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Theorem cdlemftr1 29907
Description: Part of proof of Lemma G of [Crawley] p. 116, sixth line of third paragraph on p. 117: there is "a translation h, different from the identity, such that tr h  =/= tr f." (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
cdlemftr1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. f  e.  T  ( f  =/=  (  _I  |`  B )  /\  ( R `  f )  =/=  X ) )
Distinct variable groups:    f, X    f, H    f, K    R, f    T, f    f, W
Allowed substitution hint:    B( f)

Proof of Theorem cdlemftr1
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, 4cdlemftr2 29906 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. f  e.  T  ( f  =/=  (  _I  |`  B )  /\  ( R `  f )  =/=  X  /\  ( R `  f )  =/=  X ) )
6 3simpa 957 . . 3  |-  ( ( f  =/=  (  _I  |`  B )  /\  ( R `  f )  =/=  X  /\  ( R `
 f )  =/= 
X )  ->  (
f  =/=  (  _I  |`  B )  /\  ( R `  f )  =/=  X ) )
76reximi 2623 . 2  |-  ( E. f  e.  T  ( f  =/=  (  _I  |`  B )  /\  ( R `  f )  =/=  X  /\  ( R `
 f )  =/= 
X )  ->  E. f  e.  T  ( f  =/=  (  _I  |`  B )  /\  ( R `  f )  =/=  X
) )
85, 7syl 17 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. f  e.  T  ( f  =/=  (  _I  |`  B )  /\  ( R `  f )  =/=  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   E.wrex 2517    _I cid 4262    |` cres 4649   ` cfv 4659   Basecbs 13096   HLchlt 28691   LHypclh 29324   LTrncltrn 29441   trLctrl 29498
This theorem is referenced by:  cdlemftr0  29908  cdlemg48  30077  cdlemk19x  30283
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-undef 6250  df-riota 6258  df-map 6728  df-poset 14028  df-plt 14040  df-lub 14056  df-glb 14057  df-join 14058  df-meet 14059  df-p0 14093  df-p1 14094  df-lat 14100  df-clat 14162  df-oposet 28517  df-ol 28519  df-oml 28520  df-covers 28607  df-ats 28608  df-atl 28639  df-cvlat 28663  df-hlat 28692  df-llines 28838  df-lplanes 28839  df-lvols 28840  df-lines 28841  df-psubsp 28843  df-pmap 28844  df-padd 29136  df-lhyp 29328  df-laut 29329  df-ldil 29444  df-ltrn 29445  df-trl 29499
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