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Theorem isldil 28988
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 28987 . . 3  |-  ( ( K  e.  C  /\  W  e.  H )  ->  D  =  { f  e.  I  |  A. x  e.  B  (
x  .<_  W  ->  (
f `  x )  =  x ) } )
76eleq2d 2320 . 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 5376 . . . . . 6  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
98eqeq1d 2261 . . . . 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 2527 . . 3  |-  ( f  =  F  ->  ( A. x  e.  B  ( x  .<_  W  -> 
( f `  x
)  =  x )  <->  A. x  e.  B  ( x  .<_  W  -> 
( F `  x
)  =  x ) ) )
1211elrab 2860 . 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 2509   {crab 2512   class class class wbr 3920   ` cfv 4592   Basecbs 13022   lecple 13089   LHypclh 28862   LAutclaut 28863   LDilcldil 28978
This theorem is referenced by:  ldillaut  28989  ldilval  28991  idldil  28992  ldilcnv  28993  ldilco  28994  cdleme50ldil  29426
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ldil 28982
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