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Theorem cdlemf 31297
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 2435 . . 3  |-  ( join `  K )  =  (
join `  K )
3 cdlemf.a . . 3  |-  A  =  ( Atoms `  K )
4 cdlemf.h . . 3  |-  H  =  ( LHyp `  K
)
5 eqid 2435 . . 3  |-  ( meet `  K )  =  (
meet `  K )
61, 2, 3, 4, 5cdlemf2 31296 . 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 31293 . . . . . 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 6089 . . . . . . . . . . . 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 6088 . . . . . . . . . . 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 30897 . . . . . . . . . . . 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 2477 . . . . . . . . . 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 2808 . . . . 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 2828 . 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 1652    e. wcel 1725   E.wrex 2698   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   lecple 13528   joincjn 14393   meetcmee 14394   Atomscatm 29998   HLchlt 30085   LHypclh 30718   LTrncltrn 30835   trLctrl 30892
This theorem is referenced by:  cdlemfnid  31298  trlord  31303  dih1dimb2  31976
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-map 7012  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233  df-lvols 30234  df-lines 30235  df-psubsp 30237  df-pmap 30238  df-padd 30530  df-lhyp 30722  df-laut 30723  df-ldil 30838  df-ltrn 30839  df-trl 30893
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