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Theorem cdlemftr3 34517
Description: Special case of cdlemf 34515 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 2451 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2451 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 cdlemftr.h . . . . 5  |-  H  =  ( LHyp `  K
)
41, 2, 3lhpexle3 33964 . . . 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 2801 . . . 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 34516 . . . . . . . 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 724 . . . . . . 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 2460 . . . . . . . . 9  |-  ( ( R `  f )  =  u  <->  u  =  ( R `  f ) )
1312anbi1i 695 . . . . . . . 8  |-  ( ( ( R `  f
)  =  u  /\  f  =/=  (  _I  |`  B ) )  <->  ( u  =  ( R `  f
)  /\  f  =/=  (  _I  |`  B ) ) )
1413rexbii 2852 . . . . . . 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 532 . . . . 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 434 . . . 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 1677 . . 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 3090 . . 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 649 . . . . . 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 1635 . . . . 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 5801 . . . . . 6  |-  ( R `
 f )  e. 
_V
25 neeq1 2729 . . . . . . . 8  |-  ( u  =  ( R `  f )  ->  (
u  =/=  X  <->  ( R `  f )  =/=  X
) )
26 neeq1 2729 . . . . . . . 8  |-  ( u  =  ( R `  f )  ->  (
u  =/=  Y  <->  ( R `  f )  =/=  Y
) )
27 neeq1 2729 . . . . . . . 8  |-  ( u  =  ( R `  f )  ->  (
u  =/=  Z  <->  ( R `  f )  =/=  Z
) )
2825, 26, 273anbi123d 1290 . . . . . . 7  |-  ( u  =  ( R `  f )  ->  (
( u  =/=  X  /\  u  =/=  Y  /\  u  =/=  Z
)  <->  ( ( R `
 f )  =/= 
X  /\  ( R `  f )  =/=  Y  /\  ( R `  f
)  =/=  Z ) ) )
2928anbi2d 703 . . . . . 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 3107 . . . . 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 2852 . . 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 2971 . . . 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 1635 . . 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 369    /\ w3a 965    = wceq 1370   E.wex 1587    e. wcel 1758    =/= wne 2644   E.wrex 2796   class class class wbr 4392    _I cid 4731    |` cres 4942   ` cfv 5518   Basecbs 14278   lecple 14349   Atomscatm 33216   HLchlt 33303   LHypclh 33936   LTrncltrn 34053   trLctrl 34110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-riotaBAD 32912
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-iin 4274  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-undef 6894  df-map 7318  df-poset 15220  df-plt 15232  df-lub 15248  df-glb 15249  df-join 15250  df-meet 15251  df-p0 15313  df-p1 15314  df-lat 15320  df-clat 15382  df-oposet 33129  df-ol 33131  df-oml 33132  df-covers 33219  df-ats 33220  df-atl 33251  df-cvlat 33275  df-hlat 33304  df-llines 33450  df-lplanes 33451  df-lvols 33452  df-lines 33453  df-psubsp 33455  df-pmap 33456  df-padd 33748  df-lhyp 33940  df-laut 33941  df-ldil 34056  df-ltrn 34057  df-trl 34111
This theorem is referenced by:  cdlemftr2  34518  cdlemk26-3  34858  cdlemk11t  34898
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