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Theorem lhpat4N 30903
Description: Property of an atom under a co-atom. (Contributed by NM, 24-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
lhpat.l  |-  .<_  =  ( le `  K )
lhpat.j  |-  .\/  =  ( join `  K )
lhpat.m  |-  ./\  =  ( meet `  K )
lhpat.a  |-  A  =  ( Atoms `  K )
lhpat.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
lhpat4N  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( U  e.  A  /\  U  .<_  W ) )  ->  ( ( P 
.\/  U )  ./\  W )  =  U )

Proof of Theorem lhpat4N
StepHypRef Expression
1 simp1 958 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( U  e.  A  /\  U  .<_  W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp2 959 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( U  e.  A  /\  U  .<_  W ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
3 simp3l 986 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( U  e.  A  /\  U  .<_  W ) )  ->  U  e.  A
)
4 eqid 2438 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
5 lhpat.a . . . 4  |-  A  =  ( Atoms `  K )
64, 5atbase 30149 . . 3  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
73, 6syl 16 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( U  e.  A  /\  U  .<_  W ) )  ->  U  e.  (
Base `  K )
)
8 simp3r 987 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( U  e.  A  /\  U  .<_  W ) )  ->  U  .<_  W )
9 lhpat.l . . 3  |-  .<_  =  ( le `  K )
10 lhpat.j . . 3  |-  .\/  =  ( join `  K )
11 lhpat.m . . 3  |-  ./\  =  ( meet `  K )
12 lhpat.h . . 3  |-  H  =  ( LHyp `  K
)
134, 9, 10, 11, 5, 12lhple 30901 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( U  e.  ( Base `  K )  /\  U  .<_  W ) )  -> 
( ( P  .\/  U )  ./\  W )  =  U )
141, 2, 7, 8, 13syl112anc 1189 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( U  e.  A  /\  U  .<_  W ) )  ->  ( ( P 
.\/  U )  ./\  W )  =  U )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403   meetcmee 14404   Atomscatm 30123   HLchlt 30210   LHypclh 30843
This theorem is referenced by:  cdlemm10N  31978
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-p1 14471  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-psubsp 30362  df-pmap 30363  df-padd 30655  df-lhyp 30847
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