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Theorem lhpmcvr2 30031
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 2316 . . 3  |-  (  <o  `  K )  =  ( 
<o  `  K )
5 lhpmcvr2.h . . 3  |-  H  =  ( LHyp `  K
)
61, 2, 3, 4, 5lhpmcvr 30030 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( X  ./\  W
) (  <o  `  K
) X )
7 simpll 730 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  K  e.  HL )
8 simprl 732 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  X  e.  B )
91, 5lhpbase 30005 . . . 4  |-  ( W  e.  H  ->  W  e.  B )
109ad2antlr 707 . . 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 29422 . . 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 1182 . 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 201 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1633    e. wcel 1701   E.wrex 2578   class class class wbr 4060   ` cfv 5292  (class class class)co 5900   Basecbs 13195   lecple 13262   joincjn 14127   meetcmee 14128    <o ccvr 29270   Atomscatm 29271   HLchlt 29358   LHypclh 29991
This theorem is referenced by:  lhpmcvr5N  30034  cdleme29ex  30381  cdleme29c  30383  cdlemefrs29cpre1  30405  cdlemefr29exN  30409  cdleme32d  30451  cdleme32f  30453  cdleme48gfv1  30543  cdlemg7fvbwN  30614  cdlemg7aN  30632  dihlsscpre  31242  dihvalcqpre  31243  dihord6apre  31264  dihord4  31266  dihord5b  31267  dihord5apre  31270  dihmeetlem1N  31298  dihglblem5apreN  31299  dihglbcpreN  31308
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-undef 6340  df-riota 6346  df-poset 14129  df-plt 14141  df-lub 14157  df-glb 14158  df-join 14159  df-meet 14160  df-p0 14194  df-p1 14195  df-lat 14201  df-clat 14263  df-oposet 29184  df-ol 29186  df-oml 29187  df-covers 29274  df-ats 29275  df-atl 29306  df-cvlat 29330  df-hlat 29359  df-lhyp 29995
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