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Theorem List for Metamath Proof Explorer - 25301-25400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdomidmor 25301 Domain of an identity morphism. (Contributed by FL, 6-Nov-2013.)
 |-  (
 ( U  e.  Univ  /\  A  e.  U ) 
 ->  ( ( dom SetCat `  U ) `  ( ( Id SetCat `
  U ) `  A ) )  =  A )
 
Theoremdomidmor2 25302 Domain of an identity morphism. (Contributed by FL, 6-Nov-2013.)
 |- .dom  =  ( dom SetCat `  U )   &    |- .id  =  ( Id SetCat `
  U )   =>    |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (.dom  `  (.id  `  A ) )  =  A )
 
Theoremcodidmor 25303 Domain of an identity morphism. (Contributed by FL, 6-Nov-2013.)
 |-  (
 ( U  e.  Univ  /\  A  e.  U ) 
 ->  ( ( cod SetCat `  U ) `  ( ( Id SetCat `
  U ) `  A ) )  =  A )
 
Theoremcodidmor2 25304 Domain of an identity morphism. (Contributed by FL, 6-Nov-2013.)
 |- .cod  =  ( cod SetCat `  U )   &    |- .id  =  ( Id SetCat `
  U )   =>    |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (.cod  `  (.id  `  A ) )  =  A )
 
Theoremgrphidmor 25305 Graph of an identity morphism. (Contributed by FL, 6-Nov-2013.)
 |-  (
 ( U  e.  Univ  /\  A  e.  U ) 
 ->  ( ( graph SetCat `  U ) `  ( ( Id SetCat `
  U ) `  A ) )  =  (  _I  |`  A ) )
 
Theoremgrphidmor2 25306 Graph of an identity morphism. (Contributed by FL, 6-Nov-2013.)
 |- .graph  =  2nd   &    |- .id  =  ( Id SetCat `
  U )   =>    |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (.graph  `  (.id  `  A ) )  =  (  _I  |`  A ) )
 
Theoremgrphidmor3 25307 Graph of an identity morphism. (Contributed by FL, 6-Nov-2013.)
 |- .graph  =  ( graph SetCat `  U )   &    |- .id  =  ( Id SetCat `  U )   =>    |-  ( ( U  e.  Univ  /\  A  e.  U ) 
 ->  (.graph  `  (.id  `  A ) )  =  (  _I  |`  A ) )
 
Syntaxcrocase 25308 Extend class notation to include the morphisms composition in the category Set.
 class  ro SetCat
 
Definitiondf-rocatset 25309* Composition of two morphisms in the category Set. Experimental. (Contributed by FL, 6-Nov-2013.)
 |-  ro SetCat  =  ( x  e.  Univ  |->  {
 <. <. a ,  b >. ,  c >.  |  ( ( a  e.  ( Morphism SetCat `  x )  /\  b  e.  ( Morphism SetCat `  x )  /\  ( ( 1st  o.  1st ) `  a )  =  ( ( 2nd 
 o.  1st ) `  b
 ) )  /\  c  =  <. <. ( ( 1st 
 o.  1st ) `  b
 ) ,  ( ( 2nd  o.  1st ) `  a ) >. ,  (
 ( 2nd `  a )  o.  ( 2nd `  b
 ) ) >. ) }
 )
 
Theoremisrocatset 25310* Definition of the composition of two morphisms in the category Set . (Contributed by FL, 6-Nov-2013.) (Revised by Mario Carneiro, 20-Dec-2013.)
 |-  ( U  e.  Univ  ->  ( ro
 SetCat `  U )  =  { <. <. a ,  b >. ,  c >.  |  ( ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  ( ( 1st  o.  1st ) `  a )  =  ( ( 2nd 
 o.  1st ) `  b
 ) )  /\  c  =  <. <. ( ( 1st 
 o.  1st ) `  b
 ) ,  ( ( 2nd  o.  1st ) `  a ) >. ,  (
 ( 2nd `  a )  o.  ( 2nd `  b
 ) ) >. ) }
 )
 
Theoremcmp2morp 25311 Composite of two morphisms. (Contributed by FL, 6-Nov-2013.) (Revised by Mario Carneiro, 20-Dec-2013.)
 |-  O  =  ( ro SetCat `  U )   =>    |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
 )  /\  ( ( dom
 SetCat `  U ) `  A )  =  (
 ( cod SetCat `  U ) `  B ) )  ->  ( A O B )  =  <. <. ( ( 1st 
 o.  1st ) `  B ) ,  ( ( 2nd  o.  1st ) `  A ) >. ,  (
 ( 2nd `  A )  o.  ( 2nd `  B ) ) >. )
 
