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Theorem cdlemf 31045
Description: Lemma F in [Crawley] p. 116. If u is an atom under w, there exists a translation whose trace is u. (Contributed by NM, 12-Apr-2013.)
Hypotheses
Ref Expression
cdlemf.l  |-  .<_  =  ( le `  K )
cdlemf.a  |-  A  =  ( Atoms `  K )
cdlemf.h  |-  H  =  ( LHyp `  K
)
cdlemf.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemf.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
cdlemf  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  ->  E. f  e.  T  ( R `  f )  =  U )
Distinct variable groups:    A, f    f, H    f, K    .<_ , f    T, f    U, f    f, W
Allowed substitution hint:    R( f)

Proof of Theorem cdlemf
Dummy variables  p  q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cdlemf.l . . 3  |-  .<_  =  ( le `  K )
2 eqid 2404 . . 3  |-  ( join `  K )  =  (
join `  K )
3 cdlemf.a . . 3  |-  A  =  ( Atoms `  K )
4 cdlemf.h . . 3  |-  H  =  ( LHyp `  K
)
5 eqid 2404 . . 3  |-  ( meet `  K )  =  (
meet `  K )
61, 2, 3, 4, 5cdlemf2 31044 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  ->  E. p  e.  A  E. q  e.  A  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p ( join `  K ) q ) ( meet `  K
) W ) ) )
7 simp1l 981 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( p  e.  A  /\  q  e.  A )  /\  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p ( join `  K
) q ) (
meet `  K ) W ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
8 simp2l 983 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( p  e.  A  /\  q  e.  A )  /\  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p ( join `  K
) q ) (
meet `  K ) W ) ) )  ->  p  e.  A
)
9 simp3ll 1028 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( p  e.  A  /\  q  e.  A )  /\  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p ( join `  K
) q ) (
meet `  K ) W ) ) )  ->  -.  p  .<_  W )
10 simp2r 984 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( p  e.  A  /\  q  e.  A )  /\  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p ( join `  K
) q ) (
meet `  K ) W ) ) )  ->  q  e.  A
)
11 simp3lr 1029 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( p  e.  A  /\  q  e.  A )  /\  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p ( join `  K
) q ) (
meet `  K ) W ) ) )  ->  -.  q  .<_  W )
12 cdlemf.t . . . . . . 7  |-  T  =  ( ( LTrn `  K
) `  W )
131, 3, 4, 12cdleme50ex 31041 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  A  /\  -.  p  .<_  W )  /\  (
q  e.  A  /\  -.  q  .<_  W ) )  ->  E. f  e.  T  ( f `  p )  =  q )
147, 8, 9, 10, 11, 13syl122anc 1193 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( p  e.  A  /\  q  e.  A )  /\  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p ( join `  K
) q ) (
meet `  K ) W ) ) )  ->  E. f  e.  T  ( f `  p
)  =  q )
15 simp3r 986 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( p  e.  A  /\  q  e.  A
) )  /\  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p ( join `  K
) q ) (
meet `  K ) W ) )  /\  ( f  e.  T  /\  ( f `  p
)  =  q ) )  ->  ( f `  p )  =  q )
1615oveq2d 6056 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( p  e.  A  /\  q  e.  A
) )  /\  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p ( join `  K
) q ) (
meet `  K ) W ) )  /\  ( f  e.  T  /\  ( f `  p
)  =  q ) )  ->  ( p
( join `  K )
( f `  p
) )  =  ( p ( join `  K
) q ) )
1716oveq1d 6055 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( p  e.  A  /\  q  e.  A
) )  /\  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p ( join `  K
) q ) (
meet `  K ) W ) )  /\  ( f  e.  T  /\  ( f `  p
)  =  q ) )  ->  ( (
p ( join `  K
) ( f `  p ) ) (
meet `  K ) W )  =  ( ( p ( join `  K ) q ) ( meet `  K
) W ) )
18 simp11 987 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( p  e.  A  /\  q  e.  A
) )  /\  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p ( join `  K
) q ) (
meet `  K ) W ) )  /\  ( f  e.  T  /\  ( f `  p
)  =  q ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
19 simp3l 985 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( p  e.  A  /\  q  e.  A
) )  /\  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p ( join `  K
) q ) (
meet `  K ) W ) )  /\  ( f  e.  T  /\  ( f `  p
)  =  q ) )  ->  f  e.  T )
