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

Theorem ldilfset 30297
Description: The mapping from fiducial co-atom  w to its set of lattice dilations. (Contributed by NM, 11-May-2012.)
Hypotheses
Ref Expression
ldilset.b  |-  B  =  ( Base `  K
)
ldilset.l  |-  .<_  =  ( le `  K )
ldilset.h  |-  H  =  ( LHyp `  K
)
ldilset.i  |-  I  =  ( LAut `  K
)
Assertion
Ref Expression
ldilfset  |-  ( K  e.  C  ->  ( LDil `  K )  =  ( w  e.  H  |->  { f  e.  I  |  A. x  e.  B  ( x  .<_  w  -> 
( f `  x
)  =  x ) } ) )
Distinct variable groups:    x, B    w, H    f, I    w, f, x, K
Allowed substitution hints:    B( w, f)    C( x, w, f)    H( x, f)    I( x, w)    .<_ ( x, w, f)

Proof of Theorem ldilfset
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 2796 . 2  |-  ( K  e.  C  ->  K  e.  _V )
2 fveq2 5525 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 ldilset.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2333 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5525 . . . . . 6  |-  ( k  =  K  ->  ( LAut `  k )  =  ( LAut `  K
) )
6 ldilset.i . . . . . 6  |-  I  =  ( LAut `  K
)
75, 6syl6eqr 2333 . . . . 5  |-  ( k  =  K  ->  ( LAut `  k )  =  I )
8 fveq2 5525 . . . . . . 7  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
9 ldilset.b . . . . . . 7  |-  B  =  ( Base `  K
)
108, 9syl6eqr 2333 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  B )
11 fveq2 5525 . . . . . . . . 9  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
12 ldilset.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
1311, 12syl6eqr 2333 . . . . . . . 8  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1413breqd 4034 . . . . . . 7  |-  ( k  =  K  ->  (
x ( le `  k ) w  <->  x  .<_  w ) )
1514imbi1d 308 . . . . . 6  |-  ( k  =  K  ->  (
( x ( le
`  k ) w  ->  ( f `  x )  =  x )  <->  ( x  .<_  w  ->  ( f `  x )  =  x ) ) )
1610, 15raleqbidv 2748 . . . . 5  |-  ( k  =  K  ->  ( A. x  e.  ( Base `  k ) ( x ( le `  k ) w  -> 
( f `  x
)  =  x )  <->  A. x  e.  B  ( x  .<_  w  -> 
( f `  x
)  =  x ) ) )
177, 16rabeqbidv 2783 . . . 4  |-  ( k  =  K  ->  { f  e.  ( LAut `  k
)  |  A. x  e.  ( Base `  k
) ( x ( le `  k ) w  ->  ( f `  x )  =  x ) }  =  {
f  e.  I  | 
A. x  e.  B  ( x  .<_  w  -> 
( f `  x
)  =  x ) } )
184, 17mpteq12dv 4098 . . 3  |-  ( k  =  K  ->  (
w  e.  ( LHyp `  k )  |->  { f  e.  ( LAut `  k
)  |  A. x  e.  ( Base `  k
) ( x ( le `  k ) w  ->  ( f `  x )  =  x ) } )  =  ( w  e.  H  |->  { f  e.  I  |  A. x  e.  B  ( x  .<_  w  -> 
( f `  x
)  =  x ) } ) )
19 df-ldil 30293 . . 3  |-  LDil  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  { f  e.  ( LAut `  k )  |  A. x  e.  ( Base `  k ) ( x ( le `  k
) w  ->  (
f `  x )  =  x ) } ) )
20 fvex 5539 . . . . 5  |-  ( LHyp `  K )  e.  _V
213, 20eqeltri 2353 . . . 4  |-  H  e. 
_V
2221mptex 5746 . . 3  |-  ( w  e.  H  |->  { f  e.  I  |  A. x  e.  B  (
x  .<_  w  ->  (
f `  x )  =  x ) } )  e.  _V
2318, 19, 22fvmpt 5602 . 2  |-  ( K  e.  _V  ->  ( LDil `  K )  =  ( w  e.  H  |->  { f  e.  I  |  A. x  e.  B  ( x  .<_  w  -> 
( f `  x
)  =  x ) } ) )
241, 23syl 15 1  |-  ( K  e.  C  ->  ( LDil `  K )  =  ( w  e.  H  |->  { f  e.  I  |  A. x  e.  B  ( x  .<_  w  -> 
( f `  x
)  =  x ) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788   class class class wbr 4023    e. cmpt 4077   ` cfv 5255   Basecbs 13148   lecple 13215   LHypclh 30173   LAutclaut 30174   LDilcldil 30289
This theorem is referenced by:  ldilset  30298
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ldil 30293
  Copyright terms: Public domain W3C validator