Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  df-pautN Unicode version

Definition df-pautN 29447
Description: Define set of all projective automorphisms. This is the intended definition of automorphism in [Crawley] p. 112. (Contributed by NM, 26-Jan-2012.)
Assertion
Ref Expression
df-pautN  |-  PAut  =  ( k  e.  _V  |->  { f  |  ( f : ( PSubSp `  k ) -1-1-onto-> ( PSubSp `  k )  /\  A. x  e.  (
PSubSp `  k ) A. y  e.  ( PSubSp `  k ) ( x 
C_  y  <->  ( f `  x )  C_  (
f `  y )
) ) } )
Distinct variable group:    f, k, x, y

Detailed syntax breakdown of Definition df-pautN
StepHypRef Expression
1 cpautN 29443 . 2  class  PAut
2 vk . . 3  set  k
3 cvv 2789 . . 3  class  _V
42cv 1623 . . . . . . 7  class  k
5 cpsubsp 28952 . . . . . . 7  class  PSubSp
64, 5cfv 5221 . . . . . 6  class  ( PSubSp `  k )
7 vf . . . . . . 7  set  f
87cv 1623 . . . . . 6  class  f
96, 6, 8wf1o 5220 . . . . 5  wff  f : ( PSubSp `  k ) -1-1-onto-> ( PSubSp `
 k )
10 vx . . . . . . . . . 10  set  x
1110cv 1623 . . . . . . . . 9  class  x
12 vy . . . . . . . . . 10  set  y
1312cv 1623 . . . . . . . . 9  class  y
1411, 13wss 3153 . . . . . . . 8  wff  x  C_  y
1511, 8cfv 5221 . . . . . . . . 9  class  ( f `
 x )
1613, 8cfv 5221 . . . . . . . . 9  class  ( f `
 y )
1715, 16wss 3153 . . . . . . . 8  wff  ( f `
 x )  C_  ( f `  y
)
1814, 17wb 178 . . . . . . 7  wff  ( x 
C_  y  <->  ( f `  x )  C_  (
f `  y )
)
1918, 12, 6wral 2544 . . . . . 6  wff  A. y  e.  ( PSubSp `  k )
( x  C_  y  <->  ( f `  x ) 
C_  ( f `  y ) )
2019, 10, 6wral 2544 . . . . 5  wff  A. x  e.  ( PSubSp `  k ) A. y  e.  ( PSubSp `
 k ) ( x  C_  y  <->  ( f `  x )  C_  (
f `  y )
)
219, 20wa 360 . . . 4  wff  ( f : ( PSubSp `  k
)
-1-1-onto-> ( PSubSp `  k )  /\  A. x  e.  (
PSubSp `  k ) A. y  e.  ( PSubSp `  k ) ( x 
C_  y  <->  ( f `  x )  C_  (
f `  y )
) )
2221, 7cab 2270 . . 3  class  { f  |  ( f : ( PSubSp `  k ) -1-1-onto-> ( PSubSp `
 k )  /\  A. x  e.  ( PSubSp `  k ) A. y  e.  ( PSubSp `  k )
( x  C_  y  <->  ( f `  x ) 
C_  ( f `  y ) ) ) }
232, 3, 22cmpt 4078 . 2  class  ( k  e.  _V  |->  { f  |  ( f : ( PSubSp `  k ) -1-1-onto-> ( PSubSp `
 k )  /\  A. x  e.  ( PSubSp `  k ) A. y  e.  ( PSubSp `  k )
( x  C_  y  <->  ( f `  x ) 
C_  ( f `  y ) ) ) } )
241, 23wceq 1624 1  wff  PAut  =  ( k  e.  _V  |->  { f  |  ( f : ( PSubSp `  k ) -1-1-onto-> ( PSubSp `  k )  /\  A. x  e.  (
PSubSp `  k ) A. y  e.  ( PSubSp `  k ) ( x 
C_  y  <->  ( f `  x )  C_  (
f `  y )
) ) } )
Colors of variables: wff set class
This definition is referenced by:  pautsetN  29554
  Copyright terms: Public domain W3C validator