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Theorem cdlemftr3 31362
Description: Special case of cdlemf 31360 showing existence of non-identity translation with trace different from any 3 given lattice elements. (Contributed by NM, 24-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
cdlemftr3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. f  e.  T  ( f  =/=  (  _I  |`  B )  /\  ( ( R `  f )  =/=  X  /\  ( R `  f
)  =/=  Y  /\  ( R `  f )  =/=  Z ) ) )
Distinct variable groups:    f, X    f, Y    f, Z    f, H    f, K    R, f    T, f    f, W
Allowed substitution hint:    B( f)

Proof of Theorem cdlemftr3
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 eqid 2436 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2436 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 cdlemftr.h . . . . 5  |-  H  =  ( LHyp `  K
)
41, 2, 3lhpexle3 30809 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. u  e.  (
Atoms `  K ) ( u ( le `  K ) W  /\  ( u  =/=  X  /\  u  =/=  Y  /\  u  =/=  Z
) ) )
5 df-rex 2711 . . . 4  |-  ( E. u  e.  ( Atoms `  K ) ( u ( le `  K
) W  /\  (
u  =/=  X  /\  u  =/=  Y  /\  u  =/=  Z ) )  <->  E. u
( u  e.  (
Atoms `  K )  /\  ( u ( le
`  K ) W  /\  ( u  =/= 
X  /\  u  =/=  Y  /\  u  =/=  Z
) ) ) )
64, 5sylib 189 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. u ( u  e.  ( Atoms `  K
)  /\  ( u
( le `  K
) W  /\  (
u  =/=  X  /\  u  =/=  Y  /\  u  =/=  Z ) ) ) )
7 cdlemftr.b . . . . . . . . 9  |-  B  =  ( Base `  K
)
8 cdlemftr.t . . . . . . . . 9  |-  T  =  ( ( LTrn `  K
) `  W )
9 cdlemftr.r . . . . . . . . 9  |-  R  =  ( ( trL `  K
) `  W )
107, 1, 2, 3, 8, 9cdlemfnid 31361 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( u  e.  ( Atoms `  K )  /\  u ( le `  K ) W ) )  ->  E. f  e.  T  ( ( R `  f )  =  u  /\  f  =/=  (  _I  |`  B ) ) )
1110adantrrr 706 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( u  e.  ( Atoms `  K )  /\  ( u ( le
`  K ) W  /\  ( u  =/= 
X  /\  u  =/=  Y  /\  u  =/=  Z
) ) ) )  ->  E. f  e.  T  ( ( R `  f )  =  u  /\  f  =/=  (  _I  |`  B ) ) )
12 eqcom 2438 . . . . . . . . 9  |-  ( ( R `  f )  =  u  <->  u  =  ( R `  f ) )
1312anbi1i 677 . . . . . . . 8  |-  ( ( ( R `  f
)  =  u  /\  f  =/=  (  _I  |`  B ) )  <->  ( u  =  ( R `  f
)  /\  f  =/=  (  _I  |`  B ) ) )
1413rexbii 2730 . . . . . . 7  |-  ( E. f  e.  T  ( ( R `  f
)  =  u  /\  f  =/=  (  _I  |`  B ) )  <->  E. f  e.  T  ( u  =  ( R `  f )  /\  f  =/=  (  _I  |`  B ) ) )
