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Theorem lhpmcvr2 33974
Description: Alternate way to express that the meet of a lattice hyperplane with an element not under it is covered by the element. (Contributed by NM, 9-Apr-2013.)
Hypotheses
Ref Expression
lhpmcvr2.b  |-  B  =  ( Base `  K
)
lhpmcvr2.l  |-  .<_  =  ( le `  K )
lhpmcvr2.j  |-  .\/  =  ( join `  K )
lhpmcvr2.m  |-  ./\  =  ( meet `  K )
lhpmcvr2.a  |-  A  =  ( Atoms `  K )
lhpmcvr2.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
lhpmcvr2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  E. p  e.  A  ( -.  p  .<_  W  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )
Distinct variable groups:    A, p    B, p    K, p    .<_ , p    ./\ , p    X, p    W, p
Allowed substitution hints:    H( p)    .\/ ( p)

Proof of Theorem lhpmcvr2
StepHypRef Expression
1 lhpmcvr2.b . . 3  |-  B  =  ( Base `  K
)
2 lhpmcvr2.l . . 3  |-  .<_  =  ( le `  K )
3 lhpmcvr2.m . . 3  |-  ./\  =  ( meet `  K )
4 eqid 2451 . . 3  |-  (  <o  `  K )  =  ( 
<o  `  K )
5 lhpmcvr2.h . . 3  |-  H  =  ( LHyp `  K
)
61, 2, 3, 4, 5lhpmcvr 33973 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( X  ./\  W
) (  <o  `  K
) X )
7 simpll 753 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  K  e.  HL )
8 simprl 755 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  X  e.  B )
91, 5lhpbase 33948 . . . 4  |-  ( W  e.  H  ->  W  e.  B )
109ad2antlr 726 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  W  e.  B )
11 lhpmcvr2.j . . . 4  |-  .\/  =  ( join `  K )
12 lhpmcvr2.a . . . 4  |-  A  =  ( Atoms `  K )
131, 2, 11, 3, 4, 12cvrval5 33365 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  W  e.  B )  ->  ( ( X  ./\  W ) (  <o  `  K
) X  <->  E. p  e.  A  ( -.  p  .<_  W  /\  (
p  .\/  ( X  ./\ 
W ) )  =  X ) ) )
147, 8, 10, 13syl3anc 1219 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( ( X  ./\  W ) (  <o  `  K
) X  <->  E. p  e.  A  ( -.  p  .<_  W  /\  (
p  .\/  ( X  ./\ 
W ) )  =  X ) ) )
156, 14mpbid 210 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  E. p  e.  A  ( -.  p  .<_  W  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2796   class class class wbr 4390   ` cfv 5516  (class class class)co 6190   Basecbs 14276   lecple 14347   joincjn 15216   meetcmee 15217    <o ccvr 33213   Atomscatm 33214   HLchlt 33301   LHypclh 33934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-poset 15218  df-plt 15230  df-lub 15246  df-glb 15247  df-join 15248  df-meet 15249  df-p0 15311  df-p1 15312  df-lat 15318  df-clat 15380  df-oposet 33127  df-ol 33129  df-oml 33130  df-covers 33217  df-ats 33218  df-atl 33249  df-cvlat 33273  df-hlat 33302  df-lhyp 33938
This theorem is referenced by:  lhpmcvr5N  33977  cdleme29ex  34324  cdleme29c  34326  cdlemefrs29cpre1  34348  cdlemefr29exN  34352  cdleme32d  34394  cdleme32f  34396  cdleme48gfv1  34486  cdlemg7fvbwN  34557  cdlemg7aN  34575  dihlsscpre  35185  dihvalcqpre  35186  dihord6apre  35207  dihord4  35209  dihord5b  35210  dihord5apre  35213  dihmeetlem1N  35241  dihglblem5apreN  35242  dihglbcpreN  35251
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