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Theorem cdlemg7fvbwN 31406
Description: Properties of a translation of an element not under 
W. TODO: Fix comment. Can this be simplified? Perhaps derived from cdleme48bw 31301? Done with a *ltrn* theorem? (Contributed by NM, 28-Apr-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemg4.l  |-  .<_  =  ( le `  K )
cdlemg4.a  |-  A  =  ( Atoms `  K )
cdlemg4.h  |-  H  =  ( LHyp `  K
)
cdlemg4.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg4.b  |-  B  =  ( Base `  K
)
Assertion
Ref Expression
cdlemg7fvbwN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T )  ->  (
( F `  X
)  e.  B  /\  -.  ( F `  X
)  .<_  W ) )

Proof of Theorem cdlemg7fvbwN
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 cdlemg4.b . . . 4  |-  B  =  ( Base `  K
)
2 cdlemg4.l . . . 4  |-  .<_  =  ( le `  K )
3 eqid 2438 . . . 4  |-  ( join `  K )  =  (
join `  K )
4 eqid 2438 . . . 4  |-  ( meet `  K )  =  (
meet `  K )
5 cdlemg4.a . . . 4  |-  A  =  ( Atoms `  K )
6 cdlemg4.h . . . 4  |-  H  =  ( LHyp `  K
)
71, 2, 3, 4, 5, 6lhpmcvr2 30823 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  E. r  e.  A  ( -.  r  .<_  W  /\  ( r (
join `  K )
( X ( meet `  K ) W ) )  =  X ) )
873adant3 978 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T )  ->  E. r  e.  A  ( -.  r  .<_  W  /\  (
r ( join `  K
) ( X (
meet `  K ) W ) )  =  X ) )
9 simp11 988 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T
)  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( r ( join `  K
) ( X (
meet `  K ) W ) )  =  X ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
10 simp2 959 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T
)  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( r ( join `  K
) ( X (
meet `  K ) W ) )  =  X ) )  -> 
r  e.  A )
11 simp3l 986 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T
)  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( r ( join `  K
) ( X (
meet `  K ) W ) )  =  X ) )  ->  -.  r  .<_  W )
1210, 11jca 520 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T
)  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( r ( join `  K
) ( X (
meet `  K ) W ) )  =  X ) )  -> 
( r  e.  A  /\  -.  r  .<_  W ) )
13 simp12 989 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T
)  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( r ( join `  K
) ( X (
meet `  K ) W ) )  =  X ) )  -> 
( X  e.  B  /\  -.  X  .<_  W ) )
14 simp13 990 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T
)  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( r ( join `  K
) ( X (
meet `  K ) W ) )  =  X ) )  ->  F  e.  T )
15 simp3r 987 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T
)  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( r ( join `  K
) ( X (
meet `  K ) W ) )  =  X ) )  -> 
( r ( join `  K ) ( X ( meet `  K
) W ) )  =  X )
16 cdlemg4.t . . . . . . 7  |-  T  =  ( ( LTrn `  K
) `  W )
176, 16, 2, 3, 5, 4, 1cdlemg2fv 31398 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( r  e.  A  /\  -.  r  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  (
r ( join `  K
) ( X (
meet `  K ) W ) )  =  X ) )  -> 
( F `  X
)  =  ( ( F `  r ) ( join `  K
) ( X (
meet `  K ) W ) ) )
189, 12, 13, 14, 15, 17syl122anc 1194 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T
)  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( r ( join `  K
) ( X (
meet `  K ) W ) )  =  X ) )  -> 
( F `  X
)  =  ( ( F `  r ) ( join `  K
) ( X (
meet `  K ) W ) ) )
19 simp11l 1069 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T
)  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( r ( join `  K
) ( X (
meet `  K ) W ) )  =  X ) )  ->  K  e.  HL )
20 hllat 30163 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
2119, 20syl 16 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T
)  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( r ( join `  K
) ( X (
meet `  K ) W ) )  =  X ) )  ->  K  e.  Lat )
222, 5, 6, 16ltrnel 30938 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( r  e.  A  /\  -.  r  .<_  W ) )  ->  ( ( F `  r )  e.  A  /\  -.  ( F `  r )  .<_  W ) )
2322simpld 447 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( r  e.  A  /\  -.  r  .<_  W ) )  ->  ( F `  r )  e.  A
)
249, 14, 12, 23syl3anc 1185 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T
)  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( r ( join `  K
) ( X (
meet `  K ) W ) )  =  X ) )  -> 
( F `  r
)  e.  A )
251, 5atbase 30089 . . . . . . 7  |-  ( ( F `  r )  e.  A  ->  ( F `  r )  e.  B )
2624, 25syl 16 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T
)  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( r ( join `  K
) ( X (
meet `  K ) W ) )  =  X ) )  -> 
( F `  r
)  e.  B )
27 simp12l 1071 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T
)  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( r ( join `  K
) ( X (
meet `  K ) W ) )  =  X ) )  ->  X  e.  B )
28 simp11r 1070 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T
)  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( r ( join `  K
) ( X (
meet `  K ) W ) )  =  X ) )  ->  W  e.  H )
