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Theorem ldilset 30906
Description: The set of lattice dilations for a fiducial co-atom  W. (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
)
ldilset.d  |-  D  =  ( ( LDil `  K
) `  W )
Assertion
Ref Expression
ldilset  |-  ( ( K  e.  C  /\  W  e.  H )  ->  D  =  { f  e.  I  |  A. x  e.  B  (
x  .<_  W  ->  (
f `  x )  =  x ) } )
Distinct variable groups:    x, B    f, I    x, f, K   
f, W, x
Allowed substitution hints:    B( f)    C( x, f)    D( x, f)    H( x, f)    I( x)    .<_ ( x, f)

Proof of Theorem ldilset
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 ldilset.d . 2  |-  D  =  ( ( LDil `  K
) `  W )
2 ldilset.b . . . . 5  |-  B  =  ( Base `  K
)
3 ldilset.l . . . . 5  |-  .<_  =  ( le `  K )
4 ldilset.h . . . . 5  |-  H  =  ( LHyp `  K
)
5 ldilset.i . . . . 5  |-  I  =  ( LAut `  K
)
62, 3, 4, 5ldilfset 30905 . . . 4  |-  ( K  e.  C  ->  ( LDil `  K )  =  ( w  e.  H  |->  { f  e.  I  |  A. x  e.  B  ( x  .<_  w  -> 
( f `  x
)  =  x ) } ) )
76fveq1d 5730 . . 3  |-  ( K  e.  C  ->  (
( LDil `  K ) `  W )  =  ( ( w  e.  H  |->  { f  e.  I  |  A. x  e.  B  ( x  .<_  w  -> 
( f `  x
)  =  x ) } ) `  W
) )
8 breq2 4216 . . . . . . 7  |-  ( w  =  W  ->  (
x  .<_  w  <->  x  .<_  W ) )
98imbi1d 309 . . . . . 6  |-  ( w  =  W  ->  (
( x  .<_  w  -> 
( f `  x
)  =  x )  <-> 
( x  .<_  W  -> 
( f `  x
)  =  x ) ) )
109ralbidv 2725 . . . . 5  |-  ( w  =  W  ->  ( A. x  e.  B  ( x  .<_  w  -> 
( f `  x
)  =  x )  <->  A. x  e.  B  ( x  .<_  W  -> 
( f `  x
)  =  x ) ) )
1110rabbidv 2948 . . . 4  |-  ( w  =  W  ->  { f  e.  I  |  A. x  e.  B  (
x  .<_  w  ->  (
f `  x )  =  x ) }  =  { f  e.  I  |  A. x  e.  B  ( x  .<_  W  -> 
( f `  x
)  =  x ) } )
12 eqid 2436 . . . 4  |-  ( w  e.  H  |->  { f  e.  I  |  A. x  e.  B  (
x  .<_  w  ->  (
f `  x )  =  x ) } )  =  ( w  e.  H  |->  { f  e.  I  |  A. x  e.  B  ( x  .<_  w  ->  ( f `  x )  =  x ) } )
13 fvex 5742 . . . . . 6  |-  ( LAut `  K )  e.  _V
145, 13eqeltri 2506 . . . . 5  |-  I  e. 
_V
1514rabex 4354 . . . 4  |-  { f  e.  I  |  A. x  e.  B  (
x  .<_  W  ->  (
f `  x )  =  x ) }  e.  _V
1611, 12, 15fvmpt 5806 . . 3  |-  ( W  e.  H  ->  (
( w  e.  H  |->  { f  e.  I  |  A. x  e.  B  ( x  .<_  w  -> 
( f `  x
)  =  x ) } ) `  W
)  =  { f  e.  I  |  A. x  e.  B  (
x  .<_  W  ->  (
f `  x )  =  x ) } )
177, 16sylan9eq 2488 . 2  |-  ( ( K  e.  C  /\  W  e.  H )  ->  ( ( LDil `  K
) `  W )  =  { f  e.  I  |  A. x  e.  B  ( x  .<_  W  -> 
( f `  x
)  =  x ) } )
181, 17syl5eq 2480 1  |-  ( ( K  e.  C  /\  W  e.  H )  ->  D  =  { f  e.  I  |  A. x  e.  B  (
x  .<_  W  ->  (
f `  x )  =  x ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   {crab 2709   _Vcvv 2956   class class class wbr 4212    e. cmpt 4266   ` cfv 5454   Basecbs 13469   lecple 13536   LHypclh 30781   LAutclaut 30782   LDilcldil 30897
This theorem is referenced by:  isldil  30907
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ldil 30901
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