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Theorem isldil 29567
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
Dummy variable  f is distinct from all other variables.
Allowed substitution hints:    C( x)    D( x)    H( x)    I( x)    .<_ ( x)

Proof of Theorem isldil
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 29566 . . 3  |-  ( ( K  e.  C  /\  W  e.  H )  ->  D  =  { f  e.  I  |  A. x  e.  B  (
x  .<_  W  ->  (
f `  x )  =  x ) } )
76eleq2d 2352 . 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 5485 . . . . . 6  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
98eqeq1d 2293 . . . . 5  |-  ( f  =  F  ->  (
( f `  x
)  =  x  <->  ( F `  x )  =  x ) )
109imbi2d 309 . . . 4  |-  ( f  =  F  ->  (
( x  .<_  W  -> 
( f `  x
)  =  x )  <-> 
( x  .<_  W  -> 
( F `  x
)  =  x ) ) )
1110ralbidv 2565 . . 3  |-  ( f  =  F  ->  ( A. x  e.  B  ( x  .<_  W  -> 
( f `  x
)  =  x )  <->  A. x  e.  B  ( x  .<_  W  -> 
( F `  x
)  =  x ) ) )
1211elrab 2925 . 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 254 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 set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1624    e. wcel 1685   A.wral 2545   {crab 2549   class class class wbr 4025   ` cfv 5222   Basecbs 13143   lecple 13210   LHypclh 29441   LAutclaut 29442   LDilcldil 29557
This theorem is referenced by:  ldillaut  29568  ldilval  29570  idldil  29571  ldilcnv  29572  ldilco  29573  cdleme50ldil  30005
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  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-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ldil 29561
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