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Theorem isldil 29466
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
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 29465 . . 3  |-  ( ( K  e.  C  /\  W  e.  H )  ->  D  =  { f  e.  I  |  A. x  e.  B  (
x  .<_  W  ->  (
f `  x )  =  x ) } )
76eleq2d 2325 . 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 5457 . . . . . 6  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
98eqeq1d 2266 . . . . 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 2538 . . 3  |-  ( f  =  F  ->  ( A. x  e.  B  ( x  .<_  W  -> 
( f `  x
)  =  x )  <->  A. x  e.  B  ( x  .<_  W  -> 
( F `  x
)  =  x ) ) )
1211elrab 2898 . 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 1619    e. wcel 1621   A.wral 2518   {crab 2522   class class class wbr 3997   ` cfv 4673   Basecbs 13110   lecple 13177   LHypclh 29340   LAutclaut 29341   LDilcldil 29456
This theorem is referenced by:  ldillaut  29467  ldilval  29469  idldil  29470  ldilcnv  29471  ldilco  29472  cdleme50ldil  29904
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ldil 29460
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