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Theorem ltrnldil 30311
Description: A lattice translation is a lattice dilation. (Contributed by NM, 20-May-2012.)
Hypotheses
Ref Expression
ltrnldil.h  |-  H  =  ( LHyp `  K
)
ltrnldil.d  |-  D  =  ( ( LDil `  K
) `  W )
ltrnldil.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrnldil  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T )  ->  F  e.  D )

Proof of Theorem ltrnldil
Dummy variables  q  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . 3  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2283 . . 3  |-  ( join `  K )  =  (
join `  K )
3 eqid 2283 . . 3  |-  ( meet `  K )  =  (
meet `  K )
4 eqid 2283 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
5 ltrnldil.h . . 3  |-  H  =  ( LHyp `  K
)
6 ltrnldil.d . . 3  |-  D  =  ( ( LDil `  K
) `  W )
7 ltrnldil.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
81, 2, 3, 4, 5, 6, 7isltrn 30308 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( F  e.  T  <->  ( F  e.  D  /\  A. p  e.  ( Atoms `  K ) A. q  e.  ( Atoms `  K )
( ( -.  p
( le `  K
) W  /\  -.  q ( le `  K ) W )  ->  ( ( p ( join `  K
) ( F `  p ) ) (
meet `  K ) W )  =  ( ( q ( join `  K ) ( F `
 q ) ) ( meet `  K
) W ) ) ) ) )
98simprbda 606 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T )  ->  F  e.  D )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   meetcmee 14079   Atomscatm 29453   LHypclh 30173   LDilcldil 30289   LTrncltrn 30290
This theorem is referenced by:  ltrnlaut  30312  ltrnval1  30323  ltrncnv  30335  ltrnco  30908
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-ltrn 30294
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