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Theorem cdlemftr2 31300
Description: Special case of cdlemf 31297 showing existence of non-identity translation with trace different from any 2 given lattice elements. (Contributed by NM, 25-Jul-2013.)
Hypotheses
Ref Expression
cdlemftr.b  |-  B  =  ( Base `  K
)
cdlemftr.h  |-  H  =  ( LHyp `  K
)
cdlemftr.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemftr.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
cdlemftr2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. f  e.  T  ( f  =/=  (  _I  |`  B )  /\  ( R `  f )  =/=  X  /\  ( R `  f )  =/=  Y ) )
Distinct variable groups:    f, X    f, Y    f, H    f, K    R, f    T, f   
f, W
Allowed substitution hint:    B( f)

Proof of Theorem cdlemftr2
StepHypRef Expression
1 cdlemftr.b . . 3  |-  B  =  ( Base `  K
)
2 cdlemftr.h . . 3  |-  H  =  ( LHyp `  K
)
3 cdlemftr.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
4 cdlemftr.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
51, 2, 3, 4cdlemftr3 31299 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. f  e.  T  ( f  =/=  (  _I  |`  B )  /\  ( ( R `  f )  =/=  X  /\  ( R `  f
)  =/=  Y  /\  ( R `  f )  =/=  Y ) ) )
6 simpl 444 . . . 4  |-  ( ( f  =/=  (  _I  |`  B )  /\  (
( R `  f
)  =/=  X  /\  ( R `  f )  =/=  Y  /\  ( R `  f )  =/=  Y ) )  -> 
f  =/=  (  _I  |`  B ) )
7 simpr1 963 . . . 4  |-  ( ( f  =/=  (  _I  |`  B )  /\  (
( R `  f
)  =/=  X  /\  ( R `  f )  =/=  Y  /\  ( R `  f )  =/=  Y ) )  -> 
( R `  f
)  =/=  X )
8 simpr2 964 . . . 4  |-  ( ( f  =/=  (  _I  |`  B )  /\  (
( R `  f
)  =/=  X  /\  ( R `  f )  =/=  Y  /\  ( R `  f )  =/=  Y ) )  -> 
( R `  f
)  =/=  Y )
96, 7, 83jca 1134 . . 3  |-  ( ( f  =/=  (  _I  |`  B )  /\  (
( R `  f
)  =/=  X  /\  ( R `  f )  =/=  Y  /\  ( R `  f )  =/=  Y ) )  -> 
( f  =/=  (  _I  |`  B )  /\  ( R `  f )  =/=  X  /\  ( R `  f )  =/=  Y ) )
109reximi 2805 . 2  |-  ( E. f  e.  T  ( f  =/=  (  _I  |`  B )  /\  (
( R `  f
)  =/=  X  /\  ( R `  f )  =/=  Y  /\  ( R `  f )  =/=  Y ) )  ->  E. f  e.  T  ( f  =/=  (  _I  |`  B )  /\  ( R `  f )  =/=  X  /\  ( R `  f )  =/=  Y ) )
115, 10syl 16 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. f  e.  T  ( f  =/=  (  _I  |`  B )  /\  ( R `  f )  =/=  X  /\  ( R `  f )  =/=  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698    _I cid 4485    |` cres 4872   ` cfv 5446   Basecbs 13461   HLchlt 30085   LHypclh 30718   LTrncltrn 30835   trLctrl 30892
This theorem is referenced by:  cdlemftr1  31301  cdlemk26b-3  31639  cdlemk29-3  31645  cdlemk38  31649  cdlemkid5  31669  cdlemkid  31670  cdlemk55b  31694
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-map 7012  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233  df-lvols 30234  df-lines 30235  df-psubsp 30237  df-pmap 30238  df-padd 30530  df-lhyp 30722  df-laut 30723  df-ldil 30838  df-ltrn 30839  df-trl 30893
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