1511, 14sylib 189 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( u  e.  ( Atoms `  K )  /\  ( u ( le
`  K ) W  /\  ( u  =/= 
X  /\  u  =/=  Y  /\  u  =/=  Z
) ) ) )  ->  E. f  e.  T  ( u  =  ( R `  f )  /\  f  =/=  (  _I  |`  B ) ) )
16 simprrr 742 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( u  e.  ( Atoms `  K )  /\  ( u ( le
`  K ) W  /\  ( u  =/= 
X  /\  u  =/=  Y  /\  u  =/=  Z
) ) ) )  ->  ( u  =/= 
X  /\  u  =/=  Y  /\  u  =/=  Z
) )
1715, 16jca 519 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( u  e.  ( Atoms `  K )  /\  ( u ( le
`  K ) W  /\  ( u  =/= 
X  /\  u  =/=  Y  /\  u  =/=  Z
) ) ) )  ->  ( E. f  e.  T  ( u  =  ( R `  f )  /\  f  =/=  (  _I  |`  B ) )  /\  ( u  =/=  X  /\  u  =/=  Y  /\  u  =/= 
Z ) ) )
1817ex 424 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( u  e.  ( Atoms `  K )  /\  ( u ( le
`  K ) W  /\  ( u  =/= 
X  /\  u  =/=  Y  /\  u  =/=  Z
) ) )  -> 
( E. f  e.  T  ( u  =  ( R `  f
)  /\  f  =/=  (  _I  |`  B ) )  /\  ( u  =/=  X  /\  u  =/=  Y  /\  u  =/= 
Z ) ) ) )
1918eximdv 1632 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( E. u ( u  e.  ( Atoms `  K )  /\  (
u ( le `  K ) W  /\  ( u  =/=  X  /\  u  =/=  Y  /\  u  =/=  Z
) ) )  ->  E. u ( E. f  e.  T  ( u  =  ( R `  f )  /\  f  =/=  (  _I  |`  B ) )  /\  ( u  =/=  X  /\  u  =/=  Y  /\  u  =/= 
Z ) ) ) )
206, 19mpd 15 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. u ( E. f  e.  T  ( u  =  ( R `
 f )  /\  f  =/=  (  _I  |`  B ) )  /\  ( u  =/=  X  /\  u  =/=  Y  /\  u  =/= 
Z ) ) )
21 rexcom4 2975 . . 3  |-  ( E. f  e.  T  E. u ( ( u  =  ( R `  f )  /\  f  =/=  (  _I  |`  B ) )  /\  ( u  =/=  X  /\  u  =/=  Y  /\  u  =/= 
Z ) )  <->  E. u E. f  e.  T  ( ( u  =  ( R `  f
)  /\  f  =/=  (  _I  |`  B ) )  /\  ( u  =/=  X  /\  u  =/=  Y  /\  u  =/= 
Z ) ) )
22 anass 631 . . . . . 6  |-  ( ( ( u  =  ( R `  f )  /\  f  =/=  (  _I  |`  B ) )  /\  ( u  =/= 
X  /\  u  =/=  Y  /\  u  =/=  Z
) )  <->  ( u  =  ( R `  f )  /\  (
f  =/=  (  _I  |`  B )  /\  (
u  =/=  X  /\  u  =/=  Y  /\  u  =/=  Z ) ) ) )
2322exbii 1592 . . . . 5  |-  ( E. u ( ( u  =  ( R `  f )  /\  f  =/=  (  _I  |`  B ) )  /\  ( u  =/=  X  /\  u  =/=  Y  /\  u  =/= 
Z ) )  <->  E. u
( u  =  ( R `  f )  /\  ( f  =/=  (  _I  |`  B )  /\  ( u  =/= 
X  /\  u  =/=  Y  /\  u  =/=  Z
) ) ) )
24 fvex 5742 . . . . . 6  |-  ( R `
 f )  e. 
