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Theorem List for Metamath Proof Explorer - 25201-25300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxcors 25201 Extend class notation with the function that returns two propositions joined with .

Definitiondf-orc 25202 Function that returns two propositions joined with . Experimental. (Contributed by FL, 2-Feb-2014.)

Syntaxcimpc 25203 Extend class notation with the function that returns two propositions joined with .

Definitiondf-impc 25204 Function that returns two propositions joined with . Experimental. (Contributed by FL, 2-Feb-2014.)

Syntaxcbic 25205 Extend class notation with the function that returns two propositions joined with .

Definitiondf-bic 25206 Function that returns two propositions joined with . Experimental. (Contributed by FL, 2-Feb-2014.)

Syntaxcprop 25207 Extend class notation to include the set of all propositioanl formulas.

Definitiondf-prop 25208 The set of propositional formulas. Gallier p. 32. (Contributed by FL, 2-Feb-2014.)

Theoremsmbkle 25209 The symbols and variables of belong to the Kleene star of . (Contributed by FL, 2-Feb-2014.)

Theoremintset 25210 The interval in terms of its elements. (Contributed by FL, 2-Feb-2014.)

Theoremfnckle 25211* The functions of . (Contributed by FL, 2-Feb-2014.)

Theoremfnckleb 25212 The functions of . (Contributed by FL, 2-Feb-2014.)

Theorempfsubkl 25213 Propositional formulas are a subset of the Kleene star of . (Contributed by FL, 2-Feb-2014.)

Theorempvp 25214 Propositional variables are propositions . (Contributed by FL, 2-Feb-2014.)

Theoremcndpv 25215 Condition to be a propositional variable. (Contributed by FL, 2-Feb-2014.)

Theoremphpf 25216 is a propositional formula. (Contributed by FL, 2-Feb-2014.)

Theorempspf 25217 is a propositional formula. (Contributed by FL, 2-Feb-2014.)

Theorempgapspf 25218 is a propositional formula. We use variables. (Contributed by FL, 2-Feb-2014.)

Theorempgapspf2 25219 is a propositional formula. Here we use meta-variables. (Contributed by FL, 2-Feb-2014.)

Syntaxcderv 25220 Extend class notation with the relation "derives in one step".

Definitiondf-derv 25221* The relation derives in one step in the grammar . Experimental. (Contributed by FL, 15-Jul-2012.)

16.11.61  Planar geometry

Syntaxcpoints 25222 Extend class notation with the class of all Points.
PPoints

Definitiondf-points 25223 Definition of PPoints. (Contributed by FL, 1-Apr-2016.)
PPoints

Syntaxcplines 25224 Extend class notation with the class of all Lines.
PLines

Definitiondf-plines 25225 Definition of PLines. (Contributed by FL, 1-Apr-2016.)
PLines Scalar

Syntaxcig 25226 Extend class notation with the class of all planar incidence geometries.
Ig

Definitiondf-ig2 25227* Definition of a geometry that can build on the axioms of incidence. Definition of an Incidence-Betweenness Geometry in [AitkenIBG] p. 1-2. (For my private use only. Don't use.) (Contributed by FL, 1-Apr-2016.)
Ig PPoints PLines

Theorembisig0 25228* Definition of a geometry that can build on the axioms of incidence. Definition of an Incidence-Betweenness Geometry in [AitkenIBG] p. 1-2. (For my private use only. Don't use.) (Contributed by FL, 1-Apr-2016.)
PPoints       PLines       Ig

Theoremisig1a2 25229* A line is a set of points. This axiom is not needed. (Let's recall that the incidence relation can be formalized as an abstract relation. And that the belonging relationship is only an interpretation.) However Wayne Aitken adds this axiom to his system and I will follow him. The definitions below will take advantage of it. (For my private use only. Don't use.) (Contributed by FL, 2-Apr-2016.)
PPoints       PLines       Ig

Theoremisig12 25230 A line is a set of points. (For my private use only. Don't use.) (Contributed by FL, 2-Apr-2016.)
PPoints       PLines       Ig

Theoremisig22 25231* There is only one line passing through two distinct points. (For my private use only. Don't use.) (Contributed by FL, 2-Apr-2016.)
PPoints       PLines       Ig

Theoremisig2a2 25232* There is only one line passing through two distinct points. (For my private use only. Don't use.) (Contributed by FL, 2-Apr-2016.)
PPoints       PLines       Ig

Theoremelhaltdp 25233* Every line has at least two distinct points. (For my private use only. Don't use.) (Contributed by FL, 28-Apr-2016.)
PPoints       PLines       Ig

