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Theorem ltrnldil 30363
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 2358 . . 3  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2358 . . 3  |-  ( join `  K )  =  (
join `  K )
3 eqid 2358 . . 3  |-  ( meet `  K )  =  (
meet `  K )
4 eqid 2358 . . 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 30360 . 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 1642    e. wcel 1710   A.wral 2619   class class class wbr 4102   ` cfv 5334  (class class class)co 5942   lecple 13306   joincjn 14171   meetcmee 14172   Atomscatm 29505   LHypclh 30225   LDilcldil 30341   LTrncltrn 30342
This theorem is referenced by:  ltrnlaut  30364  ltrnval1  30375  ltrncnv  30387  ltrnco  30960
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-ltrn 30346
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