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Theorem ltrnle 30926
Description: Less-than or equal property of a lattice translation. (Contributed by NM, 20-May-2012.)
Hypotheses
Ref Expression
ltrnle.b  |-  B  =  ( Base `  K
)
ltrnle.l  |-  .<_  =  ( le `  K )
ltrnle.h  |-  H  =  ( LHyp `  K
)
ltrnle.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrnle  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X  .<_  Y  <->  ( F `  X )  .<_  ( F `
 Y ) ) )

Proof of Theorem ltrnle
StepHypRef Expression
1 simp1l 981 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  K  e.  V )
2 ltrnle.h . . . 4  |-  H  =  ( LHyp `  K
)
3 eqid 2436 . . . 4  |-  ( LAut `  K )  =  (
LAut `  K )
4 ltrnle.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
52, 3, 4ltrnlaut 30920 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T )  ->  F  e.  ( LAut `  K
) )
653adant3 977 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  F  e.  ( LAut `  K ) )
7 simp3l 985 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  X  e.  B )
8 simp3r 986 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  Y  e.  B )
9 ltrnle.b . . 3  |-  B  =  ( Base `  K
)
10 ltrnle.l . . 3  |-  .<_  =  ( le `  K )
119, 10, 3lautle 30881 . 2  |-  ( ( ( K  e.  V  /\  F  e.  ( LAut `  K ) )  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X  .<_  Y  <->  ( F `  X )  .<_  ( F `
 Y ) ) )
121, 6, 7, 8, 11syl22anc 1185 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X  .<_  Y  <->  ( F `  X )  .<_  ( F `
 Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454   Basecbs 13469   lecple 13536   LHypclh 30781   LAutclaut 30782   LTrncltrn 30898
This theorem is referenced by:  ltrnel  30936  ltrncnvel  30939  cdlemc2  30989  cdlemg17h  31465
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-laut 30786  df-ldil 30901  df-ltrn 30902
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