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Theorem ltrnnidn 30290
Description: If a lattice translation is not the identity, then the translation of any atom not under the fiducial co-atom  W is different from the atom. Remark above Lemma C in [Crawley] p. 112. (Contributed by NM, 24-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
ltrnnidn  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  P )  =/=  P
)

Proof of Theorem ltrnnidn
StepHypRef Expression
1 simp1l 981 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  HL )
2 hlatl 29477 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
31, 2syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  AtLat )
4 simp1 957 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
5 simp2l 983 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  F  e.  T
)
6 simp2r 984 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  F  =/=  (  _I  |`  B ) )
7 ltrnnidn.b . . . . 5  |-  B  =  ( Base `  K
)
8 ltrnnidn.a . . . . 5  |-  A  =  ( Atoms `  K )
9 ltrnnidn.h . . . . 5  |-  H  =  ( LHyp `  K
)
10 ltrnnidn.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
11 eqid 2389 . . . . 5  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
127, 8, 9, 10, 11trlnidat 30289 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  ( ( ( trL `  K ) `
 W ) `  F )  e.  A
)
134, 5, 6, 12syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( ( trL `  K ) `
 W ) `  F )  e.  A
)
14 eqid 2389 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
1514, 8atn0 29425 . . 3  |-  ( ( K  e.  AtLat  /\  (
( ( trL `  K
) `  W ) `  F )  e.  A
)  ->  ( (
( trL `  K
) `  W ) `  F )  =/=  ( 0. `  K ) )
163, 13, 15syl2anc 643 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( ( trL `  K ) `
 W ) `  F )  =/=  ( 0. `  K ) )
17 simpl1 960 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `  P )  =  P )  ->  ( K  e.  HL  /\  W  e.  H ) )
18 simpl3 962 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `  P )  =  P )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
19 simpl2l 1010 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `  P )  =  P )  ->  F  e.  T )
20 simpr 448 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `  P )  =  P )  ->  ( F `  P )  =  P )
21 ltrnnidn.l . . . . . 6  |-  .<_  =  ( le `  K )
2221, 14, 8, 9, 10, 11trl0 30286 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =  P ) )  ->  ( ( ( trL `  K ) `
 W ) `  F )  =  ( 0. `  K ) )
2317, 18, 19, 20, 22syl112anc 1188 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `  P )  =  P )  ->  ( (
( trL `  K
) `  W ) `  F )  =  ( 0. `  K ) )
2423ex 424 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `
 P )  =  P  ->  ( (
( trL `  K
) `  W ) `  F )  =  ( 0. `  K ) ) )
2524necon3d 2590 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( ( ( trL `  K
) `  W ) `  F )  =/=  ( 0. `  K )  -> 
( F `  P
)  =/=  P ) )
2616, 25mpd 15 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  P )  =/=  P
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2552   class class class wbr 4155    _I cid 4436    |` cres 4822   ` cfv 5396   Basecbs 13398   lecple 13465   0.cp0 14395   Atomscatm 29380   AtLatcal 29381   HLchlt 29467   LHypclh 30100   LTrncltrn 30217   trLctrl 30274
This theorem is referenced by:  ltrnideq  30291
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-map 6958  df-poset 14332  df-plt 14344  df-lub 14360  df-glb 14361  df-join 14362  df-meet 14363  df-p0 14397  df-p1 14398  df-lat 14404  df-clat 14466  df-oposet 29293  df-ol 29295  df-oml 29296  df-covers 29383  df-ats 29384  df-atl 29415  df-cvlat 29439  df-hlat 29468  df-lhyp 30104  df-laut 30105  df-ldil 30220  df-ltrn 30221  df-trl 30275
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