_V
25 neeq1 2609 . . . . . . . 8  |-  ( u  =  ( R `  f )  ->  (
u  =/=  X  <->  ( R `  f )  =/=  X
) )
26 neeq1 2609 . . . . . . . 8  |-  ( u  =  ( R `  f )  ->  (
u  =/=  Y  <->  ( R `  f )  =/=  Y
) )
27 neeq1 2609 . . . . . . . 8  |-  ( u  =  ( R `  f )  ->  (
u  =/=  Z  <->  ( R `  f )  =/=  Z
) )
2825, 26, 273anbi123d 1254 . . . . . . 7  |-  ( u  =  ( R `  f )  ->  (
( u  =/=  X  /\  u  =/=  Y  /\  u  =/=  Z
)  <->  ( ( R `
 f )  =/= 
X  /\  ( R `  f )  =/=  Y  /\  ( R `  f
)  =/=  Z ) ) )
2928anbi2d 685 . . . . . 6  |-  ( u  =  ( R `  f )  ->  (
( f  =/=  (  _I  |`  B )  /\  ( u  =/=  X  /\  u  =/=  Y  /\  u  =/=  Z
) )  <->  ( f  =/=  (  _I  |`  B )  /\  ( ( R `
 f )  =/= 
X  /\  ( R `  f )  =/=  Y  /\  ( R `  f
)  =/=  Z ) ) ) )
3024, 29ceqsexv 2991 . . . . 5  |-  ( E. u ( u  =  ( R `  f
)  /\  ( f  =/=  (  _I  |`  B )  /\  ( u  =/= 
X  /\  u  =/=  Y  /\  u  =/=  Z
) ) )  <->  ( f  =/=  (  _I  |`  B )  /\  ( ( R `
 f )  =/= 
X  /\  ( R `  f )  =/=  Y  /\  ( R `  f
)  =/=  Z ) ) )
3123, 30bitri 241 . . . 4  |-  ( E. u ( ( u  =  ( R `  f )  /\  f  =/=  (  _I  |`  B ) )  /\  ( u  =/=  X  /\  u  =/=  Y  /\  u  =/= 
Z ) )  <->  ( f  =/=  (  _I  |`  B )  /\  ( ( R `
 f )  =/= 
X  /\  ( R `  f )  =/=  Y  /\  ( R `  f
)  =/=  Z ) ) )
3231rexbii 2730 . . 3  |-  ( E. f  e.  T  E. u ( ( u  =  ( R `  f )  /\  f  =/=  (  _I  |`  B ) )  /\  ( u  =/=  X  /\  u  =/=  Y  /\  u  =/= 
Z ) )  <->  E. f  e.  T  ( f  =/=  (  _I  |`  B )  /\  ( ( R `
 f )  =/= 
X  /\  ( R `  f )  =/=  Y  /\  ( R `  f
)  =/=  Z ) ) )
33 r19.41v 2861 . . . 4  |-  ( E. f  e.  T  ( ( u  =  ( R `  f )  /\  f  =/=  (  _I  |`  B ) )  /\  ( u  =/= 
X  /\  u  =/=  Y  /\  u  =/=  Z
) )  <->  ( E. f  e.  T  (
u  =  ( R `
 f )  /\  f  =/=  (  _I  |`  B ) )  /\  ( u  =/=  X  /\  u  =/=  Y  /\  u  =/= 
Z ) ) )
3433exbii 1592 . . 3  |-  ( E. u E. f  e.  T  ( ( u  =  ( R `  f )  /\  f  =/=  (  _I  |`  B ) )  /\  ( u  =/=  X  /\  u  =/=  Y  /\  u  =/= 
Z ) )  <->  E. u
( E. f  e.  T  ( u  =  ( R `  f
)  /\  f  =/=  (  _I  |`  B ) )  /\  ( u  =/=  X  /\  u  =/=  Y  /\  u  =/= 
Z ) ) )
3521, 32, 343bitr3ri 268 . 2  |-  ( E. u ( E. f  e.  T  ( u  =  ( R `  f )  /\  f  =/=  (  _I  |`  B ) )  /\  ( u  =/=  X  /\  u  =/=  Y  /\  u  =/= 
Z ) )  <->  E. f  e.  T  ( f  =/=  (  _I  |`  B )  /\  ( ( R `
 f )  =/= 
X  /\  ( R `  f )  =/=  Y  /\  ( R `  f
)  =/=  Z ) ) )
3620, 35sylib 189 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. f  e.  T  ( f  =/=  (  _I  |`  B )  /\  ( ( R `  f )  =/=  X  /\  ( R `  f
)  =/=  Y  /\  ( R `  f )  =/=  Z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   class class class wbr 4212    _I cid 4493    |` cres 4880   ` cfv 5454   Basecbs 13469   lecple 13536   Atomscatm 30061   HLchlt 30148   LHypclh 30781   LTrncltrn 30898   trLctrl 30955
This theorem is referenced by:  cdlemftr2  31363  cdlemk26-3  31703  cdlemk11t  31743
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-map 7020  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-llines 30295  df-lplanes 30296  df-lvols 30297  df-lines 30298  df-psubsp 30300  df-pmap 30301  df-padd 30593  df-lhyp 30785  df-laut 30786  df-ldil 30901  df-ltrn 30902  df-trl 30956
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