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Theorem ldilfset 35975
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 3118 . 2  |-  ( K  e.  C  ->  K  e.  _V )
2 fveq2 5872 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 ldilset.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2516 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5872 . . . . . 6  |-  ( k  =  K  ->  ( LAut `  k )  =  ( LAut `  K
) )
6 ldilset.i . . . . . 6  |-  I  =  ( LAut `  K
)
75, 6syl6eqr 2516 . . . . 5  |-  ( k  =  K  ->  ( LAut `  k )  =  I )
8 fveq2 5872 . . . . . . 7  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
9 ldilset.b . . . . . . 7  |-  B  =  ( Base `  K
)
108, 9syl6eqr 2516 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  B )
11 fveq2 5872 . . . . . . . . 9  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
12 ldilset.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
1311, 12syl6eqr 2516 . . . . . . . 8  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1413breqd 4467 . . . . . . 7  |-  ( k  =  K  ->  (
x ( le `  k ) w  <->  x  .<_  w ) )
1514imbi1d 317 . . . . . 6  |-  ( k  =  K  ->  (
( x ( le
`  k ) w  ->  ( f `  x )  =  x )  <->  ( x  .<_  w  ->  ( f `  x )  =  x ) ) )
1610, 15raleqbidv 3068 . . . . 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 3104 . . . 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 4535 . . 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 35971 . . 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 5882 . . . . 5  |-  ( LHyp `  K )  e.  _V
213, 20eqeltri 2541 . . . 4  |-  H  e. 
_V
2221mptex 6144 . . 3  |-  ( w  e.  H  |->  { f  e.  I  |  A. x  e.  B  (
x  .<_  w  ->  (
f `  x )  =  x ) } )  e.  _V
2318, 19, 22fvmpt 5956 . 2  |-  ( K  e.  _V  ->  ( LDil `  K )  =  ( w  e.  H  |->  { f  e.  I  |  A. x  e.  B  ( x  .<_  w  -> 
( f `  x
)  =  x ) } ) )
241, 23syl 16 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 setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   A.wral 2807   {crab 2811   _Vcvv 3109   class class class wbr 4456    |-> cmpt 4515   ` cfv 5594   Basecbs 14644   lecple 14719   LHypclh 35851   LAutclaut 35852   LDilcldil 35967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ldil 35971
This theorem is referenced by:  ldilset  35976
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