Theoremelhaltdp2 25234* Every line has at least two distinct points. (For my private use only. Don't use.) (Contributed by FL, 28-Apr-2016.)
PPoints       PLines       Ig

Theoremelhalop2 25235* Every line has at least one point. (For my private use only. Don't use.) (Contributed by FL, 10-Aug-2016.)
PPoints       PLines       Ig

Theoremtethpnc 25236* There exists three non colinear points. (For my private use only. Don't use.) (Contributed by FL, 28-Apr-2016.)
PPoints       PLines       Ig

Theoremtethpnc2 25237* There exists three non colinear points. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
PPoints       PLines       Ig

Theoremgltpntl 25238* Given a line, there exists a point not on this line. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
PPoints       PLines       Ig

Theoremgltpntl2 25239* Given a line, there exists a point not on this line. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
PPoints       PLines       Ig

Theoremaline 25240 A line is not empty. (Contributed by FL, 10-Aug-2016.)
Ig       PLines

Theoremtpne 25241 The plane is not empty. Exercise 5 of [AitkenIBG] p. 4. (For my private use only. Don't use.) (Contributed by FL, 29-Apr-2016.)
PPoints       Ig

Syntaxcline 25242 Extend class notation with the class of all lines.

Definitiondf-li 25243* Definition of the line xy. It also defines a degenerate line. Definition 4 of [AitkenIBG] p. 2. (For my private use only. Don't use.) (Contributed by FL, 2-Apr-2016.)
Ig PPoints PPoints PLines

Theoremlinevala2 25244* Definition of the line xy. It also defines a degenerate line. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
PPoints       PLines              Ig

Theoremlineval222 25245* The line passing through two distinct points and . (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
PPoints       PLines              Ig

Theoremlineval42 25246 Any line to which and are incident is the line . (For my private use only. Don't use.) (Contributed by FL, 1-Apr-2016.)
PPoints       PLines              Ig

Theoremlineval12 25247 The line passing through two distinct points and is a line. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
PPoints       PLines              Ig

Theoremlineval22 25248 The points and belong to the line passing through two distinct points and . (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
PPoints       PLines              Ig

Theoremlineval3a 25249 Value of a degenerate line. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
PPoints              Ig

Theoremlineval12a 25250 The line passing through two distinct points and is a set of points . (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
PPoints              Ig

Theoremlineval2a 25251 The point belongs to the line passing through it . (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
PPoints              Ig

Theoremlineval2b 25252 The point belongs to the line passing through it . (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
PPoints              Ig

Theoremlineval4a 25253 The line AB is the line BA. (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints              Ig

Theoremlineval5a 25254 If is a point of AB, AB = CB. (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints              Ig

Theoremlineval6a 25255 If is a point of AB, AB = AC (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints              Ig

Syntaxccol 25256 Extend class notation with the class of all collinear points sets.
coln

Definitiondf-col 25257* Definition of collinear points. Definition 5 of [AitkenIBG] p. 3. (Contributed by FL, 1-Apr-2016.)
coln Ig PLines

Theoremiscola2 25258* The predicate "being collinear points". (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
Ig       coln PLines

Theoremiscol2 25259* The predicate "being collinear points". (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
Ig              coln PLines

Theoremiscol3 25260* Collinear points are points. (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
Ig       coln       PPoints

Syntaxccon2 25261 Extend class notation with the class of all concurrent lines.
con

Definitiondf-con2 25262* Definition of concurrent lines. Definition 6 of [AitkenIBG] p. 3. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
con Ig PLines

Theoremisconcl1b 25263* The predicate "are concurrent lines". (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
Ig       con PLines

Theoremisconcl2b 25264 The predicate "are concurrent lines". (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
Ig       con PLines

Theoremisconcl3b 25265 Only lines can be concurrent. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
Ig       con       PLines

Theoremisconcl4b 25266 Concurrent lines have at least one point in common. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
Ig       con

Theoremisconcl5a 25267* Two distinct lines intersect in at most one point. Proposition 4 of [AitkenIBG] p. 3. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
PLines       PPoints       Ig

Theoremisconcl5ab 25268* Two distinct lines intersect in at most one point. Proposition 4 of [AitkenIBG] p. 3. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
PLines       PPoints       Ig

Theoremisconcl6a 25269* Two distinct non-parallel lines intersect in one and only point. Proposition 4 of [AitkenIBG] p. 3. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
PLines       PPoints       Ig

Theoremisconcl6ab 25270* Two distinct non-parallel lines intersect in one and only point. Proposition 4 of [AitkenIBG] p. 3. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
PLines       PPoints       Ig

Theoremisconcl7a 25271 Two distinct non-parallel lines intersect in one and only point. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
PLines       Ig

Syntaxcbtw 25272 Extend class notation with the betweenness relation.
btw

Syntaxcibg 25273 Extend class notation with the class of all planar incidence betweenness geometries.
Ibg

Definitiondf-btw 25274 Definition of btw. (Contributed by FL, 1-Apr-2016.)
btw

Definitiondf-ibg2 25275* Definition of a geometry that can build on the axioms of incidence and betweenness. Axioms B-1, B-2, B-3, B-4 of [AitkenIBG] p. 3-4. (For my private use only. Don't use.) (Contributed by FL, 1-Apr-2016.)
Ibg Ig PPoints PLines coln btw

Theoremisibg2 25276* The predicate "is an incidence-betweenness geometry".

