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Theorem ltrnnidn 29630
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 28817 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
31, 2syl 17 . . 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 2284 . . . . 5  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
127, 8, 9, 10, 11trlnidat 29629 . . . 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 2284 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
1514, 8atn0 28765 . . 3  |-  ( ( K  e.  AtLat  /\  (
( ( trL `  K
) `  W ) `  F )  e.  A
)  ->  ( (
( trL `  K
) `  W ) `  F )  =/=  ( 0. `  K ) )
163, 13, 15syl2anc 644 . 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 449 . . . . 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 29626 . . . . 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 425 . . 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 2485 . 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 16 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 5    -> wi 6    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685    =/= wne 2447   class class class wbr 4024    _I cid 4303    |` cres 4690   ` cfv 5221   Basecbs 13142   lecple 13209   0.cp0 14137   Atomscatm 28720   AtLatcal 28721   HLchlt 28807   LHypclh 29440   LTrncltrn 29557   trLctrl 29614
This theorem is referenced by:  ltrnideq  29631
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-map 6769  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-p1 14140  df-lat 14146  df-clat 14208  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-lhyp 29444  df-laut 29445  df-ldil 29560  df-ltrn 29561  df-trl 29615
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