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Theorem List for Metamath Proof Explorer - 25401-25500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremselsubf 25401 A way of selecting a subset of functions so that their values belong to  B. (Contributed by FL, 14-Jan-2014.)
 |-  A  e.  _V   &    |-  C  e.  _V   =>    |-  (
 ( A  ^m  C )  i^i  ~P ( C  X.  B ) )  =  ( ( A  i^i  B )  ^m  C )
 
Theoremselsubf3 25402 A way of selecting a subset of functions so that their values belong to  B. (Contributed by FL, 14-Jan-2014.)
 |-  A  e.  _V   &    |-  C  e.  _V   =>    |-  (
 ( A  ^m  C )  i^i  ~P ( _V 
 X.  B ) )  =  ( ( A  i^i  B )  ^m  C )
 
Theoremselsubf3g 25403 A way of selecting a subset of functions so that their values belong to  B. (Contributed by FL, 14-Jan-2014.)
 |-  (
 ( A  e.  D  /\  C  e.  E ) 
 ->  ( ( A  ^m  C )  i^i  ~P ( _V  X.  B ) )  =  ( ( A  i^i  B )  ^m  C ) )
 
Syntaxclincl 25404 Extend class notation with the class of inductive closures.
 class  IndCls
 
Definitiondf-indcls 25405* Definition of an inductive closure. Top down definition. Gallier p. 19 (Contributed by FL, 14-Jan-2014.)
 |-  IndCls  =  ( x  e.  _V ,  y  e.  _V  |->  |^| { a  |  ( x  C_  a  /\  A. f  e.  y  A. j  e.  (  dom  f  i^i  ~P ( _V  X.  a ) ) ( f `  j
 )  e.  a ) } )
 
Theoremlemindclsbu 25406* Lemma for indcls2 25407. (Contributed by FL, 14-Jan-2014.)
 |-  (
 ( U  e.  A  /\  A. f  e.  F  E. n  e.  NN  f  e.  ( U  ^m  ( U  ^m  (
 1 ... n ) ) ) )  ->  F  e.  _V )
 
Theoremindcls2 25407* The inductive closure of  X under  F. (Contributed by FL, 14-Jan-2014.)
 |-  (
 ( U  e.  A  /\  X  C_  U  /\  A. f  e.  F  E. n  e.  NN  f  e.  ( U  ^m  ( U  ^m  ( 1 ... n ) ) ) )  ->  ( X  IndCls  F )  =  |^| { a  |  ( X 
 C_  a  /\  A. f  e.  F  A. j  e.  (  dom  f  i^i 
 ~P ( _V  X.  a ) ) ( f `  j )  e.  a ) }
 )
 
Theoremxindcls 25408* X is a part of the inductive closure of  X under  F. (Contributed by FL, 15-Jan-2014.)
 |-  (
 ( U  e.  A  /\  X  C_  U  /\  A. f  e.  F  E. n  e.  NN  f  e.  ( U  ^m  ( U  ^m  ( 1 ... n ) ) ) )  ->  X  C_  ( X  IndCls  F ) )
 
Syntaxcgrm 25409 Extend class notation with the class of all grammars.
 class  Grammar
 
Definitiondf-grm 25410* A grammar is a structure composed of a set of non-terminal symbols  n, of terminal symbols  t, a set of productions  p and a distinguished element of  n, the start symbol  s. Experimental. (Contributed by FL, 15-Jul-2012.)
 |-  Grammar  =  { x  |  E. n E. t E. p E. s ( ( x  =  <. <. n ,  t >. ,  <. p ,  s >.
 >.  /\  n  e.  Fin  /\  t  e.  Fin )  /\  ( ( n  i^i  t )  =  (/)  /\  p  =  { <. l ,  r >.  |  ( l  e.  ( Kleene `  ( n  u.  t ) )  /\  r  e.  ( Kleene `  ( n  u.  t
 ) )  /\  E. a  e.  NN  (
 l `  a )  e.  n ) }  /\  s  e.  n )
 ) }
 
Syntaxcsym 25411 Extend class notation with a function returning the symbols of a grammar.
 class  sym
 