Theoremrocatval 25312 The composite of two morphisms in the category Set is a morphism. (Contributed by FL, 6-Nov-2013.)
 |-  O  =  ( ro SetCat `  U )   =>    |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
 )  /\  ( ( dom
 SetCat `  U ) `  A )  =  (
 ( cod SetCat `  U ) `  B ) )  ->  ( A O B )  e.  ( Morphism SetCat `  U ) )
 
Theoremrocatval2 25313 The composite of two morphisms in the category Set is a morphism. (Contributed by FL, 7-Nov-2013.)
 |-  O  =  ( ro SetCat `  U )   &    |- .Morphism  =  ( Morphism SetCat `  U )   &    |- .dom  =  ( dom SetCat `  U )   &    |- .cod  =  ( cod SetCat `
  U )   =>    |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( A O B )  e. .Morphism  )
 
Theoremcmp2morpcats 25314 Composite of two morphisms. (Contributed by FL, 7-Nov-2013.)
 |-  O  =  ( ro SetCat `  U )   &    |- .Morphism  =  ( Morphism SetCat `  U )   &    |- .dom  =  ( dom SetCat `  U )   &    |- .cod  =  ( cod SetCat `
  U )   =>    |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( A O B )  =  <. <.
 (.dom  `  B ) ,  (.cod  `  A ) >. ,  (
 ( 2nd `  A )  o.  ( 2nd `  B ) ) >. )
 
Theoremcmp2morpcatt 25315 Composite of two morphisms. (Contributed by FL, 7-Nov-2013.)
 |-  O  =  ( ro SetCat `  U )   &    |- .Morphism  =  ( Morphism SetCat `  U )   &    |- .dom  =  ( dom SetCat `  U )   &    |- .cod  =  ( cod SetCat `
  U )   &    |- .graph  =  ( graph SetCat `  U )   =>    |-  (
 ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( A O B )  =  <. <. (.dom  `  B ) ,  (.cod  `  A ) >. ,  (
 (.graph  `  A )  o.  (.graph  `  B ) ) >. )
 
Theoremcmp2morpgrp 25316 Graph of the composite of two morphisms. (Contributed by FL, 7-Nov-2013.)
 |-  O  =  ( ro SetCat `  U )   &    |- .Morphism  =  ( Morphism SetCat `  U )   &    |- .dom  =  ( dom SetCat `  U )   &    |- .cod  =  ( cod SetCat `
  U )   &    |- .graph  =  ( graph SetCat `  U )   =>    |-  (
 ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  (.graph  `  ( A O B ) )  =  ( (.graph  `  A )  o.  (.graph  `  B ) ) )
 
Theoremcmp2morpdom 25317 Domain of the composite of two morphisms. (Contributed by FL, 7-Nov-2013.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  O  =  ( ro SetCat `  U )   &    |- .Morphism  =  ( Morphism SetCat `  U )   &    |- .dom  =  ( dom SetCat `  U )   &    |- .cod  =  ( cod SetCat `
  U )   =>    |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  (.dom  `  ( A O B ) )  =  (.dom  `  B ) )
 
Theoremcmp2morpcod 25318 Codomain of the composite of two morphisms. (Contributed by FL, 7-Nov-2013.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  O  =  ( ro SetCat `  U )   &    |- .Morphism  =  ( Morphism SetCat `  U )   &    |- .dom  =  ( dom SetCat `  U )   &    |- .cod  =  ( cod SetCat `
  U )   =>    |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  (.cod  `  ( A O B ) )  =  (.cod  `  A ) )
 
Theoremcmpmorass 25319 Associativity of composition in category Set. (Contributed by FL, 7-Nov-2013.)
 |-  O  =  ( ro SetCat `  U )   &    |- .Morphism  =  ( Morphism SetCat `  U )   &    |- .dom  =  ( dom SetCat `  U )   &    |- .cod  =  ( cod SetCat `
  U )   =>    |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  /\  C  e. .Morphism  )  /\  ( (.dom  `  C )  =  (.cod  `  B ) 
 /\  (.dom  `  B )  =  (.cod  `  A ) ) ) 
 ->  ( C O ( B O A ) )  =  ( ( C O B ) O A ) )
 
Theoremmorexcmp 25320 A morphism expressed thanks to its components. (Contributed by FL, 8-Nov-2013.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
 |- .Morphism  =  ( Morphism SetCat `  U )   &    |- .dom  =  ( dom SetCat `  U )   &    |- .cod  =  ( cod SetCat `
  U )   &    |- .graph  =  2nd   =>    |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  F  =  <. <. (.dom  `  F ) ,  (.cod  `  F ) >. ,  (.graph  `  F ) >. )
 
