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Theorem cdlemftr3 30754
Description: Special case of cdlemf 30752 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 2283 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2283 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 cdlemftr.h . . . . 5  |-  H  =  ( LHyp `  K
)
41, 2, 3lhpexle3 30201 . . . 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 2549 . . . 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 188 . . 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 30753 . . . . . . . 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 705 . . . . . . 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 2285 . . . . . . . . 9  |-  ( ( R `  f )  =  u  <->  u  =  ( R `  f ) )
1312anbi1i 676 . . . . . . . 8  |-  ( ( ( R `  f
)  =  u  /\  f  =/=  (  _I  |`  B ) )  <->  ( u  =  ( R `  f
)  /\  f  =/=  (  _I  |`  B ) ) )
1413rexbii 2568 . . . . . . 7  |-  ( E. f  e.  T  ( ( R `  f
)  =  u  /\  f  =/=  (  _I  |`  B ) )  <->  E. f  e.  T  ( u  =  ( R `  f )  /\  f  =/=  (  _I  |`  B ) ) )
1511, 14sylib 188 . . . . . 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 741 . . . . . 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 518 . . . . 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 423 . . . 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 1608 . . 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 14 . 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 2807 . . 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 630 . . . . . 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 1569 . . . . 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 5539 . . . . . 6  |-  ( R `
 f )  e. 
_V
25 neeq1 2454 . . . . . . . 8  |-  ( u  =  ( R `  f )  ->  (
u  =/=  X  <->  ( R `  f )  =/=  X
) )
26 neeq1 2454 . . . . . . . 8  |-  ( u  =  ( R `  f )  ->  (
u  =/=  Y  <->  ( R `  f )  =/=  Y
) )
27 neeq1 2454 . . . . . . . 8  |-  ( u  =  ( R `  f )  ->  (
u  =/=  Z  <->  ( R `  f )  =/=  Z
) )
2825, 26, 273anbi123d 1252 . . . . . . 7  |-  ( u  =  ( R `  f )  ->  (
( u  =/=  X  /\  u  =/=  Y  /\  u  =/=  Z
)  <->  ( ( R `
 f )  =/= 
X  /\  ( R `  f )  =/=  Y  /\  ( R `  f
)  =/=  Z ) ) )
2928anbi2d 684 . . . . . 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 2823 . . . . 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 240 . . . 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 2568 . . 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 2693 . . . 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 1569 . . 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 267 . 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 188 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 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   class class class wbr 4023    _I cid 4304    |` cres 4691   ` cfv 5255   Basecbs 13148   lecple 13215   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347
This theorem is referenced by:  cdlemftr2  30755  cdlemk26-3  31095  cdlemk11t  31135
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348
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