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Theorem cdlemftr3 31424
Description: Special case of cdlemf 31422 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 2438 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2438 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 cdlemftr.h . . . . 5  |-  H  =  ( LHyp `  K
)
41, 2, 3lhpexle3 30871 . . . 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 2713 . . . 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 190 . . 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 31423 . . . . . . . 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 707 . . . . . . 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 2440 . . . . . . . . 9  |-  ( ( R `  f )  =  u  <->  u  =  ( R `  f ) )
1312anbi1i 678 . . . . . . . 8  |-  ( ( ( R `  f
)  =  u  /\  f  =/=  (  _I  |`  B ) )  <->  ( u  =  ( R `  f
)  /\  f  =/=  (  _I  |`  B ) ) )
1413rexbii 2732 . . . . . . 7  |-  ( E. f  e.  T  ( ( R `  f
)  =  u  /\  f  =/=  (  _I  |`  B ) )  <->  E. f  e.  T  ( u  =  ( R `  f )  /\  f  =/=  (  _I  |`  B ) ) )
1511, 14sylib 190 . . . . . 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 743 . . . . . 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 520 . . . . 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 425 . . . 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 1633 . . 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 2977 . . 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 632 . . . . . 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 1593 . . . . 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 5744 . . . . . 6  |-  ( R `
 f )  e. 
_V
25 neeq1 2611 . . . . . . . 8  |-  ( u  =  ( R `  f )  ->  (
u  =/=  X  <->  ( R `  f )  =/=  X
) )
26 neeq1 2611 . . . . . . . 8  |-  ( u  =  ( R `  f )  ->  (
u  =/=  Y  <->  ( R `  f )  =/=  Y
) )
27 neeq1 2611 . . . . . . . 8  |-  ( u  =  ( R `  f )  ->  (
u  =/=  Z  <->  ( R `  f )  =/=  Z
) )
2825, 26, 273anbi123d 1255 . . . . . . 7  |-  ( u  =  ( R `  f )  ->  (
( u  =/=  X  /\  u  =/=  Y  /\  u  =/=  Z
)  <->  ( ( R `
 f )  =/= 
X  /\  ( R `  f )  =/=  Y  /\  ( R `  f
)  =/=  Z ) ) )
2928anbi2d 686 . . . . . 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 2993 . . . . 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 242 . . . 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 2732 . . 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 2863 . . . 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 1593 . . 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 269 . 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 190 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 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   class class class wbr 4214    _I cid 4495    |` cres 4882   ` cfv 5456   Basecbs 13471   lecple 13538   Atomscatm 30123   HLchlt 30210   LHypclh 30843   LTrncltrn 30960   trLctrl 31017
This theorem is referenced by:  cdlemftr2  31425  cdlemk26-3  31765  cdlemk11t  31805
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-map 7022  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-p1 14471  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-llines 30357  df-lplanes 30358  df-lvols 30359  df-lines 30360  df-psubsp 30362  df-pmap 30363  df-padd 30655  df-lhyp 30847  df-laut 30848  df-ldil 30963  df-ltrn 30964  df-trl 31018
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