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Theorem ldilval 33130
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 2402 . . . . 5  |-  ( LAut `  K )  =  (
LAut `  K )
5 ldilval.d . . . . 5  |-  D  =  ( ( LDil `  K
) `  W )
61, 2, 3, 4, 5isldil 33127 . . . 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 459 . . . 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 228 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( F  e.  D  ->  A. x  e.  B  ( x  .<_  W  -> 
( F `  x
)  =  x ) ) )
9 breq1 4398 . . . . . 6  |-  ( x  =  X  ->  (
x  .<_  W  <->  X  .<_  W ) )
10 fveq2 5849 . . . . . . 7  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
11 id 22 . . . . . . 7  |-  ( x  =  X  ->  x  =  X )
1210, 11eqeq12d 2424 . . . . . 6  |-  ( x  =  X  ->  (
( F `  x
)  =  x  <->  ( F `  X )  =  X ) )
139, 12imbi12d 318 . . . . 5  |-  ( x  =  X  ->  (
( x  .<_  W  -> 
( F `  x
)  =  x )  <-> 
( X  .<_  W  -> 
( F `  X
)  =  X ) ) )
1413rspccv 3157 . . . 4  |-  ( A. x  e.  B  (
x  .<_  W  ->  ( F `  x )  =  x )  ->  ( X  e.  B  ->  ( X  .<_  W  ->  ( F `  X )  =  X ) ) )
1514impd 429 . . 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 1191 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  D  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( F `  X )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2754   class class class wbr 4395   ` cfv 5569   Basecbs 14841   lecple 14916   LHypclh 33001   LAutclaut 33002   LDilcldil 33117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ldil 33121
This theorem is referenced by:  ldilcnv  33132  ldilco  33133  ltrnval1  33151
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