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Theorem ldilval 30910
Description: Value of a lattice dilation under its co-atom. (Contributed by NM, 20-May-2012.)
Hypotheses
Ref Expression
ldilval.b  |-  B  =  ( Base `  K
)
ldilval.l  |-  .<_  =  ( le `  K )
ldilval.h  |-  H  =  ( LHyp `  K
)
ldilval.d  |-  D  =  ( ( LDil `  K
) `  W )
Assertion
Ref Expression
ldilval  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  D  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( F `  X )  =  X )

Proof of Theorem ldilval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ldilval.b . . . . 5  |-  B  =  ( Base `  K
)
2 ldilval.l . . . . 5  |-  .<_  =  ( le `  K )
3 ldilval.h . . . . 5  |-  H  =  ( LHyp `  K
)
4 eqid 2436 . . . . 5  |-  ( LAut `  K )  =  (
LAut `  K )
5 ldilval.d . . . . 5  |-  D  =  ( ( LDil `  K
) `  W )
61, 2, 3, 4, 5isldil 30907 . . . 4  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( F  e.  D  <->  ( F  e.  ( LAut `  K )  /\  A. x  e.  B  (
x  .<_  W  ->  ( F `  x )  =  x ) ) ) )
7 simpr 448 . . . 4  |-  ( ( F  e.  ( LAut `  K )  /\  A. x  e.  B  (
x  .<_  W  ->  ( F `  x )  =  x ) )  ->  A. x  e.  B  ( x  .<_  W  -> 
( F `  x
)  =  x ) )
86, 7syl6bi 220 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( F  e.  D  ->  A. x  e.  B  ( x  .<_  W  -> 
( F `  x
)  =  x ) ) )
9 breq1 4215 . . . . . 6  |-  ( x  =  X  ->  (
x  .<_  W  <->  X  .<_  W ) )
10 fveq2 5728 . . . . . . 7  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
11 id 20 . . . . . . 7  |-  ( x  =  X  ->  x  =  X )
1210, 11eqeq12d 2450 . . . . . 6  |-  ( x  =  X  ->  (
( F `  x
)  =  x  <->  ( F `  X )  =  X ) )
139, 12imbi12d 312 . . . . 5  |-  ( x  =  X  ->  (
( x  .<_  W  -> 
( F `  x
)  =  x )  <-> 
( X  .<_  W  -> 
( F `  X
)  =  X ) ) )
1413rspccv 3049 . . . 4  |-  ( A. x  e.  B  (
x  .<_  W  ->  ( F `  x )  =  x )  ->  ( X  e.  B  ->  ( X  .<_  W  ->  ( F `  X )  =  X ) ) )
1514imp3a 421 . . 3  |-  ( A. x  e.  B  (
x  .<_  W  ->  ( F `  x )  =  x )  ->  (
( X  e.  B  /\  X  .<_  W )  ->  ( F `  X )  =  X ) )
168, 15syl6 31 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( F  e.  D  ->  ( ( X  e.  B  /\  X  .<_  W )  ->  ( F `  X )  =  X ) ) )
17163imp 1147 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  D  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( F `  X )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   class class class wbr 4212   ` cfv 5454   Basecbs 13469   lecple 13536   LHypclh 30781   LAutclaut 30782   LDilcldil 30897
This theorem is referenced by:  ldilcnv  30912  ldilco  30913  ltrnval1  30931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ldil 30901
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