Theoremmorexcmp2 25321 A morphism expressed thanks to its components. (Contributed by FL, 8-Nov-2013.)
 |- .Morphism  =  ( Morphism SetCat `  U )   &    |- .dom  =  ( dom SetCat `  U )   &    |- .cod  =  ( cod SetCat `
  U )   &    |- .graph  =  ( graph SetCat `  U )   =>    |-  (
 ( U  e.  Univ  /\  F  e. .Morphism  )  ->  F  =  <. <. (.dom  `  F ) ,  (.cod  `  F ) >. ,  (.graph  `  F ) >. )
 
Theoremcmpidmor2 25322 Composition with an identity. (Contributed by FL, 8-Nov-2013.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
 |-  O  =  ( ro SetCat `  U )   &    |- .Morphism  =  ( Morphism SetCat `  U )   &    |- .cod  =  ( cod SetCat `  U )   &    |- .id  =  ( Id SetCat `
  U )   =>    |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  ( (.id  `  (.cod  `  F ) ) O F )  =  F )
 
Theoremcmpidmor3 25323 Composition with an identity. (Contributed by FL, 8-Nov-2013.)
 |-  O  =  ( ro SetCat `  U )   &    |- .Morphism  =  ( Morphism SetCat `  U )   &    |- .dom  =  ( dom SetCat `  U )   &    |- .id  =  ( Id SetCat `
  U )   =>    |-  ( ( U  e.  Univ  /\  F  e. .Morphism  )  ->  ( F O (.id  `  (.dom  `  F ) ) )  =  F )
 
Theoremcmpmorfun 25324 Composition of morphisms is a function. (Contributed by FL, 8-Nov-2013.) (Revised by Mario Carneiro, 20-Dec-2013.)
 |-  ( U  e.  Univ  ->  Fun  ( ro
 SetCat `  U ) )
 
Theoremcmppar2 25325* Morphisms composition is defined every time the codomain of the second operand matches the domain of the first one. (Contributed by FL, 8-Nov-2013.) (Revised by Mario Carneiro, 20-Dec-2013.)
 |-  ( U  e.  Univ  ->  dom  ( ro
 SetCat `  U )  =  { <. a ,  b >.  |  ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  (
 ( 1st  o.  1st ) `  a )  =  ( ( 2nd  o.  1st ) `  b ) ) } )
 
Theoremcmppar 25326 Composition of morphisms is a partial operation in the set of morphisms. (Contributed by FL, 8-Nov-2013.) (Revised by Mario Carneiro, 20-Dec-2013.)
 |-  ( U  e.  Univ  ->  dom  ( ro
 SetCat `  U )  C_  ( ( Morphism SetCat `  U )  X.  ( Morphism SetCat `  U ) ) )
 
Theoremcmppar3 25327 Morphisms composition is defined every time the codomain of the second operand matches the domain of the first one. (Contributed by FL, 8-Nov-2013.)
 |-  (
 ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
 )  ->  ( <. A ,  B >.  e.  dom  ( ro SetCat `  U )  <->  ( ( dom SetCat `  U ) `  A )  =  ( ( cod SetCat `  U ) `  B ) ) )
 
Theoremcmpmor 25328 The composite of two morphisms is a morphism. (Contributed by FL, 8-Nov-2013.)
 |-  ( U  e.  Univ  ->  ran  ( ro
 SetCat `  U )  C_  ( Morphism SetCat `  U )
 )
 
Syntaxccaset 25329 Extend class notation to include the category Set.
 class  SetCat OLD
 
Definitiondf-catset 25330 Definition of the category Set. (We should say "the categories Set" since there is such a category per universe but for our purpose they are equivalent obviously.) Experimental. (Contributed by FL, 8-Nov-2013.)
 |-  SetCat OLD  =  ( x  e.  Univ  |->  <. <. ( dom SetCat `
  x ) ,  ( cod SetCat `  x ) >. ,  <. ( Id SetCat `
  x ) ,  ( ro SetCat `  x ) >. >. )
 
Theoremiscatset 25331 The category Set. (Contributed by FL, 8-Nov-2013.)
 |-  ( U  e.  Univ  ->  ( SetCat OLD `  U )  =  <. <. ( dom SetCat `  U ) ,  ( cod SetCat `  U ) >. ,  <. ( Id SetCat `  U ) ,  ( ro SetCat `  U ) >. >. )
 
