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Theorem ltrn2ateq 33717
Description: Property of the equality of a lattice translation with its value. (Contributed by NM, 27-May-2012.)
Hypotheses
Ref Expression
ltrn2eq.l  |-  .<_  =  ( le `  K )
ltrn2eq.a  |-  A  =  ( Atoms `  K )
ltrn2eq.h  |-  H  =  ( LHyp `  K
)
ltrn2eq.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrn2ateq  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  (
( F `  P
)  =  P  <->  ( F `  Q )  =  Q ) )

Proof of Theorem ltrn2ateq
StepHypRef Expression
1 eqid 2438 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 ltrn2eq.l . . . 4  |-  .<_  =  ( le `  K )
3 ltrn2eq.a . . . 4  |-  A  =  ( Atoms `  K )
4 ltrn2eq.h . . . 4  |-  H  =  ( LHyp `  K
)
5 ltrn2eq.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
61, 2, 3, 4, 5ltrnideq 33712 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F  =  (  _I  |`  ( Base `  K ) )  <-> 
( F `  P
)  =  P ) )
763adant3r3 1198 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( F  =  (  _I  |`  ( Base `  K
) )  <->  ( F `  P )  =  P ) )
81, 2, 3, 4, 5ltrnideq 33712 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( F  =  (  _I  |`  ( Base `  K ) )  <-> 
( F `  Q
)  =  Q ) )
983adant3r2 1197 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( F  =  (  _I  |`  ( Base `  K
) )  <->  ( F `  Q )  =  Q ) )
107, 9bitr3d 255 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  (
( F `  P
)  =  P  <->  ( F `  Q )  =  Q ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4287    _I cid 4626    |` cres 4837   ` cfv 5413   Basecbs 14166   lecple 14237   Atomscatm 32801   HLchlt 32888   LHypclh 33521   LTrncltrn 33638
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-map 7208  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p0 15201  df-p1 15202  df-lat 15208  df-clat 15270  df-oposet 32714  df-ol 32716  df-oml 32717  df-covers 32804  df-ats 32805  df-atl 32836  df-cvlat 32860  df-hlat 32889  df-lhyp 33525  df-laut 33526  df-ldil 33641  df-ltrn 33642  df-trl 33696
This theorem is referenced by:  ltrnateq  33718  ltrnatneq  33719  trlval3  33724
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