Definitiondf-sym 25412 The symbols of a grammar  g. Experimental. (Contributed by FL, 15-Jul-2012.)
 |-  sym  =  ( g  e.  Grammar  |->  ( ( ( 1st  o.  1st ) `  g )  u.  ( ( 2nd 
 o.  1st ) `  g
 ) ) )
 
Syntaxcprdct 25413 Extend class notation with a function returning the productions of a grammar.
 class  prdct
 
Definitiondf-prodct 25414 The productions of a grammar. (Contributed by FL, 15-Jul-2012.)
 |-  prdct  =  ( 1st  o.  2nd )
 
Syntaxcconc 25415 Extend class notation with an operation concatenating two sequences of symbols.
 class  conc
 
Definitiondf-conc 25416* Concatenation of two words. Experimental. (Contributed by FL, 14-Jan-2014.)
 |-  conc  =  ( x  e.  _V ,  y  e.  _V  |->  if ( x  =  (/) ,  y ,  if (
 y  =  (/) ,  x ,  ( x  u.  (
 a  e.  ( ( ( # `  x )  +  1 ) ... ( ( # `  x )  +  ( # `  y
 ) ) )  |->  ( y `  ( a  -  ( # `  x ) ) ) ) ) ) ) )
 
Theoremisconc1 25417 Concatenation with the empty set. (Contributed by FL, 14-Jan-2014.)
 |-  ( A  e.  B  ->  ( (/)  conc  A )  =  A )
 
Theoremisconc2 25418 Concatenation with the empty set. (Contributed by FL, 14-Jan-2014.)
 |-  ( A  e.  B  ->  ( A  conc  (/) )  =  A )
 
Theoremisconc3 25419* Definition of a concatenation. (Contributed by FL, 14-Jan-2014.)
 |-  (
 ( A  e.  C  /\  B  e.  C  /\  ( A  =/=  (/)  /\  B  =/= 
 (/) ) )  ->  ( A  conc  B )  =  ( A  u.  ( a  e.  (
 ( ( # `  A )  +  1 ) ... ( ( # `  A )  +  ( # `  B ) ) )  |->  ( B `  ( a  -  ( # `  A ) ) ) ) ) )
 
Theoremempklst 25420 The empty set is an element of the kleene star of  A. (Contributed by FL, 2-Feb-2014.)
 |-  ( A  e.  B  ->  (/)  e.  ( Kleene `  A )
 )
 
Theoremclscnc 25421 Closure of concatenation. (Contributed by FL, 2-Feb-2014.)
 |-  (
 ( A  e.  ( Kleene `
  C )  /\  B  e.  ( Kleene `  C ) )  ->  ( A  conc  B )  e.  ( Kleene `  C ) )
 
Syntaxcnots 25422 Extend class notation with the symbol  -..
 class  -. c
 
Definitiondf-nots 25423 Definition of the symbol  -.. Experimental. (Contributed by FL, 2-Feb-2014.)
 |-  -. c  =  { <. 1 ,  1 >. }
 
Syntaxcands 25424 Extend class notation with the symbol  /\.
 class  /\ c
 
Definitiondf-ands 25425 Definition of the symbol  /\. Experimental. (Contributed by FL, 2-Feb-2014.)
 |-  /\ c  =  { <. 1 ,  2 >. }
 
Syntaxclors 25426 Extend class notation with the symbol  \/.
 class  \/ c
 
Definitiondf-ors 25427 Definition of the symbol  \/. Experimental. (Contributed by FL, 2-Feb-2014.)
 |-  \/ c  =  { <. 1 ,  3 >. }
 
Syntaxcimps 25428 Extend class notation with the symbol  ->.
 class  => c
 
Definitiondf-imps 25429 Definition of the symbol  ->. Experimental. (Contributed by FL, 2-Feb-2014.)
 |-  => c  =  { <. 1 ,  4 >. }
 
