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Theorem ldilval 30302
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 2283 . . . . 5  |-  ( LAut `  K )  =  (
LAut `  K )
5 ldilval.d . . . . 5  |-  D  =  ( ( LDil `  K
) `  W )
61, 2, 3, 4, 5isldil 30299 . . . 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 4026 . . . . . 6  |-  ( x  =  X  ->  (
x  .<_  W  <->  X  .<_  W ) )
10 fveq2 5525 . . . . . . 7  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
11 id 19 . . . . . . 7  |-  ( x  =  X  ->  x  =  X )
1210, 11eqeq12d 2297 . . . . . 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 2881 . . . 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 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   LHypclh 30173   LAutclaut 30174   LDilcldil 30289
This theorem is referenced by:  ldilcnv  30304  ldilco  30305  ltrnval1  30323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ldil 30293
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