20 simp13l 1072 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( p  e.  A  /\  q  e.  A
) )  /\  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p ( join `  K
) q ) (
meet `  K ) W ) )  /\  ( f  e.  T  /\  ( f `  p
)  =  q ) )  ->  p  e.  A )
21 simp2ll 1024 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( p  e.  A  /\  q  e.  A
) )  /\  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p ( join `  K
) q ) (
meet `  K ) W ) )  /\  ( f  e.  T  /\  ( f `  p
)  =  q ) )  ->  -.  p  .<_  W )
22 cdlemf.r . . . . . . . . . . . . 13  |-  R  =  ( ( trL `  K
) `  W )
231, 2, 5, 3, 4, 12, 22trlval2 30645 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  ->  ( R `  f )  =  ( ( p ( join `  K ) ( f `
 p ) ) ( meet `  K
) W ) )
2418, 19, 20, 21, 23syl112anc 1188 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( p  e.  A  /\  q  e.  A
) )  /\  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p ( join `  K
) q ) (
meet `  K ) W ) )  /\  ( f  e.  T  /\  ( f `  p
)  =  q ) )  ->  ( R `  f )  =  ( ( p ( join `  K ) ( f `
 p ) ) ( meet `  K
) W ) )
25 simp2r 984 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( p  e.  A  /\  q  e.  A
) )  /\  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p ( join `  K
) q ) (
meet `  K ) W ) )  /\  ( f  e.  T  /\  ( f `  p
)  =  q ) )  ->  U  =  ( ( p (
join `  K )
q ) ( meet `  K ) W ) )
2617, 24, 253eqtr4d 2446 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( p  e.  A  /\  q  e.  A
) )  /\  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p ( join `  K
) q ) (
meet `  K ) W ) )  /\  ( f  e.  T  /\  ( f `  p
)  =  q ) )  ->  ( R `  f )  =  U )
27263exp 1152 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( p  e.  A  /\  q  e.  A ) )  -> 
( ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p ( join `  K ) q ) ( meet `  K
) W ) )  ->  ( ( f  e.  T  /\  (
f `  p )  =  q )  -> 
( R `  f
)  =  U ) ) )
28273expia 1155 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  ->  (
( p  e.  A  /\  q  e.  A
)  ->  ( (
( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p ( join `  K
) q ) (
meet `  K ) W ) )  -> 
( ( f  e.  T  /\  ( f `
 p )  =  q )  ->  ( R `  f )  =  U ) ) ) )
29283imp 1147 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( p  e.  A  /\  q  e.  A )  /\  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p ( join `  K
) q ) (
meet `  K ) W ) ) )  ->  ( ( f  e.  T  /\  (
f `  p )  =  q )  -> 
( R `  f
)  =  U ) )
3029exp3a 426 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( p  e.  A  /\  q  e.  A )  /\  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p ( join `  K
) q ) (
meet `  K ) W ) ) )  ->  ( f  e.  T  ->  ( (
f `  p )  =  q  ->  ( R `
 f )  =  U ) ) )
3130reximdvai 2776 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( p  e.  A  /\  q  e.  A )  /\  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p ( join `  K
) q ) (
meet `  K ) W ) ) )  ->  ( E. f  e.  T  ( f `  p )  =  q  ->  E. f  e.  T  ( R `  f )  =  U ) )
3214, 31mpd 15 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  ( p  e.  A  /\  q  e.  A )  /\  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p ( join `  K
) q ) (
meet `  K ) W ) ) )  ->  E. f  e.  T  ( R `  f )  =  U )
33323exp 1152 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  ->  (
( p  e.  A  /\  q  e.  A
)  ->  ( (
( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p ( join `  K
) q ) (
meet `  K ) W ) )  ->  E. f  e.  T  ( R `  f )  =  U ) ) )
3433rexlimdvv 2796 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  ->  ( E. p  e.  A  E. q  e.  A  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  U  =  ( ( p ( join `  K
) q ) (
meet `  K ) W ) )  ->  E. f  e.  T  ( R `  f )  =  U ) )
356, 34mpd 15 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  ->  E. f  e.  T  ( R `  f )  =  U )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   E.wrex 2667   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   lecple 13491   joincjn 14356   meetcmee 14357   Atomscatm 29746   HLchlt 29833   LHypclh 30466   LTrncltrn 30583   trLctrl 30640
This theorem is referenced by:  cdlemfnid  31046  trlord  31051  dih1dimb2  31724
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-map 6979  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641
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