

Theoremeleesubd 25856* Membership of a subtraction mapping in a Euclidean space. Deduction form of eleesub 25855. (Contributed by Scott Fenton, 17-Jul-2013.)

19.7.41  Tarski's axioms for geometry

Theoremaxdimuniq 25857 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 25858 is as far from as is from . Axiom A1 of [Schwabhauser] p. 10. (Contributed by Scott Fenton, 3-Jun-2013.)
Cgr

Theoremaxcgrtr 25859 Congruence is transitive. Axiom A2 of [Schwabhauser] p. 10. (Contributed by Scott Fenton, 3-Jun-2013.)
Cgr Cgr Cgr

Theoremaxcgrid 25860 If there is no distance between and , then . Axiom A3 of [Schwabhauser] p. 10. (Contributed by Scott Fenton, 3-Jun-2013.)
Cgr

Theoremaxsegconlem1 25861* Lemma for axsegcon 25871. Handle the degenerate case. (Contributed by Scott Fenton, 7-Jun-2013.)

Theoremaxsegconlem2 25862* Lemma for axsegcon 25871. Show that the square of the distance between two points is a real number. (Contributed by Scott Fenton, 17-Sep-2013.)

Theoremaxsegconlem3 25863* Lemma for axsegcon 25871. Show that the square of the distance between two points is non-negative. (Contributed by Scott Fenton, 17-Sep-2013.)

Theoremaxsegconlem4 25864* Lemma for axsegcon 25871. Show that the distance between two points is a real number. (Contributed by Scott Fenton, 17-Sep-2013.)

Theoremaxsegconlem5 25865* Lemma for axsegcon 25871. Show that the distance between two points is non-negative. (Contributed by Scott Fenton, 17-Sep-2013.)

Theoremaxsegconlem6 25866* Lemma for axsegcon 25871. Show that the distance between two distinct points is positive. (Contributed by Scott Fenton, 17-Sep-2013.)

Theoremaxsegconlem7 25867* Lemma for axsegcon 25871. Show that a particular ratio of distances is in the closed unit interval. (Contributed by Scott Fenton, 18-Sep-2013.)

Theoremaxsegconlem8 25868* Lemma for axsegcon 25871. Show that a particular mapping generates a point. (Contributed by Scott Fenton, 18-Sep-2013.)

Theoremaxsegconlem9 25869* Lemma for axsegcon 25871. Show that is congruent to . (Contributed by Scott Fenton, 19-Sep-2013.)

Theoremaxsegconlem10 25870* Lemma for axsegcon 25871. Show that the scaling constant from axsegconlem7 25867 produces the betweenness condition for , and . (Contributed by Scott Fenton, 21-Sep-2013.)

Theoremaxsegcon 25871* 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 25872* Lemma for ax5seg 25882. Rexpress a one congruence sum given betweenness. (Contributed by Scott Fenton, 11-Jun-2013.)

Theoremax5seglem2 25873* Lemma for ax5seg 25882. Rexpress another congruence sum given betweenness. (Contributed by Scott Fenton, 11-Jun-2013.)

Theoremax5seglem3a 25874 Lemma for ax5seg 25882. (Contributed by Scott Fenton, 7-May-2015.)

Theoremax5seglem3 25875* Lemma for ax5seg 25882. Combine congruences for points on a line. (Contributed by Scott Fenton, 11-Jun-2013.)
Cgr Cgr

Theoremax5seglem4 25876* Lemma for ax5seg 25882. Given two distinct points, the scaling constant in a betweenness statement is non-zero. (Contributed by Scott Fenton, 11-Jun-2013.)

Theoremax5seglem5 25877* Lemma for ax5seg 25882. If is between and , and is distinct from , then is distinct from . (Contributed by Scott Fenton, 11-Jun-2013.)

Theoremax5seglem6 25878* Lemma for ax5seg 25882. 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 25879 Lemma for ax5seg 25882. An algebraic calculation needed further down the line. (Contributed by Scott Fenton, 12-Jun-2013.)

Theoremax5seglem8 25880 Lemma for ax5seg 25882. Use the weak deduction theorem to eliminate the hypotheses from ax5seglem7 25879. (Contributed by Scott Fenton, 11-Jun-2013.)

Theoremax5seglem9 25881* Lemma for ax5seg 25882. Take the calculation in ax5seglem8 25880 and turn it into a series of measurements. (Contributed by Scott Fenton, 12-Jun-2013.) (Revised by Mario Carneiro, 22-May-2014.)

Theoremax5seg 25882 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 25883 Points are indivisible. That is, if lies between and , then . Axiom A6 of [Schwabhauser] p. 11. (Contributed by Scott Fenton, 3-Jun-2013.)

Theoremaxpaschlem 25884* Lemma for axpasch 25885. Set up coefficents used in the proof. (Contributed by Scott Fenton, 5-Jun-2013.)

Theoremaxpasch 25885* 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 25886 Lemma for axlowdim 25905. Establish a particular constant function as a function. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremaxlowdimlem2 25887 Lemma for axlowdim 25905. Show that two sets are disjoint. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremaxlowdimlem3 25888 Lemma for axlowdim 25905. Set up a union property for an interval of integers. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremaxlowdimlem4 25889 Lemma for axlowdim 25905. Set up a particular constant function. (Contributed by Scott Fenton, 17-Apr-2013.)

Theoremaxlowdimlem5 25890 Lemma for axlowdim 25905. Show that a particular union is a point in Euclidean space. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremaxlowdimlem6 25891 Lemma for axlowdim 25905. Show that three points are non-colinear. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremaxlowdimlem7 25892 Lemma for axlowdim 25905. Set up a point in Euclidean space. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremaxlowdimlem8 25893 Lemma for axlowdim 25905. Calulate the value of at three. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem9 25894 Lemma for axlowdim 25905. Calulate the value of away from three. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem10 25895 Lemma for axlowdim 25905. Set up a family of points in Euclidean space. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem11 25896 Lemma for axlowdim 25905. Calculate the value of at its distinguished point. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem12 25897 Lemma for axlowdim 25905. Calculate the value of away from its distunguished point. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem13 25898 Lemma for axlowdim 25905. Establish that and are different points. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem14 25899 Lemma for axlowdim 25905. Take two possible from axlowdimlem10 25895. They are the same iff their distunguished values are the same. (Contributed by Scott Fenton, 21-Apr-2013.)

Theoremaxlowdimlem15 25900* Lemma for axlowdim 25905. Set up a one-to-one function of points. (Contributed by Scott Fenton, 21-Apr-2013.)

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