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Theorem cdlemg7fvbwN 35620
Description: Properties of a translation of an element not under 
W. TODO: Fix comment. Can this be simplified? Perhaps derived from cdleme48bw 35515? 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 2467 . . . 4  |-  ( join `  K )  =  (
join `  K )
4 eqid 2467 . . . 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 35037 . . 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 1016 . 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 1026 . . . . . 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 997 . . . . . . 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 1024 . . . . . . 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 532 . . . . . 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 1027 . . . . . 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 1028 . . . . . 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 1025 . . . . . 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 35612 . . . . . 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 1237 . . . . 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 1107 . . . . . . 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 34377 . . . . . . 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 35152 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( r  e.  A  /\  -.  r  .<_  W ) )  ->  ( ( F `  r )  e.  A  /\  -.  ( F `  r )  .<_  W ) )
2322simpld 459 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( r  e.  A  /\  -.  r  .<_  W ) )  ->  ( F `  r )  e.  A
)
249, 14, 12, 23syl3anc 1228 . . . . . . 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 34303 . . . . . . 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 1109 . . . . . . 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 1108 . . . . . . . 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 35011 . . . . . . . 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 15542 . . . . . . 7  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X ( meet `  K ) W )  e.  B )
3221, 27, 30, 31syl3anc 1228 . . . . . 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 15541 . . . . . 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 1228 . . . . 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 2555 . . . 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 463 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( r  e.  A  /\  -.  r  .<_  W ) )  ->  -.  ( F `  r )  .<_  W )
379, 14, 12, 36syl3anc 1228 . . . . . 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 15550 . . . . . . . 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 1228 . . . . . . 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 15546 . . . . . . . 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 1230 . . . . . . 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 675 . . . . . 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 177 . . . . 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 4457 . . . . 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 301 . . . 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 532 . . 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 2957 . 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2815   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   Basecbs 14493   lecple 14565   joincjn 15434   meetcmee 15435   Latclat 15535   Atomscatm 34277   HLchlt 34364   LHypclh 34997   LTrncltrn 35114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-riotaBAD 33973
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-undef 7003  df-map 7423  df-poset 15436  df-plt 15448  df-lub 15464  df-glb 15465  df-join 15466  df-meet 15467  df-p0 15529  df-p1 15530  df-lat 15536  df-clat 15598  df-oposet 34190  df-ol 34192  df-oml 34193  df-covers 34280  df-ats 34281  df-atl 34312  df-cvlat 34336  df-hlat 34365  df-llines 34511  df-lplanes 34512  df-lvols 34513  df-lines 34514  df-psubsp 34516  df-pmap 34517  df-padd 34809  df-lhyp 35001  df-laut 35002  df-ldil 35117  df-ltrn 35118  df-trl 35172
This theorem is referenced by:  cdlemg7fvN  35637
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