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Theorem ltrnle 33771
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 1012 . 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 2442 . . . 4  |-  ( LAut `  K )  =  (
LAut `  K )
4 ltrnle.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
52, 3, 4ltrnlaut 33765 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T )  ->  F  e.  ( LAut `  K
) )
653adant3 1008 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  F  e.  ( LAut `  K ) )
7 simp3l 1016 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  X  e.  B )
8 simp3r 1017 . 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 33726 . 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 1219 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 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4291   ` cfv 5417   Basecbs 14173   lecple 14244   LHypclh 33626   LAutclaut 33627   LTrncltrn 33743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-map 7215  df-laut 33631  df-ldil 33746  df-ltrn 33747
This theorem is referenced by:  ltrnel  33781  ltrncnvel  33784  cdlemc2  33834  cdlemg17h  34310
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