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Theorem ltrnideq 33816
Description: Property of the identity lattice translation. (Contributed by NM, 27-May-2012.)
Hypotheses
Ref Expression
ltrnnidn.b  |-  B  =  ( Base `  K
)
ltrnnidn.l  |-  .<_  =  ( le `  K )
ltrnnidn.a  |-  A  =  ( Atoms `  K )
ltrnnidn.h  |-  H  =  ( LHyp `  K
)
ltrnnidn.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrnideq  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F  =  (  _I  |`  B )  <-> 
( F `  P
)  =  P ) )

Proof of Theorem ltrnideq
StepHypRef Expression
1 simpr 461 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  (  _I  |`  B ) )  ->  F  =  (  _I  |`  B ) )
21fveq1d 5691 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  (  _I  |`  B ) )  ->  ( F `  P )  =  ( (  _I  |`  B ) `
 P ) )
3 simpl3l 1043 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  (  _I  |`  B ) )  ->  P  e.  A )
4 ltrnnidn.b . . . . . 6  |-  B  =  ( Base `  K
)
5 ltrnnidn.a . . . . . 6  |-  A  =  ( Atoms `  K )
64, 5atbase 32931 . . . . 5  |-  ( P  e.  A  ->  P  e.  B )
7 fvresi 5902 . . . . 5  |-  ( P  e.  B  ->  (
(  _I  |`  B ) `
 P )  =  P )
83, 6, 73syl 20 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  (  _I  |`  B ) )  ->  ( (  _I  |`  B ) `  P )  =  P )
92, 8eqtrd 2473 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  (  _I  |`  B ) )  ->  ( F `  P )  =  P )
109ex 434 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F  =  (  _I  |`  B )  ->  ( F `  P )  =  P ) )
11 simpl1 991 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/=  (  _I  |`  B ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
12 simpl2 992 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/=  (  _I  |`  B ) )  ->  F  e.  T )
13 simpr 461 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/=  (  _I  |`  B ) )  ->  F  =/=  (  _I  |`  B ) )
14 simpl3 993 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/=  (  _I  |`  B ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
15 ltrnnidn.l . . . . . 6  |-  .<_  =  ( le `  K )
16 ltrnnidn.h . . . . . 6  |-  H  =  ( LHyp `  K
)
17 ltrnnidn.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
184, 15, 5, 16, 17ltrnnidn 33815 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  P )  =/=  P
)
1911, 12, 13, 14, 18syl121anc 1223 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/=  (  _I  |`  B ) )  ->  ( F `  P )  =/=  P
)
2019ex 434 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F  =/=  (  _I  |`  B )  ->  ( F `  P )  =/=  P
) )
2120necon4d 2672 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  =  P  ->  F  =  (  _I  |`  B ) ) )
2210, 21impbid 191 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F  =  (  _I  |`  B )  <-> 
( F `  P
)  =  P ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2604   class class class wbr 4290    _I cid 4629    |` cres 4840   ` cfv 5416   Basecbs 14172   lecple 14243   Atomscatm 32905   HLchlt 32992   LHypclh 33625   LTrncltrn 33742
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 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-map 7214  df-poset 15114  df-plt 15126  df-lub 15142  df-glb 15143  df-join 15144  df-meet 15145  df-p0 15207  df-p1 15208  df-lat 15214  df-clat 15276  df-oposet 32818  df-ol 32820  df-oml 32821  df-covers 32908  df-ats 32909  df-atl 32940  df-cvlat 32964  df-hlat 32993  df-lhyp 33629  df-laut 33630  df-ldil 33745  df-ltrn 33746  df-trl 33800
This theorem is referenced by:  trlid0  33817  trlnidatb  33818  ltrn2ateq  33821  cdlemd8  33846  ltrniotaidvalN  34224  cdlemkid4  34575  dia2dimlem7  34712
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