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Theorem cdlemftr2 30681
Description: Special case of cdlemf 30678 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 30680 . 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 2757 . 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 1649    e. wcel 1717    =/= wne 2551   E.wrex 2651    _I cid 4435    |` cres 4821   ` cfv 5395   Basecbs 13397   HLchlt 29466   LHypclh 30099   LTrncltrn 30216   trLctrl 30273
This theorem is referenced by:  cdlemftr1  30682  cdlemk26b-3  31020  cdlemk29-3  31026  cdlemk38  31030  cdlemkid5  31050  cdlemkid  31051  cdlemk55b  31075
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 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-undef 6480  df-riota 6486  df-map 6957  df-poset 14331  df-plt 14343  df-lub 14359  df-glb 14360  df-join 14361  df-meet 14362  df-p0 14396  df-p1 14397  df-lat 14403  df-clat 14465  df-oposet 29292  df-ol 29294  df-oml 29295  df-covers 29382  df-ats 29383  df-atl 29414  df-cvlat 29438  df-hlat 29467  df-llines 29613  df-lplanes 29614  df-lvols 29615  df-lines 29616  df-psubsp 29618  df-pmap 29619  df-padd 29911  df-lhyp 30103  df-laut 30104  df-ldil 30219  df-ltrn 30220  df-trl 30274
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