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Theorem ldillaut 33108
Description: A lattice dilation is an automorphism. (Contributed by NM, 20-May-2012.)
Hypotheses
Ref Expression
ldillaut.h  |-  H  =  ( LHyp `  K
)
ldillaut.i  |-  I  =  ( LAut `  K
)
ldillaut.d  |-  D  =  ( ( LDil `  K
) `  W )
Assertion
Ref Expression
ldillaut  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  D )  ->  F  e.  I )

Proof of Theorem ldillaut
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2402 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2402 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 ldillaut.h . . 3  |-  H  =  ( LHyp `  K
)
4 ldillaut.i . . 3  |-  I  =  ( LAut `  K
)
5 ldillaut.d . . 3  |-  D  =  ( ( LDil `  K
) `  W )
61, 2, 3, 4, 5isldil 33107 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( F  e.  D  <->  ( F  e.  I  /\  A. x  e.  ( Base `  K ) ( x ( le `  K
) W  ->  ( F `  x )  =  x ) ) ) )
76simprbda 621 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  D )  ->  F  e.  I )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2753   class class class wbr 4394   ` cfv 5568   Basecbs 14839   lecple 14914   LHypclh 32981   LAutclaut 32982   LDilcldil 33097
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 4506  ax-sep 4516  ax-nul 4524  ax-pr 4629
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 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ldil 33101
This theorem is referenced by:  ldil1o  33109  ldilcnv  33112  ldilco  33113  ltrnlaut  33120
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