Syntaxcbis 25430 Extend class notation with the symbol  <->.
 class  <=> c
 
Definitiondf-bis 25431 Definition of the symbol 
<->. Experimental. (Contributed by FL, 2-Feb-2014.)
 |-  <=> c  =  { <. 1 ,  5 >. }
 
Syntaxcfals 25432 Extend class notation with the symbol _|_ .
 class  _|_ c
 
Definitiondf-fals 25433 Definition of the symbol _|_ . Experimental. (Contributed by FL, 2-Feb-2014.)
 |-  _|_ c  =  { <. 1 ,  6 >. }
 
Syntaxcphc 25434 Extend class notation with the symbol  ph.
 class  ph c
 
Definitiondf-phc 25435 Definition of the symbol  ph. Experimental. (Contributed by FL, 2-Feb-2014.)
 |-  ph c  =  { <. 1 ,  7 >. }
 
Syntaxclpsc 25436 Extend class notation with the symbol  ps.
 class  ps c
 
Definitiondf-psc 25437 Definition of the symbol  ps. Experimental. (Contributed by FL, 2-Feb-2014.)
 |-  ps c  =  { <. 1 ,  8 >. }
 
Theoremphckle 25438 The variable  ph c belongs to the Kleene star of  NN. (Contributed by FL, 2-Feb-2014.)
 |-  ph c  e.  ( Kleene `  NN )
 
Theorempsckle 25439 The variable  ps c belongs to the Kleene star of  NN. (Contributed by FL, 2-Feb-2014.)
 |-  ps c  e.  ( Kleene `  NN )
 
SyntaxcPc 25440 Extend class notation with a function that returns a propositional variable.
 class  P c
 
Definitiondf-propvar 25441 Function that returns a propositional variable. Experimental. (Contributed by FL, 2-Feb-2014.)
 |-  P c  =  ( x  e.  ( ZZ>= `  7 )  |->  { <. 1 ,  x >. } )
 
Syntaxcnotc 25442 Extend class notation with a function that returns a negated proposition.
 class  not c
 
Definitiondf-notc 25443 Function that returns a negated proposition. Experimental. (Contributed by FL, 2-Feb-2014.)
 |-  not c  =  ( x  e.  ( ( Kleene `  NN )  ^m  { 1 } )  |->  (  -. c  conc  ( x `  1
 ) ) )
 
Syntaxcandc 25444 Extend class notation with the class of function that returns two propositions joined with  /\.
 class  and c
 
Definitiondf-andc 25445 Function that returns two propositions joined with  /\. Experimental. (Contributed by FL, 2-Feb-2014.)
 |-  and c  =  ( x  e.  ( ( Kleene `  NN )  ^m  { 1 ,  2 } )  |->  ( (  /\ c  conc  ( x `  1 ) )  conc  ( x `  2 ) ) )
 
Syntaxcors 25446 Extend class notation with the function that returns two propositions joined with  \/.
 class  or s
 
Definitiondf-orc 25447 Function that returns two propositions joined with  \/. Experimental. (Contributed by FL, 2-Feb-2014.)
 |-  or s  =  ( x  e.  ( ( Kleene `  NN )  ^m  { 1 ,  2 } )  |->  ( (  \/ c  conc  ( x `  1 ) )  conc  ( x `  2 ) ) )
 
Syntaxcimpc 25448 Extend class notation with the function that returns two propositions joined with  ->.
 class  imp c
 
Definitiondf-impc 25449 Function that returns two propositions joined with  ->. Experimental. (Contributed by FL, 2-Feb-2014.)
 |-  imp c  =  ( x  e.  ( ( Kleene `  NN )  ^m  { 1 ,  2 } )  |->  ( (  => c  conc  ( x `  1 ) ) 
 conc  ( x `  2
 ) ) )
 
Syntaxcbic 25450 Extend class notation with the function that returns two propositions joined with 
<->.
 class  bi c
 
Definitiondf-bic 25451 Function that returns two propositions joined with  <->. Experimental. (Contributed by FL, 2-Feb-2014.)
 |-  bi c  =  ( x  e.  ( ( Kleene `  NN )  ^m  { 1 ,  2 } )  |->  ( (  <=> c  conc  ( x `
  1 ) ) 
 conc  ( x `  2
 ) ) )
 
