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Theorem ldilval 34784
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 2460 . . . . 5  |-  ( LAut `  K )  =  (
LAut `  K )
5 ldilval.d . . . . 5  |-  D  =  ( ( LDil `  K
) `  W )
61, 2, 3, 4, 5isldil 34781 . . . 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 461 . . . 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 4443 . . . . . 6  |-  ( x  =  X  ->  (
x  .<_  W  <->  X  .<_  W ) )
10 fveq2 5857 . . . . . . 7  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
11 id 22 . . . . . . 7  |-  ( x  =  X  ->  x  =  X )
1210, 11eqeq12d 2482 . . . . . 6  |-  ( x  =  X  ->  (
( F `  x
)  =  x  <->  ( F `  X )  =  X ) )
139, 12imbi12d 320 . . . . 5  |-  ( x  =  X  ->  (
( x  .<_  W  -> 
( F `  x
)  =  x )  <-> 
( X  .<_  W  -> 
( F `  X
)  =  X ) ) )
1413rspccv 3204 . . . 4  |-  ( A. x  e.  B  (
x  .<_  W  ->  ( F `  x )  =  x )  ->  ( X  e.  B  ->  ( X  .<_  W  ->  ( F `  X )  =  X ) ) )
1514impd 431 . . 3  |-  ( A. x  e.  B  (
x  .<_  W  ->  ( F `  x )  =  x )  ->  (
( X  e.  B  /\  X  .<_  W )  ->  ( F `  X )  =  X ) )
168, 15syl6 33 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( F  e.  D  ->  ( ( X  e.  B  /\  X  .<_  W )  ->  ( F `  X )  =  X ) ) )
17163imp 1185 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 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2807   class class class wbr 4440   ` cfv 5579   Basecbs 14479   lecple 14551   LHypclh 34655   LAutclaut 34656   LDilcldil 34771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ldil 34775
This theorem is referenced by:  ldilcnv  34786  ldilco  34787  ltrnval1  34805
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