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Theorem cdlemg7fvbwN 31341
Description: Properties of a translation of an element not under 
W. TODO: Fix comment. Can this be simplified? Perhaps derived from cdleme48bw 31236? 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 2435 . . . 4  |-  ( join `  K )  =  (
join `  K )
4 eqid 2435 . . . 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 30758 . . 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 977 . 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 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 ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
10 simp2 958 . . . . . . 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 985 . . . . . . 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 519 . . . . . 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 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 ) )  -> 
( X  e.  B  /\  -.  X  .<_  W ) )
14 simp13 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 ) )  ->  F  e.  T )
15 simp3r 986 . . . . . 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 31333 . . . . . 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 1193 . . . . 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 1068 . . . . . . 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 30098 . . . . . . 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 30873 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( r  e.  A  /\  -.  r  .<_  W ) )  ->  ( ( F `  r )  e.  A  /\  -.  ( F `  r )  .<_  W ) )
2322simpld 446 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( r  e.  A  /\  -.  r  .<_  W ) )  ->  ( F `  r )  e.  A
)
249, 14, 12, 23syl3anc 1184 . . . . . . 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 30024 . . . . . . 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 1070 . . . . . . 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 1069 . . . . . . . 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 30732 . . . . . . . 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 14472 . . . . . . 7  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X ( meet `  K ) W )  e.  B )
3221, 27, 30, 31syl3anc 1184 . . . . . 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 14471 . . . . . 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 1184 . . . . 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 2509 . . . 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 450 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( r  e.  A  /\  -.  r  .<_  W ) )  ->  -.  ( F `  r )  .<_  W )
379, 14, 12, 36syl3anc 1184 . . . . . 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 14481 . . . . . . . 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 1184 . . . . . . 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 14477 . . . . . . . 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 1186 . . . . . . 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 657 . . . . . 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 170 . . . . 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 4214 . . . . 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 293 . . . 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 519 . . 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 2824 . 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2698   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   meetcmee 14394   Latclat 14466   Atomscatm 29998   HLchlt 30085   LHypclh 30718   LTrncltrn 30835
This theorem is referenced by:  cdlemg7fvN  31358
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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