Syntaxcprop 25452 Extend class notation to include the set of all propositioanl formulas.
 class  Prop
 
Definitiondf-prop 25453 The set of propositional formulas. Gallier p. 32. (Contributed by FL, 2-Feb-2014.)
 |-  Prop  =  ( ( ( P c " ( ZZ>= `  7 ) )  u. 
 { _|_ c } )  IndCls  ( { not c ,  and c ,  or s }  u.  { imp c ,  bi c } )
 )
 
Theoremsmbkle 25454 The symbols and variables of 
Prop belong to the Kleene star of  NN. (Contributed by FL, 2-Feb-2014.)
 |-  (
 ( P c "
 ( ZZ>= `  7 )
 )  u.  { _|_ c } )  C_  ( Kleene `
  NN )
 
Theoremintset 25455 The interval  ( 1 ... 2 ) in terms of its elements. (Contributed by FL, 2-Feb-2014.)
 |-  (
 1 ... 2 )  =  { 1 ,  2 }
 
Theoremfnckle 25456* The functions of  Prop. (Contributed by FL, 2-Feb-2014.)
 |-  A. f  e.  ( { not c ,  and c ,  or s }  u.  { imp c ,  bi c }
 ) E. n  e. 
 NN  f  e.  (
 ( Kleene `  NN )  ^m  ( ( Kleene `  NN )  ^m  ( 1 ... n ) ) )
 
Theoremfnckleb 25457 The functions of  Prop. (Contributed by FL, 2-Feb-2014.)
 |-  A. f  e.  ( { not c ,  and c ,  or s }  u.  { imp c ,  bi c }
 ) ( Fun  f  /\  ran  f  C_  ( Kleene `
  NN ) )
 
Theorempfsubkl 25458 Propositional formulas are a subset of the Kleene star of  NN. (Contributed by FL, 2-Feb-2014.)
 |-  Prop  C_  ( Kleene `  NN )
 
Theorempvp 25459 Propositional variables are propositions . (Contributed by FL, 2-Feb-2014.)
 |-  ( P c " ( ZZ>= `  7 ) )  C_  Prop
 
Theoremcndpv 25460 Condition to be a propositional variable. (Contributed by FL, 2-Feb-2014.)
 |-  (
 ( N  e.  ZZ  /\  7  <_  N )  ->  { <. 1 ,  N >. }  e.  ( P c " ( ZZ>= `  7 ) ) )
 
Theoremphpf 25461  ph c is a propositional formula. (Contributed by FL, 2-Feb-2014.)
 |-  ph c  e.  Prop
 
Theorempspf 25462  ps c is a propositional formula. (Contributed by FL, 2-Feb-2014.)
 |-  ps c  e.  Prop
 
Theorempgapspf 25463  ( ph  /\ 
ps ) is a propositional formula. We use variables. (Contributed by FL, 2-Feb-2014.)
 |-  (
 (  /\ c  conc  ph c
 )  conc  ps c )  e.  Prop
 
Theorempgapspf2 25464  ( ph  /\ 
ps ) is a propositional formula. Here we use meta-variables. (Contributed by FL, 2-Feb-2014.)
 |-  (
 ( P  e.  Prop  /\  Q  e.  Prop )  ->  ( (  /\ c  conc  P )  conc  Q )  e.  Prop )
 
Syntaxcderv 25465 Extend class notation with the relation "derives in one step".
 class  derv
 
Definitiondf-derv 25466* The relation  u derives  v in one step in the grammar  g. Experimental. (Contributed by FL, 15-Jul-2012.)
 |-  derv  =  ( g  e.  Grammar  |->  {
 <. x ,  y >.  |  ( x  e.  ( Kleene `
  g )  /\  y  e.  ( Kleene `  g )  /\  E. u  e.  ( Kleene `  g ) E. v  e.  ( Kleene `
  g ) E. p  e.  ( Kleene `  g ) E. q  e.  ( Kleene `  g )
 ( x  =  ( ( u (  conc  `  g ) p ) (  conc  `  g ) v )  /\  <. p ,  q >.  e.  ( prdct `  g )  /\  y  =  ( ( u (  conc  `  g
 ) q ) ( 
 conc  `  g ) v ) ) ) }
 )
 
