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Theorem ldilset 30920
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 30919 . . . 4  |-  ( K  e.  C  ->  ( LDil `  K )  =  ( w  e.  H  |->  { f  e.  I  |  A. x  e.  B  ( x  .<_  w  -> 
( f `  x
)  =  x ) } ) )
76fveq1d 5543 . . 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 4043 . . . . . . 7  |-  ( w  =  W  ->  (
x  .<_  w  <->  x  .<_  W ) )
98imbi1d 308 . . . . . 6  |-  ( w  =  W  ->  (
( x  .<_  w  -> 
( f `  x
)  =  x )  <-> 
( x  .<_  W  -> 
( f `  x
)  =  x ) ) )
109ralbidv 2576 . . . . 5  |-  ( w  =  W  ->  ( A. x  e.  B  ( x  .<_  w  -> 
( f `  x
)  =  x )  <->  A. x  e.  B  ( x  .<_  W  -> 
( f `  x
)  =  x ) ) )
1110rabbidv 2793 . . . 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 2296 . . . 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 5555 . . . . . 6  |-  ( LAut `  K )  e.  _V
145, 13eqeltri 2366 . . . . 5  |-  I  e. 
_V
1514rabex 4181 . . . 4  |-  { f  e.  I  |  A. x  e.  B  (
x  .<_  W  ->  (
f `  x )  =  x ) }  e.  _V
1611, 12, 15fvmpt 5618 . . 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 2348 . 2  |-  ( ( K  e.  C  /\  W  e.  H )  ->  ( ( LDil `  K
) `  W )  =  { f  e.  I  |  A. x  e.  B  ( x  .<_  W  -> 
( f `  x
)  =  x ) } )
181, 17syl5eq 2340 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 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801   class class class wbr 4039    e. cmpt 4093   ` cfv 5271   Basecbs 13164   lecple 13231   LHypclh 30795   LAutclaut 30796   LDilcldil 30911
This theorem is referenced by:  isldil  30921
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ldil 30915
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