Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ldillaut Unicode version

Theorem ldillaut 30276
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 2380 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2380 . . 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 30275 . 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 607 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  D )  ->  F  e.  I )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2642   class class class wbr 4146   ` cfv 5387   Basecbs 13389   lecple 13456   LHypclh 30149   LAutclaut 30150   LDilcldil 30265
This theorem is referenced by:  ldil1o  30277  ldilcnv  30280  ldilco  30281  ltrnlaut  30288
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ldil 30269
  Copyright terms: Public domain W3C validator