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Theorem cdlemg7fvbwN 36730
Description: Properties of a translation of an element not under 
W. TODO: Fix comment. Can this be simplified? Perhaps derived from cdleme48bw 36625? 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 2454 . . . 4  |-  ( join `  K )  =  (
join `  K )
4 eqid 2454 . . . 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 36145 . . 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 1014 . 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 1024 . . . . . 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 995 . . . . . . 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 1022 . . . . . . 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 530 . . . . . 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 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 ) )  -> 
( X  e.  B  /\  -.  X  .<_  W ) )
14 simp13 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 ) )  ->  F  e.  T )
15 simp3r 1023 . . . . . 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 36722 . . . . . 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 1235 . . . . 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 1105 . . . . . . 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 35485 . . . . . . 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 36260 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( r  e.  A  /\  -.  r  .<_  W ) )  ->  ( ( F `  r )  e.  A  /\  -.  ( F `  r )  .<_  W ) )
2322simpld 457 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( r  e.  A  /\  -.  r  .<_  W ) )  ->  ( F `  r )  e.  A
)
249, 14, 12, 23syl3anc 1226 . . . . . . 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 35411 . . . . . . 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 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 ) )  ->  X  e.  B )
28 simp11r 1106 . . . . . . . 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 36119 . . . . . . . 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 15881 . . . . . . 7  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X ( meet `  K ) W )  e.  B )
3221, 27, 30, 31syl3anc 1226 . . . . . 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 15880 . . . . . 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 1226 . . . . 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 2542 . . . 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 461 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( r  e.  A  /\  -.  r  .<_  W ) )  ->  -.  ( F `  r )  .<_  W )
379, 14, 12, 36syl3anc 1226 . . . . . 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 15889 . . . . . . . 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 1226 . . . . . . 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 15885 . . . . . . . 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 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 ) )  /\  ( ( F `
 r ) (
join `  K )
( X ( meet `  K ) W ) )  .<_  W )  ->  ( F `  r
)  .<_  W ) )
4239, 41mpand 673 . . . . . 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 4449 . . . . 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 299 . . . 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 530 . . 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 2948 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   E.wrex 2805   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   lecple 14791   joincjn 15772   meetcmee 15773   Latclat 15874   Atomscatm 35385   HLchlt 35472   LHypclh 36105   LTrncltrn 36222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-riotaBAD 35081
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-undef 6994  df-map 7414  df-preset 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-p1 15869  df-lat 15875  df-clat 15937  df-oposet 35298  df-ol 35300  df-oml 35301  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473  df-llines 35619  df-lplanes 35620  df-lvols 35621  df-lines 35622  df-psubsp 35624  df-pmap 35625  df-padd 35917  df-lhyp 36109  df-laut 36110  df-ldil 36225  df-ltrn 36226  df-trl 36281
This theorem is referenced by:  cdlemg7fvN  36747
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