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Theorem ltrnideq 36001
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 5874 . . . 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 1051 . . . . 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 35115 . . . . 5  |-  ( P  e.  A  ->  P  e.  B )
7 fvresi 6098 . . . . 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 2498 . . 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 999 . . . . 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 1000 . . . . 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 1001 . . . . 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 36000 . . . . 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 1233 . . . 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 2684 . 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 973    = wceq 1395    e. wcel 1819    =/= wne 2652   class class class wbr 4456    _I cid 4799    |` cres 5010   ` cfv 5594   Basecbs 14643   lecple 14718   Atomscatm 35089   HLchlt 35176   LHypclh 35809   LTrncltrn 35926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7440  df-preset 15683  df-poset 15701  df-plt 15714  df-lub 15730  df-glb 15731  df-join 15732  df-meet 15733  df-p0 15795  df-p1 15796  df-lat 15802  df-clat 15864  df-oposet 35002  df-ol 35004  df-oml 35005  df-covers 35092  df-ats 35093  df-atl 35124  df-cvlat 35148  df-hlat 35177  df-lhyp 35813  df-laut 35814  df-ldil 35929  df-ltrn 35930  df-trl 35985
This theorem is referenced by:  trlid0  36002  trlnidatb  36003  ltrn2ateq  36006  cdlemd8  36031  ltrniotaidvalN  36410  cdlemkid4  36761  dia2dimlem7  36898
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