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Theorem cdlemf 31049
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 2408 . . 3  |-  ( join `  K )  =  (
join `  K )
3 cdlemf.a . . 3  |-  A  =  ( Atoms `  K )
4 cdlemf.h . . 3  |-  H  =  ( LHyp `  K
)
5 eqid 2408 . . 3  |-  ( meet `  K )  =  (
meet `  K )
61, 2, 3, 4, 5cdlemf2 31048 . 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 31045 . . . . . 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 6060 . . . . . . . . . . . 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 6059 . . . . . . . . . . 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 30649 . . . . . . . . . . . 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 2450 . . . . . . . . . 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 2780 . . . . 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 2800 . 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 2671   class class class wbr 4176   ` cfv 5417  (class class class)co 6044   lecple 13495   joincjn 14360   meetcmee 14361   Atomscatm 29750   HLchlt 29837   LHypclh 30470   LTrncltrn 30587   trLctrl 30644
This theorem is referenced by:  cdlemfnid  31050  trlord  31055  dih1dimb2  31728
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 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-undef 6506  df-riota 6512  df-map 6983  df-poset 14362  df-plt 14374  df-lub 14390  df-glb 14391  df-join 14392  df-meet 14393  df-p0 14427  df-p1 14428  df-lat 14434  df-clat 14496  df-oposet 29663  df-ol 29665  df-oml 29666  df-covers 29753  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-llines 29984  df-lplanes 29985  df-lvols 29986  df-lines 29987  df-psubsp 29989  df-pmap 29990  df-padd 30282  df-lhyp 30474  df-laut 30475  df-ldil 30590  df-ltrn 30591  df-trl 30645
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