Theoremsetiscat 25332 The category set is a category. (Contributed by FL, 6-Nov-2013.)
 |-  ( U  e.  Univ  ->  ( SetCat OLD `  U )  e.  Cat OLD  )
 
16.12.59  Grammars, Logics, Machines and Automata
 
Syntaxckln 25333 Extend class notation with the Kleene star.
 class  Kleene
 
16.12.60  Words
 
Syntaxcwrd 25334 Extend class notation with the class of words of a given size.
 class  Words
 
Definitiondf-words 25335* The words of size  y over an alphabet  x are the finite sequences over  x of size  y. Their domains are fiercely set to  ( 1 ... y ) so that I can concatenate them easily. The case  y  =  0 is a bit tricky and corresponds to the unique empty word (often denoted by an epsilon or by  1 in textbooks.) Experimental. (Contributed by FL, 14-Jan-2014.)
 |-  Words  =  ( x  e.  _V ,  y  e.  NN0  |->  ( x 
 ^m  ( 1 ... y ) ) )
 
Theoremisword 25336 The words over a set  A. (Contributed by FL, 14-Jan-2014.)
 |-  (
 ( A  e.  B  /\  N  e.  NN0 )  ->  ( A  Words  N )  =  ( A  ^m  ( 1 ... N ) ) )
 
Syntaxcdwords 25337 Syntax for the dWords operator.
 class dWords
 
Definitiondf-dwords 25338* Words of size S over an alphabet  A with all the elements different. (For my private use only. Don't use.) (Contributed by FL, 26-May-2016.)
 |- dWords  =  ( a  e.  _V ,  s  e.  NN0  |->  { w  e.  ( a  Words  s )  |  A. x  e.  ( 1 ... s
 ) A. y  e.  (
 1 ... s ) ( x  =/=  y  ->  ( w `  x )  =/=  ( w `  y ) ) }
 )
 
Theoremisnword 25339* The words over a set  A. (For my private use only. Don't use.) (Contributed by FL, 26-May-2014.)
 |-  (
 ( A  e.  B  /\  S  e.  NN0 )  ->  ( W  e.  ( AdWords S )  <->  ( W  e.  ( A  ^m  ( 1
 ... S ) ) 
 /\  A. x  e.  (
 1 ... S ) A. y  e.  ( 1 ... S ) ( x  =/=  y  ->  ( W `  x )  =/=  ( W `  y
 ) ) ) ) )
 
Definitiondf-kle 25340* The Kleene star of an alphabet  x is the set of all the finite sequences of elements of this alphabet. Experimental. (Contributed by FL, 14-Jan-2014.)
 |-  Kleene  =  ( x  e.  _V  |->  U_ u  e.  NN0  ( x 
 ^m  ( 1 ... u ) ) )
 
TheoremisKleene 25341* The predicate is the Kleene star of a set  A. An element of  ( Kleene `  A
) is called a word. (Contributed by FL, 14-Jan-2014.)
 |-  ( A  e.  B  ->  (
 Kleene `  A )  = 
 U_ u  e.  NN0  ( A  ^m  ( 1
 ... u ) ) )
 
Theorem1iskle 25342 Symbols and variables belong to the Kleene star of  NN. (Contributed by FL, 2-Feb-2014.)
 |-  ( A  e.  NN  ->  {
 <. 1 ,  A >. }  e.  ( Kleene `  NN ) )
 
Theoremselsubf 25343 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 25344 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 25345 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 25346 Extend class notation with the class of inductive closures.
 class  IndCls
 
Definitiondf-indcls 25347* 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 25348* Lemma for indcls2 25349. (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 25349* 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 25350* 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 25351 Extend class notation with the class of all grammars.
 class  Grammar
 
Definitiondf-grm 25352* 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 25353 Extend class notation with a function returning the symbols of a grammar.
 class  sym
 
Definitiondf-sym 25354 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 25355 Extend class notation with a function returning the productions of a grammar.
 class  prdct
 
Definitiondf-prodct 25356 The productions of a grammar. (Contributed by FL, 15-Jul-2012.)
 |-  prdct  =  ( 1st  o.  2nd )
 
Syntaxcconc 25357 Extend class notation with an operation concatenating two sequences of symbols.
 class  conc
 
Definitiondf-conc 25358* 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 25359 Concatenation with the empty set. (Contributed by FL, 14-Jan-2014.)
 |-  ( A  e.  B  ->  ( (/)  conc  A )  =  A )
 
Theoremisconc2 25360 Concatenation with the empty set. (Contributed by FL, 14-Jan-2014.)
 |-  ( A  e.  B  ->  ( A  conc  (/) )  =  A )
 