18.13.61  Planar geometry
 
Syntaxcpoints 25467 Extend class notation with the class of all Points.
 class PPoints
 
Definitiondf-points 25468 Definition of PPoints. (Contributed by FL, 1-Apr-2016.)
 |- PPoints  =  Base
 
Syntaxcplines 25469 Extend class notation with the class of all Lines.
 class PLines
 
Definitiondf-plines 25470 Definition of PLines. (Contributed by FL, 1-Apr-2016.)
 |- PLines  = Scalar
 
Syntaxcig 25471 Extend class notation with the class of all planar incidence geometries.
 class Ig
 
Definitiondf-ig2 25472* 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  =  { f  |  [. (PPoints `  f )  /  g ]. [. (PLines `  f
 )  /  h ]. ( A. l  e.  h  l  C_  g  /\  ( A. x  e.  g  A. y  e.  g  ( x  =/=  y  ->  E! l  e.  h  ( x  e.  l  /\  y  e.  l
 ) )  /\  A. l  e.  h  E. x  e.  g  E. y  e.  g  ( x  =/=  y  /\  x  e.  l  /\  y  e.  l )  /\  E. x  e.  g  E. y  e.  g  E. z  e.  g  (
 ( x  =/=  y  /\  y  =/=  z  /\  x  =/=  z
 )  /\  A. l  e.  h  -.  ( x  e.  l  /\  y  e.  l  /\  z  e.  l ) ) ) ) }
 
Theorembisig0 25473* 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.)
 |-  P  =  (PPoints `  I )   &    |-  L  =  (PLines `  I )   =>    |-  ( I  e. Ig  <->  ( I  e. 
 _V  /\  ( A. l  e.  L  l  C_  P  /\  A. x  e.  P  A. y  e.  P  ( x  =/=  y  ->  E! l  e.  L  ( x  e.  l  /\  y  e.  l ) )  /\  A. l  e.  L  E. x  e.  P  E. y  e.  P  ( x  =/=  y  /\  x  e.  l  /\  y  e.  l ) )  /\  E. x  e.  P  E. y  e.  P  E. z  e.  P  ( ( x  =/=  y  /\  y  =/=  z  /\  x  =/=  z )  /\  A. l  e.  L  -.  ( x  e.  l  /\  y  e.  l  /\  z  e.  l
 ) ) ) )
 
Theoremisig1a2 25474* 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.)
 |-  P  =  (PPoints `  I )   &    |-  L  =  (PLines `  I )   &    |-  ( ph  ->  I  e. Ig )   =>    |-  ( ph  ->  A. l  e.  L  l  C_  P )
 
Theoremisig12 25475 A line is a set of points. (For my private use only. Don't use.) (Contributed by FL, 2-Apr-2016.)
 |-  P  =  (PPoints `  I )   &    |-  L  =  (PLines `  I )   &    |-  ( ph  ->  I  e. Ig )   &    |-  ( ph  ->  A  e.  L )   =>    |-  ( ph  ->  A  C_  P )
 
Theoremisig22 25476* There is only one line passing through two distinct points. (For my private use only. Don't use.) (Contributed by FL, 2-Apr-2016.)
 |-  P  =  (PPoints `  I )   &    |-  L  =  (PLines `  I )   &    |-  ( ph  ->  I  e. Ig )   =>    |-  ( ph  ->  A. x  e.  P  A. y  e.  P  ( x  =/=  y  ->  E! l  e.  L  ( x  e.  l  /\  y  e.  l
 ) ) )
 
Theoremisig2a2 25477* There is only one line passing through two distinct points. (For my private use only. Don't use.) (Contributed by FL, 2-Apr-2016.)
 |-  P  =  (PPoints `  I )   &    |-  L  =  (PLines `  I )   &    |-  ( ph  ->  I  e. Ig )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  E! l  e.  L  ( A  e.  l  /\  B  e.  l )
 )
 
