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Theorem lhpmcvr 30820
Description: The meet of a lattice hyperplane with an element not under it is covered by the element. (Contributed by NM, 7-Dec-2012.)
Hypotheses
Ref Expression
lhpmcvr.b  |-  B  =  ( Base `  K
)
lhpmcvr.l  |-  .<_  =  ( le `  K )
lhpmcvr.m  |-  ./\  =  ( meet `  K )
lhpmcvr.c  |-  C  =  (  <o  `  K )
lhpmcvr.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
lhpmcvr  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( X  ./\  W
) C X )

Proof of Theorem lhpmcvr
StepHypRef Expression
1 hllat 30161 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
21ad2antrr 707 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  K  e.  Lat )
3 simprl 733 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  X  e.  B )
4 lhpmcvr.b . . . . 5  |-  B  =  ( Base `  K
)
5 lhpmcvr.h . . . . 5  |-  H  =  ( LHyp `  K
)
64, 5lhpbase 30795 . . . 4  |-  ( W  e.  H  ->  W  e.  B )
76ad2antlr 708 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  W  e.  B )
8 lhpmcvr.m . . . 4  |-  ./\  =  ( meet `  K )
94, 8latmcom 14504 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  =  ( W 
./\  X ) )
102, 3, 7, 9syl3anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( X  ./\  W
)  =  ( W 
./\  X ) )
11 eqid 2436 . . . . . 6  |-  ( 1.
`  K )  =  ( 1. `  K
)
12 lhpmcvr.c . . . . . 6  |-  C  =  (  <o  `  K )
1311, 12, 5lhp1cvr 30796 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W C ( 1.
`  K ) )
1413adantr 452 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  W C ( 1. `  K ) )
15 lhpmcvr.l . . . . 5  |-  .<_  =  ( le `  K )
16 eqid 2436 . . . . 5  |-  ( join `  K )  =  (
join `  K )
174, 15, 16, 11, 5lhpj1 30819 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( W ( join `  K ) X )  =  ( 1. `  K ) )
1814, 17breqtrrd 4238 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  W C ( W (
join `  K ) X ) )
19 simpll 731 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  K  e.  HL )
204, 16, 8, 12cvrexch 30217 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  B  /\  X  e.  B )  ->  ( ( W  ./\  X ) C X  <->  W C
( W ( join `  K ) X ) ) )
2119, 7, 3, 20syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( ( W  ./\  X ) C X  <->  W C
( W ( join `  K ) X ) ) )
2218, 21mpbird 224 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( W  ./\  X
) C X )
2310, 22eqbrtrd 4232 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( X  ./\  W
) C X )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   joincjn 14401   meetcmee 14402   1.cp1 14467   Latclat 14474    <o ccvr 30060   HLchlt 30148   LHypclh 30781
This theorem is referenced by:  lhpmcvr2  30821  lhpm0atN  30826
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-lhyp 30785
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