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Theorem ltrn11 35323
Description: One-to-one property of a lattice translation. (Contributed by NM, 20-May-2012.)
Hypotheses
Ref Expression
ltrn1o.b  |-  B  =  ( Base `  K
)
ltrn1o.h  |-  H  =  ( LHyp `  K
)
ltrn1o.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrn11  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( ( F `  X )  =  ( F `  Y )  <-> 
X  =  Y ) )

Proof of Theorem ltrn11
StepHypRef Expression
1 simp1l 1020 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  K  e.  V )
2 ltrn1o.h . . . 4  |-  H  =  ( LHyp `  K
)
3 eqid 2467 . . . 4  |-  ( LAut `  K )  =  (
LAut `  K )
4 ltrn1o.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
52, 3, 4ltrnlaut 35320 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T )  ->  F  e.  ( LAut `  K
) )
653adant3 1016 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  F  e.  ( LAut `  K ) )
7 simp3l 1024 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  X  e.  B )
8 simp3r 1025 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  Y  e.  B )
9 ltrn1o.b . . 3  |-  B  =  ( Base `  K
)
109, 3laut11 35283 . 2  |-  ( ( ( K  e.  V  /\  F  e.  ( LAut `  K ) )  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( ( F `  X )  =  ( F `  Y )  <-> 
X  =  Y ) )
111, 6, 7, 8, 10syl22anc 1229 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( ( F `  X )  =  ( F `  Y )  <-> 
X  =  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   ` cfv 5594   Basecbs 14507   LHypclh 35181   LAutclaut 35182   LTrncltrn 35298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-map 7434  df-laut 35186  df-ldil 35301  df-ltrn 35302
This theorem is referenced by:  ltrn11at  35344
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