Theoremelhaltdp 25478* Every line has at least two distinct points. (For my private use only. Don't use.) (Contributed by FL, 28-Apr-2016.)
 |-  P  =  (PPoints `  I )   &    |-  L  =  (PLines `  I )   &    |-  ( ph  ->  I  e. Ig )   =>    |-  ( ph  ->  A. l  e.  L  E. x  e.  P  E. y  e.  P  ( x  =/=  y  /\  x  e.  l  /\  y  e.  l
 ) )
 
Theoremelhaltdp2 25479* Every line has at least two distinct points. (For my private use only. Don't use.) (Contributed by FL, 28-Apr-2016.)
 |-  P  =  (PPoints `  I )   &    |-  L  =  (PLines `  I )   &    |-  ( ph  ->  I  e. Ig )   &    |-  ( ph  ->  A  e.  L )   =>    |-  ( ph  ->  E. x  e.  P  E. y  e.  P  ( x  =/=  y  /\  x  e.  A  /\  y  e.  A ) )
 
Theoremelhalop2 25480* Every line has at least one point. (For my private use only. Don't use.) (Contributed by FL, 10-Aug-2016.)
 |-  P  =  (PPoints `  I )   &    |-  L  =  (PLines `  I )   &    |-  ( ph  ->  I  e. Ig )   &    |-  ( ph  ->  M  e.  L )   =>    |-  ( ph  ->  E. x  e.  P  x  e.  M )
 
Theoremtethpnc 25481* There exists three non colinear points. (For my private use only. Don't use.) (Contributed by FL, 28-Apr-2016.)
 |-  P  =  (PPoints `  I )   &    |-  L  =  (PLines `  I )   &    |-  ( ph  ->  I  e. Ig )   =>    |-  ( ph  ->  E. x  e.  P  E. y  e.  P  E. z  e.  P  ( ( x  =/=  y  /\  y  =/=  z  /\  x  =/=  z )  /\  A. l  e.  L  -.  ( x  e.  l  /\  y  e.  l  /\  z  e.  l
 ) ) )
 
Theoremtethpnc2 25482* There exists three non colinear points. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
 |-  P  =  (PPoints `  I )   &    |-  L  =  (PLines `  I )   &    |-  ( ph  ->  I  e. Ig )   &    |-  ( ph  ->  M  e.  L )   =>    |-  ( ph  ->  E. x  e.  P  E. y  e.  P  E. z  e.  P  ( ( x  =/=  y  /\  y  =/=  z  /\  x  =/=  z )  /\  -.  ( x  e.  M  /\  y  e.  M  /\  z  e.  M ) ) )
 
Theoremgltpntl 25483* 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.)
 |-  P  =  (PPoints `  I )   &    |-  L  =  (PLines `  I )   &    |-  ( ph  ->  I  e. Ig )   &    |-  ( ph  ->  M  e.  L )   =>    |-  ( ph  ->  E. x  e.  P  x  e/  M )
 
Theoremgltpntl2 25484* 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.)
 |-  P  =  (PPoints `  I )   &    |-  L  =  (PLines `  I )   &    |-  ( ph  ->  I  e. Ig )   &    |-  ( ph  ->  M  e.  L )   =>    |-  ( ph  ->  E. x  x  e.  ( P  \  M ) )
 
Theoremaline 25485 A line is not empty. (Contributed by FL, 10-Aug-2016.)
 |-  ( ph  ->  I  e. Ig )   &    |-  L  =  (PLines `  I )   &    |-  ( ph  ->  M  e.  L )   =>    |-  ( ph  ->  M  =/= 
 (/) )
 
Theoremtpne 25486 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.)
 |-  P  =  (PPoints `  I )   &    |-  ( ph  ->  I  e. Ig )   =>    |-  ( ph  ->  P  =/=  (/) )
 
Syntaxcline 25487 Extend class notation with the class of all lines.
 class  line
 
