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Theorem ldilval 30120
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 2316 . . . . 5  |-  ( LAut `  K )  =  (
LAut `  K )
5 ldilval.d . . . . 5  |-  D  =  ( ( LDil `  K
) `  W )
61, 2, 3, 4, 5isldil 30117 . . . 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 447 . . . 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 219 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( F  e.  D  ->  A. x  e.  B  ( x  .<_  W  -> 
( F `  x
)  =  x ) ) )
9 breq1 4063 . . . . . 6  |-  ( x  =  X  ->  (
x  .<_  W  <->  X  .<_  W ) )
10 fveq2 5563 . . . . . . 7  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
11 id 19 . . . . . . 7  |-  ( x  =  X  ->  x  =  X )
1210, 11eqeq12d 2330 . . . . . 6  |-  ( x  =  X  ->  (
( F `  x
)  =  x  <->  ( F `  X )  =  X ) )
139, 12imbi12d 311 . . . . 5  |-  ( x  =  X  ->  (
( x  .<_  W  -> 
( F `  x
)  =  x )  <-> 
( X  .<_  W  -> 
( F `  X
)  =  X ) ) )
1413rspccv 2915 . . . 4  |-  ( A. x  e.  B  (
x  .<_  W  ->  ( F `  x )  =  x )  ->  ( X  e.  B  ->  ( X  .<_  W  ->  ( F `  X )  =  X ) ) )
1514imp3a 420 . . 3  |-  ( A. x  e.  B  (
x  .<_  W  ->  ( F `  x )  =  x )  ->  (
( X  e.  B  /\  X  .<_  W )  ->  ( F `  X )  =  X ) )
168, 15syl6 29 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( F  e.  D  ->  ( ( X  e.  B  /\  X  .<_  W )  ->  ( F `  X )  =  X ) ) )
17163imp 1145 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 358    /\ w3a 934    = wceq 1633    e. wcel 1701   A.wral 2577   class class class wbr 4060   ` cfv 5292   Basecbs 13195   lecple 13262   LHypclh 29991   LAutclaut 29992   LDilcldil 30107
This theorem is referenced by:  ldilcnv  30122  ldilco  30123  ltrnval1  30141
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ldil 30111
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