291, 6lhpbase 30797 . . . . . . . 8  |-  ( W  e.  H  ->  W  e.  B )
3028, 29syl 16 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T
)  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( r ( join `  K
) ( X (
meet `  K ) W ) )  =  X ) )  ->  W  e.  B )
311, 4latmcl 14482 . . . . . . 7  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X ( meet `  K ) W )  e.  B )
3221, 27, 30, 31syl3anc 1185 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T
)  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( r ( join `  K
) ( X (
meet `  K ) W ) )  =  X ) )  -> 
( X ( meet `  K ) W )  e.  B )
331, 3latjcl 14481 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( F `  r )  e.  B  /\  ( X ( meet `  K
) W )  e.  B )  ->  (
( F `  r
) ( join `  K
) ( X (
meet `  K ) W ) )  e.  B )
3421, 26, 32, 33syl3anc 1185 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T
)  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( r ( join `  K
) ( X (
meet `  K ) W ) )  =  X ) )  -> 
( ( F `  r ) ( join `  K ) ( X ( meet `  K
) W ) )  e.  B )
3518, 34eqeltrd 2512 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T
)  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( r ( join `  K
) ( X (
meet `  K ) W ) )  =  X ) )  -> 
( F `  X
)  e.  B )
3622simprd 451 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( r  e.  A  /\  -.  r  .<_  W ) )  ->  -.  ( F `  r )  .<_  W )
379, 14, 12, 36syl3anc 1185 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T
)  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( r ( join `  K
) ( X (
meet `  K ) W ) )  =  X ) )  ->  -.  ( F `  r
)  .<_  W )
381, 2, 3latlej1 14491 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( F `  r )  e.  B  /\  ( X ( meet `  K
) W )  e.  B )  ->  ( F `  r )  .<_  ( ( F `  r ) ( join `  K ) ( X ( meet `  K
) W ) ) )
3921, 26, 32, 38syl3anc 1185 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T
)  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( r ( join `  K
) ( X (
meet `  K ) W ) )  =  X ) )  -> 
( F `  r
)  .<_  ( ( F `
 r ) (
join `  K )
( X ( meet `  K ) W ) ) )
401, 2lattr 14487 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( ( F `  r )  e.  B  /\  ( ( F `  r ) ( join `  K ) ( X ( meet `  K
) W ) )  e.  B  /\  W  e.  B ) )  -> 
( ( ( F `
 r )  .<_  ( ( F `  r ) ( join `  K ) ( X ( meet `  K
) W ) )  /\  ( ( F `
 r ) (
join `  K )
( X ( meet `  K ) W ) )  .<_  W )  ->  ( F `  r
)  .<_  W ) )
4121, 26, 34, 30, 40syl13anc 1187 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T
)  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( r ( join `  K
) ( X (
meet `  K ) W ) )  =  X ) )  -> 
( ( ( F `
 r )  .<_  ( ( F `  r ) ( join `  K ) ( X ( meet `  K
) W ) )  /\  ( ( F `
 r ) (
join `  K )
( X ( meet `  K ) W ) )  .<_  W )  ->  ( F `  r
)  .<_  W ) )
4239, 41mpand 658 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T
)  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( r ( join `  K
) ( X (
meet `  K ) W ) )  =  X ) )  -> 
( ( ( F `
 r ) (
join `  K )
( X ( meet `  K ) W ) )  .<_  W  ->  ( F `  r ) 
.<_  W ) )
4337, 42mtod 171 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T
)  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( r ( join `  K
) ( X (
meet `  K ) W ) )  =  X ) )  ->  -.  ( ( F `  r ) ( join `  K ) ( X ( meet `  K
) W ) ) 
.<_  W )
4418breq1d 4224 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T
)  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( r ( join `  K
) ( X (
meet `  K ) W ) )  =  X ) )  -> 
( ( F `  X )  .<_  W  <->  ( ( F `  r )
( join `  K )
( X ( meet `  K ) W ) )  .<_  W )
)
4543, 44mtbird 294 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T
)  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( r ( join `  K
) ( X (
meet `  K ) W ) )  =  X ) )  ->  -.  ( F `  X
)  .<_  W )
4635, 45jca 520 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T
)  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( r ( join `  K
) ( X (
meet `  K ) W ) )  =  X ) )  -> 
( ( F `  X )  e.  B  /\  -.  ( F `  X )  .<_  W ) )
4746rexlimdv3a 2834 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T )  ->  ( E. r  e.  A  ( -.  r  .<_  W  /\  ( r (
join `  K )
( X ( meet `  K ) W ) )  =  X )  ->  ( ( F `
 X )  e.  B  /\  -.  ( F `  X )  .<_  W ) ) )
488, 47mpd 15 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  F  e.  T )  ->  (
( F `  X
)  e.  B  /\  -.  ( F `  X
)  .<_  W ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E.wrex 2708   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403   meetcmee 14404   Latclat 14476   Atomscatm 30063   HLchlt 30150   LHypclh 30783   LTrncltrn 30900
This theorem is referenced by:  cdlemg7fvN  31423
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-map 7022  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-p1 14471  df-lat 14477  df-clat 14539  df-oposet 29976  df-ol 29978  df-oml 29979  df-covers 30066  df-ats 30067  df-atl 30098  df-cvlat 30122  df-hlat 30151  df-llines 30297  df-lplanes 30298  df-lvols 30299  df-lines 30300  df-psubsp 30302  df-pmap 30303  df-padd 30595  df-lhyp 30787  df-laut 30788  df-ldil 30903  df-ltrn 30904  df-trl 30958
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