Definitiondf-li 25488* 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.)
 |-  line  =  ( f  e. Ig  |->  ( x  e.  (PPoints `  f
 ) ,  y  e.  (PPoints `  f )  |->  if ( x  =/=  y ,  ( iota_ l  e.  (PLines `  f
 ) ( x  e.  l  /\  y  e.  l ) ) ,  { x } )
 ) )
 
Theoremlinevala2 25489* 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.)
 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   &    |-  M  =  ( line `  G )   &    |-  ( ph  ->  G  e. Ig )   =>    |-  ( ph  ->  M  =  ( x  e.  P ,  y  e.  P  |->  if ( x  =/=  y ,  ( iota_
 l  e.  L ( x  e.  l  /\  y  e.  l )
 ) ,  { x } ) ) )
 
Theoremlineval222 25490* The line passing through two distinct points  A and 
B. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   &    |-  M  =  ( line `  G )   &    |-  ( ph  ->  G  e. Ig )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  ( A M B )  =  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l
 ) ) )
 
Theoremlineval42 25491 Any line to which  A and  B are incident is the line  ( A M B ). (For my private use only. Don't use.) (Contributed by FL, 1-Apr-2016.)
 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   &    |-  M  =  ( line `  G )   &    |-  ( ph  ->  G  e. Ig )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  A  e.  N )   &    |-  ( ph  ->  B  e.  N )   &    |-  ( ph  ->  N  e.  L )   =>    |-  ( ph  ->  N  =  ( A M B ) )
 
Theoremlineval12 25492 The line passing through two distinct points  A and 
B is a line. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   &    |-  M  =  ( line `  G )   &    |-  ( ph  ->  G  e. Ig )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  ( A M B )  e.  L )
 
Theoremlineval22 25493 The points  A and  B belong to the line passing through two distinct points  A and  B. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   &    |-  M  =  ( line `  G )   &    |-  ( ph  ->  G  e. Ig )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  ( A  e.  ( A M B )  /\  B  e.  ( A M B ) ) )
 
Theoremlineval3a 25494 Value of a degenerate line. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  P  =  (PPoints `  G )   &    |-  M  =  ( line `  G )   &    |-  ( ph  ->  G  e. Ig )   &    |-  ( ph  ->  A  e.  P )   =>    |-  ( ph  ->  ( A M A )  =  { A } )
 
Theoremlineval12a 25495 The line passing through two distinct points  A and 
B is a set of points . (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  P  =  (PPoints `  G )   &    |-  M  =  ( line `  G )   &    |-  ( ph  ->  G  e. Ig )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   =>    |-  ( ph  ->  ( A M B )  C_  P )
 
Theoremlineval2a 25496 The point  A belongs to the line passing through it . (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  P  =  (PPoints `  G )   &    |-  M  =  ( line `  G )   &    |-  ( ph  ->  G  e. Ig )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   =>    |-  ( ph  ->  A  e.  ( A M B ) )
 
Theoremlineval2b 25497 The point  B belongs to the line passing through it . (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  P  =  (PPoints `  G )   &    |-  M  =  ( line `  G )   &    |-  ( ph  ->  G  e. Ig )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   =>    |-  ( ph  ->  B  e.  ( A M B ) )
 
Theoremlineval4a 25498 The line AB is the line BA. (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  M  =  ( line `  G )   &    |-  ( ph  ->  G  e. Ig )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   =>    |-  ( ph  ->  ( A M B )  =  ( B M A ) )
 
Theoremlineval5a 25499 If  C is a point of AB, AB = CB. (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  M  =  ( line `  G )   &    |-  ( ph  ->  G  e. Ig )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  ( A M B ) )   &    |-  ( ph  ->  C  =/=  B )   =>    |-  ( ph  ->  ( A M B )  =  ( C M B ) )
 
Theoremlineval6a 25500 If  C is a point of AB, AB = AC (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  M  =  ( line `  G )   &    |-  ( ph  ->  G  e. Ig )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  ( A M B ) )   &    |-  ( ph  ->  A  =/=  C )   =>    |-  ( ph  ->  ( A M B )  =  ( A M C ) )
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