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Theorem lhpmcvr2 36164
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 2454 . . 3  |-  (  <o  `  K )  =  ( 
<o  `  K )
5 lhpmcvr2.h . . 3  |-  H  =  ( LHyp `  K
)
61, 2, 3, 4, 5lhpmcvr 36163 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( X  ./\  W
) (  <o  `  K
) X )
7 simpll 751 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  K  e.  HL )
8 simprl 754 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  X  e.  B )
91, 5lhpbase 36138 . . . 4  |-  ( W  e.  H  ->  W  e.  B )
109ad2antlr 724 . . 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 35555 . . 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 1226 . 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 367    = wceq 1398    e. wcel 1823   E.wrex 2805   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14719   lecple 14794   joincjn 15775   meetcmee 15776    <o ccvr 35403   Atomscatm 35404   HLchlt 35491   LHypclh 36124
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
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  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-ral 2809  df-rex 2810  df-reu 2811  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-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-preset 15759  df-poset 15777  df-plt 15790  df-lub 15806  df-glb 15807  df-join 15808  df-meet 15809  df-p0 15871  df-p1 15872  df-lat 15878  df-clat 15940  df-oposet 35317  df-ol 35319  df-oml 35320  df-covers 35407  df-ats 35408  df-atl 35439  df-cvlat 35463  df-hlat 35492  df-lhyp 36128
This theorem is referenced by:  lhpmcvr5N  36167  cdleme29ex  36516  cdleme29c  36518  cdlemefrs29cpre1  36540  cdlemefr29exN  36544  cdleme32d  36586  cdleme32f  36588  cdleme48gfv1  36678  cdlemg7fvbwN  36749  cdlemg7aN  36767  dihlsscpre  37377  dihvalcqpre  37378  dihord6apre  37399  dihord4  37401  dihord5b  37402  dihord5apre  37405  dihmeetlem1N  37433  dihglblem5apreN  37434  dihglbcpreN  37443
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