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Theorem cdlemf 29441
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
StepHypRef Expression
1 cdlemf.l . . 3  |-  .<_  =  ( le `  K )
2 eqid 2253 . . 3  |-  ( join `  K )  =  (
join `  K )
3 cdlemf.a . . 3  |-  A  =  ( Atoms `  K )
4 cdlemf.h . . 3  |-  H  =  ( LHyp `  K
)
5 eqid 2253 . . 3  |-  ( meet `  K )  =  (
meet `  K )
61, 2, 3, 4, 5cdlemf2 29440 . 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 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 ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
8 simp2l 986 . . . . . 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 1031 . . . . . 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 987 . . . . . 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 1032 . . . . . 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 29437 . . . . . 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 1196 . . . . 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 989 . . . . . . . . . . . . 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 5726 . . . . . . . . . . . 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 5725 . . . . . . . . . . 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 990 . . . . . . . . . . . 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 988 . . . . . . . . . . . 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 1075 . . . . . . . . . . . 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 1027 . . . . . . . . . . . 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 29041 . . . . . . . . . . . 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 1191 . . . . . . . . . . 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 987 . . . . . . . . . . 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 2295 . . . . . . . . . 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 1155 . . . . . . . . 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 1158 . . . . . . . 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 1150 . . . . . . 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 427 . . . . . 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 2615 . . . . 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 16 . . . 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 1155 . . 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 2635 . 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 16 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 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   E.wrex 2510   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   lecple 13089   joincjn 13922   meetcmee 13923   Atomscatm 28142   HLchlt 28229   LHypclh 28862   LTrncltrn 28979   trLctrl 29036
This theorem is referenced by:  cdlemfnid  29442  trlord  29447  dih1dimb2  30120
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-map 6660  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28055  df-ol 28057  df-oml 28058  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230  df-llines 28376  df-lplanes 28377  df-lvols 28378  df-lines 28379  df-psubsp 28381  df-pmap 28382  df-padd 28674  df-lhyp 28866  df-laut 28867  df-ldil 28982  df-ltrn 28983  df-trl 29037
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