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Theorem ltrnle 33414
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 1029 . 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 2429 . . . 4  |-  ( LAut `  K )  =  (
LAut `  K )
4 ltrnle.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
52, 3, 4ltrnlaut 33408 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T )  ->  F  e.  ( LAut `  K
) )
653adant3 1025 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  F  e.  ( LAut `  K ) )
7 simp3l 1033 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  X  e.  B )
8 simp3r 1034 . 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 33369 . 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 1265 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 setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   class class class wbr 4426   ` cfv 5601   Basecbs 15084   lecple 15160   LHypclh 33269   LAutclaut 33270   LTrncltrn 33386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-map 7482  df-laut 33274  df-ldil 33389  df-ltrn 33390
This theorem is referenced by:  ltrnel  33424  ltrncnvel  33427  cdlemc2  33478  cdlemg17h  33955
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