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Theorem cvr2N 30145
Description: Less-than and covers equivalence in a Hilbert lattice. (chcv2 23851 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 30098 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 978 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  ->  K  e.  Lat )
3 simp2 958 . . 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 30024 . . . 4  |-  ( P  e.  A  ->  P  e.  B )
763ad2ant3 980 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  ->  P  e.  B )
8 eqid 2435 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
9 cvr2.s . . . 4  |-  .<  =  ( lt `  K )
10 cvr2.j . . . 4  |-  .\/  =  ( join `  K )
114, 8, 9, 10latnle 14506 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  ( -.  P ( le `  K ) X  <->  X  .<  ( X 
.\/  P ) ) )
122, 3, 7, 11syl3anc 1184 . 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 30144 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  ->  ( -.  P ( le `  K ) X  <->  X C ( X 
.\/  P ) ) )
1512, 14bitr3d 247 1  |-  ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  ->  ( X  .<  ( X  .\/  P )  <->  X C
( X  .\/  P
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   ltcplt 14390   joincjn 14393   Latclat 14466    <o ccvr 29997   Atomscatm 29998   HLchlt 30085
This theorem is referenced by:  cvrval4N  30148
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086
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