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Theorem cvr2N 34613
Description: Less-than and covers equivalence in a Hilbert lattice. (chcv2 27106 analog.) (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cvr2.b  |-  B  =  ( Base `  K
)
cvr2.s  |-  .<  =  ( lt `  K )
cvr2.j  |-  .\/  =  ( join `  K )
cvr2.c  |-  C  =  (  <o  `  K )
cvr2.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cvr2N  |-  ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  ->  ( X  .<  ( X  .\/  P )  <->  X C
( X  .\/  P
) ) )

Proof of Theorem cvr2N
StepHypRef Expression
1 hllat 34566 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 1017 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  ->  K  e.  Lat )
3 simp2 997 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  ->  X  e.  B )
4 cvr2.b . . . . 5  |-  B  =  ( Base `  K
)
5 cvr2.a . . . . 5  |-  A  =  ( Atoms `  K )
64, 5atbase 34492 . . . 4  |-  ( P  e.  A  ->  P  e.  B )
763ad2ant3 1019 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  ->  P  e.  B )
8 eqid 2467 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
9 cvr2.s . . . 4  |-  .<  =  ( lt `  K )
10 cvr2.j . . . 4  |-  .\/  =  ( join `  K )
114, 8, 9, 10latnle 15588 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  ( -.  P ( le `  K ) X  <->  X  .<  ( X 
.\/  P ) ) )
122, 3, 7, 11syl3anc 1228 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  ->  ( -.  P ( le `  K ) X  <->  X  .<  ( X 
.\/  P ) ) )
13 cvr2.c . . 3  |-  C  =  (  <o  `  K )
144, 8, 10, 13, 5cvr1 34612 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  ->  ( -.  P ( le `  K ) X  <->  X C ( X 
.\/  P ) ) )
1512, 14bitr3d 255 1  |-  ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  ->  ( X  .<  ( X  .\/  P )  <->  X C
( X  .\/  P
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14506   lecple 14578   ltcplt 15444   joincjn 15447   Latclat 15548    <o ccvr 34465   Atomscatm 34466   HLchlt 34553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-poset 15449  df-plt 15461  df-lub 15477  df-glb 15478  df-join 15479  df-meet 15480  df-p0 15542  df-lat 15549  df-clat 15611  df-oposet 34379  df-ol 34381  df-oml 34382  df-covers 34469  df-ats 34470  df-atl 34501  df-cvlat 34525  df-hlat 34554
This theorem is referenced by:  cvrval4N  34616
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