Theoremisconc3 25361* 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 25362 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 25363 Closure of concatenation. (Contributed by FL, 2-Feb-2014.)
 |-  (
 ( A  e.  ( Kleene `
  C )  /\  B  e.  ( Kleene `  C ) )  ->  ( A  conc  B )  e.  ( Kleene `  C ) )
 
Syntaxcnots 25364 Extend class notation with the symbol  -..
 class  -. c
 
Definitiondf-nots 25365 Definition of the symbol  -.. Experimental. (Contributed by FL, 2-Feb-2014.)
 |-  -. c  =  { <. 1 ,  1 >. }
 
Syntaxcands 25366 Extend class notation with the symbol  /\.
 class  /\ c
 
Definitiondf-ands 25367 Definition of the symbol  /\. Experimental. (Contributed by FL, 2-Feb-2014.)
 |-  /\ c  =  { <. 1 ,  2 >. }
 
Syntaxclors 25368 Extend class notation with the symbol  \/.
 class  \/ c
 
Definitiondf-ors 25369 Definition of the symbol  \/. Experimental. (Contributed by FL, 2-Feb-2014.)
 |-  \/ c  =  { <. 1 ,  3 >. }
 
Syntaxcimps 25370 Extend class notation with the symbol  ->.
 class  => c
 
Definitiondf-imps 25371 Definition of the symbol  ->. Experimental. (Contributed by FL, 2-Feb-2014.)
 |-  => c  =  { <. 1 ,  4 >. }
 
Syntaxcbis 25372 Extend class notation with the symbol  <->.
 class  <=> c
 
Definitiondf-bis 25373 Definition of the symbol 
<->. Experimental. (Contributed by FL, 2-Feb-2014.)
 |-  <=> c  =  { <. 1 ,  5 >. }
 
Syntaxcfals 25374 Extend class notation with the symbol _|_ .
 class  _|_ c
 
Definitiondf-fals 25375 Definition of the symbol _|_ . Experimental. (Contributed by FL, 2-Feb-2014.)
 |-  _|_ c  =  { <. 1 ,  6 >. }
 
Syntaxcphc 25376 Extend class notation with the symbol  ph.
 class  ph c
 
Definitiondf-phc 25377 Definition of the symbol  ph. Experimental. (Contributed by FL, 2-Feb-2014.)
 |-  ph c  =  { <. 1 ,  7 >. }
 
Syntaxclpsc 25378 Extend class notation with the symbol  ps.
 class  ps c
 
Definitiondf-psc 25379 Definition of the symbol  ps. Experimental. (Contributed by FL, 2-Feb-2014.)
 |-  ps c  =  { <. 1 ,  8 >. }
 
Theoremphckle 25380 The variable  ph c belongs to the Kleene star of  NN. (Contributed by FL, 2-Feb-2014.)
 |-  ph c  e.  ( Kleene `  NN )
 
Theorempsckle 25381 The variable  ps c belongs to the Kleene star of  NN. (Contributed by FL, 2-Feb-2014.)
 |-  ps c  e.  ( Kleene `  NN )
 
SyntaxcPc 25382 Extend class notation with a function that returns a propositional variable.
 class  P c
 
Definitiondf-propvar 25383 Function that returns a propositional variable. Experimental. (Contributed by FL, 2-Feb-2014.)
 |-  P c  =  ( x  e.  ( ZZ>= `  7 )  |->  { <. 1 ,  x >. } )
 
Syntaxcnotc 25384 Extend class notation with a function that returns a negated proposition.
 class  not c
 
Definitiondf-notc 25385 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 25386 Extend class notation with the class of function that returns two propositions joined with  /\.
 class  and c
 
Definitiondf-andc 25387 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 25388 Extend class notation with the function that returns two propositions joined with  \/.
 class  or s
 
Definitiondf-orc 25389 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 25390 Extend class notation with the function that returns two propositions joined with  ->.
 class  imp c
 
Definitiondf-impc 25391 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 25392 Extend class notation with the function that returns two propositions joined with 
<->.
 class  bi c
 
Definitiondf-bic 25393 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 25394 Extend class notation to include the set of all propositioanl formulas.
 class  Prop
 
Definitiondf-prop 25395 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 25396 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 25397 The interval  ( 1 ... 2 ) in terms of its elements. (Contributed by FL, 2-Feb-2014.)
 |-  (
 1 ... 2 )  =  { 1 ,  2 }
 
Theoremfnckle 25398* 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 25399 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 25400 Propositional formulas are a subset of the Kleene star of  NN. (Contributed by FL, 2-Feb-2014.)
 |-  Prop  C_  ( Kleene `  NN )
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