B-1 If is between and then it is between and and , , are collinear and distinct.

B-2 If and are distinct, then there are points , , such that is before and , is between and , is after and .

B-3 If three points , , are collinear and distinct then exactly one of the followings occurs: is between and , is between and , is between and .

B-4 "Being on the same side" is a transitive relation. If and are not on the same side of and and are not on the same side of then and are on the same side of . (For my private use only. Don't use.) (Contributed by FL, 1-Apr-2016.)

PPoints       PLines       btw       coln       Ibg Ig

Theoremisibg1a 25277 An incidence-betweenness geometry is an incidence geometry. (For my private use only. Don't use.) (Contributed by FL, 2-Apr-2016.)
Ibg       Ig

Theoremisibg2aa 25278* Properties of an incidence-betweenness geometry. (Contributed by FL, 10-Aug-2016.)
PPoints       PLines       btw       coln                                   Ibg

Theoremisibg2aalem1 25279* Properties of an incidence-betweenness geometry. (Contributed by FL, 10-Aug-2016.)
PPoints       PLines

Theoremisibg2aalem2 25280* Properties of an incidence-betweenness geometry. (Contributed by FL, 10-Aug-2016.)

Theoremisibg2aalem3 25281* Properties of an incidence-betweenness geometry. (Contributed by FL, 10-Aug-2016.)

Theoremisib2g1a1 25282 If is between and , it is between and (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints       btw       Ibg

Theoremisibg1a2 25283 If is between and , then , , are collinear . (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints       btw       Ibg                     coln

Theoremisibg2a 25284* Two distinct points have a point before, between and after them. (For my private use only. Don't use.) (Contributed by FL, 1-Apr-2016.)
PPoints       btw       Ibg

Theoremisibg2a1 25285* Two distinct points , have a point before them. (For my private use only. Don't use.) (Contributed by FL, 10-Aug-2016.)
PPoints       btw       Ibg

Theoremisibg2a2 25286* Two distinct points , have a point between them. (For my private use only. Don't use.) (Contributed by FL, 10-Aug-2016.)
PPoints       btw       Ibg

Theoremisibg2a3 25287* Two distinct points , have a point after them. (For my private use only. Don't use.) (Contributed by FL, 10-Aug-2016.)
PPoints       btw       Ibg

Theoremisibg1a3a 25288 If is between and , then and , are distinct . (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints       btw       Ibg

Theoremisibg1a4a 25289 If is between and , then and , are distinct . (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints       btw       Ibg

Theoremisibg1a5a 25290 If is between and , then and , are distinct . (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints       btw       Ibg

Theoremisibg1a6 25291 If is between and , it belongs to the line XZ (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints       btw       Ibg

Theoremisibg1a7 25292 If is between and , belongs to the line YZ (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints       btw       Ibg

Theoremisibg1a8 25293 If is between and , belongs to the line XY (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints       btw       Ibg

Theorembsstr 25294 Being on the same side is a transitive relation. (For my private use only. Don't use.) (Contributed by FL, 10-Jul-2016.)
PPoints       PLines       btw       Ibg

Theoremnbssntr 25295 IF and are not on the same side, and and are not on the same side then and are on the same side. (For my private use only. Don't use.) (Contributed by FL, 10-Jul-2016.)
PPoints       PLines       btw       Ibg

Syntaxcseg 25296 Extend class notation with the segment symbol.

Definitiondf-seg2 25297* Definition of the segment xy degenerated or not. Definition 8 of [AitkenIBG] p. 4. (For my private use only. Don't use.) (Contributed by FL, 28-Feb-2016.)
Ibg PPoints PPoints PPoints btw

Theoremsgplpte21 25298* The predicate "is a non-degenerated segment". (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints              Ibg              btw

Theoremsgplpte21a 25299* The predicate "is a non-degenerated segment". (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints              Ibg              btw

Theoremsgplpte21b 25300 The predicate "is a non-degenerated segment". (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints              Ibg              btw

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