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Theorem ldillaut 34782
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 2460 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2460 . . 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 34781 . 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 623 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  D )  ->  F  e.  I )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = 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:  ldil1o  34783  ldilcnv  34786  ldilco  34787  ltrnlaut  34794
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