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Theorem cdlemftr3 36688
Description: Special case of cdlemf 36686 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 2454 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2454 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 cdlemftr.h . . . . 5  |-  H  =  ( LHyp `  K
)
41, 2, 3lhpexle3 36133 . . . 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 2810 . . . 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 196 . . 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 36687 . . . . . . . 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 722 . . . . . . 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 2463 . . . . . . . . 9  |-  ( ( R `  f )  =  u  <->  u  =  ( R `  f ) )
1312anbi1i 693 . . . . . . . 8  |-  ( ( ( R `  f
)  =  u  /\  f  =/=  (  _I  |`  B ) )  <->  ( u  =  ( R `  f
)  /\  f  =/=  (  _I  |`  B ) ) )
1413rexbii 2956 . . . . . . 7  |-  ( E. f  e.  T  ( ( R `  f
)  =  u  /\  f  =/=  (  _I  |`  B ) )  <->  E. f  e.  T  ( u  =  ( R `  f )  /\  f  =/=  (  _I  |`  B ) ) )
1511, 14sylib 196 . . . . . 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 764 . . . . . 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 530 . . . . 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 432 . . . 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 1715 . . 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 3126 . . 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 647 . . . . . 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 1672 . . . . 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 5858 . . . . . 6  |-  ( R `
 f )  e. 
_V
25 neeq1 2735 . . . . . . . 8  |-  ( u  =  ( R `  f )  ->  (
u  =/=  X  <->  ( R `  f )  =/=  X
) )
26 neeq1 2735 . . . . . . . 8  |-  ( u  =  ( R `  f )  ->  (
u  =/=  Y  <->  ( R `  f )  =/=  Y
) )
27 neeq1 2735 . . . . . . . 8  |-  ( u  =  ( R `  f )  ->  (
u  =/=  Z  <->  ( R `  f )  =/=  Z
) )
2825, 26, 273anbi123d 1297 . . . . . . 7  |-  ( u  =  ( R `  f )  ->  (
( u  =/=  X  /\  u  =/=  Y  /\  u  =/=  Z
)  <->  ( ( R `
 f )  =/= 
X  /\  ( R `  f )  =/=  Y  /\  ( R `  f
)  =/=  Z ) ) )
2928anbi2d 701 . . . . . 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 3143 . . . . 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 249 . . . 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 2956 . . 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 3006 . . . 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 1672 . . 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 276 . 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 196 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398   E.wex 1617    e. wcel 1823    =/= wne 2649   E.wrex 2805   class class class wbr 4439    _I cid 4779    |` cres 4990   ` cfv 5570   Basecbs 14716   lecple 14791   Atomscatm 35385   HLchlt 35472   LHypclh 36105   LTrncltrn 36222   trLctrl 36280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-riotaBAD 35081
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-undef 6994  df-map 7414  df-preset 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-p1 15869  df-lat 15875  df-clat 15937  df-oposet 35298  df-ol 35300  df-oml 35301  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473  df-llines 35619  df-lplanes 35620  df-lvols 35621  df-lines 35622  df-psubsp 35624  df-pmap 35625  df-padd 35917  df-lhyp 36109  df-laut 36110  df-ldil 36225  df-ltrn 36226  df-trl 36281
This theorem is referenced by:  cdlemftr2  36689  cdlemk26-3  37029  cdlemk11t  37069
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