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Theorem isldil 33107
Description: The predicate "is a lattice dilation". Similar to definition of dilation in [Crawley] p. 111. (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
isldil  |-  ( ( K  e.  C  /\  W  e.  H )  ->  ( F  e.  D  <->  ( F  e.  I  /\  A. x  e.  B  ( x  .<_  W  ->  ( F `  x )  =  x ) ) ) )
Distinct variable groups:    x, B    x, K    x, W    x, F
Allowed substitution hints:    C( x)    D( x)    H( x)    I( x)    .<_ ( x)

Proof of Theorem isldil
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 ldilset.b . . . 4  |-  B  =  ( Base `  K
)
2 ldilset.l . . . 4  |-  .<_  =  ( le `  K )
3 ldilset.h . . . 4  |-  H  =  ( LHyp `  K
)
4 ldilset.i . . . 4  |-  I  =  ( LAut `  K
)
5 ldilset.d . . . 4  |-  D  =  ( ( LDil `  K
) `  W )
61, 2, 3, 4, 5ldilset 33106 . . 3  |-  ( ( K  e.  C  /\  W  e.  H )  ->  D  =  { f  e.  I  |  A. x  e.  B  (
x  .<_  W  ->  (
f `  x )  =  x ) } )
76eleq2d 2472 . 2  |-  ( ( K  e.  C  /\  W  e.  H )  ->  ( F  e.  D  <->  F  e.  { f  e.  I  |  A. x  e.  B  ( x  .<_  W  ->  ( f `  x )  =  x ) } ) )
8 fveq1 5847 . . . . . 6  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
98eqeq1d 2404 . . . . 5  |-  ( f  =  F  ->  (
( f `  x
)  =  x  <->  ( F `  x )  =  x ) )
109imbi2d 314 . . . 4  |-  ( f  =  F  ->  (
( x  .<_  W  -> 
( f `  x
)  =  x )  <-> 
( x  .<_  W  -> 
( F `  x
)  =  x ) ) )
1110ralbidv 2842 . . 3  |-  ( f  =  F  ->  ( A. x  e.  B  ( x  .<_  W  -> 
( f `  x
)  =  x )  <->  A. x  e.  B  ( x  .<_  W  -> 
( F `  x
)  =  x ) ) )
1211elrab 3206 . 2  |-  ( F  e.  { f  e.  I  |  A. x  e.  B  ( x  .<_  W  ->  ( f `  x )  =  x ) }  <->  ( F  e.  I  /\  A. x  e.  B  ( x  .<_  W  ->  ( F `  x )  =  x ) ) )
137, 12syl6bb 261 1  |-  ( ( K  e.  C  /\  W  e.  H )  ->  ( F  e.  D  <->  ( F  e.  I  /\  A. x  e.  B  ( x  .<_  W  ->  ( F `  x )  =  x ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2753   {crab 2757   class class class wbr 4394   ` cfv 5568   Basecbs 14839   lecple 14914   LHypclh 32981   LAutclaut 32982   LDilcldil 33097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ldil 33101
This theorem is referenced by:  ldillaut  33108  ldilval  33110  idldil  33111  ldilcnv  33112  ldilco  33113  cdleme50ldil  33547
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