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Theorem lhpmcvr 33667
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 33008 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
21ad2antrr 725 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  K  e.  Lat )
3 simprl 755 . . 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 33642 . . . 4  |-  ( W  e.  H  ->  W  e.  B )
76ad2antlr 726 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  W  e.  B )
8 lhpmcvr.m . . . 4  |-  ./\  =  ( meet `  K )
94, 8latmcom 15245 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  =  ( W 
./\  X ) )
102, 3, 7, 9syl3anc 1218 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( X  ./\  W
)  =  ( W 
./\  X ) )
11 eqid 2443 . . . . . 6  |-  ( 1.
`  K )  =  ( 1. `  K
)
12 lhpmcvr.c . . . . . 6  |-  C  =  (  <o  `  K )
1311, 12, 5lhp1cvr 33643 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W C ( 1.
`  K ) )
1413adantr 465 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  W C ( 1. `  K ) )
15 lhpmcvr.l . . . . 5  |-  .<_  =  ( le `  K )
16 eqid 2443 . . . . 5  |-  ( join `  K )  =  (
join `  K )
174, 15, 16, 11, 5lhpj1 33666 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( W ( join `  K ) X )  =  ( 1. `  K ) )
1814, 17breqtrrd 4318 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  W C ( W (
join `  K ) X ) )
19 simpll 753 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  K  e.  HL )
204, 16, 8, 12cvrexch 33064 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  B  /\  X  e.  B )  ->  ( ( W  ./\  X ) C X  <->  W C
( W ( join `  K ) X ) ) )
2119, 7, 3, 20syl3anc 1218 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( ( W  ./\  X ) C X  <->  W C
( W ( join `  K ) X ) ) )
2218, 21mpbird 232 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( W  ./\  X
) C X )
2310, 22eqbrtrd 4312 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( X  ./\  W
) C X )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   Basecbs 14174   lecple 14245   joincjn 15114   meetcmee 15115   1.cp1 15208   Latclat 15215    <o ccvr 32907   HLchlt 32995   LHypclh 33628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-poset 15116  df-plt 15128  df-lub 15144  df-glb 15145  df-join 15146  df-meet 15147  df-p0 15209  df-p1 15210  df-lat 15216  df-clat 15278  df-oposet 32821  df-ol 32823  df-oml 32824  df-covers 32911  df-ats 32912  df-atl 32943  df-cvlat 32967  df-hlat 32996  df-lhyp 33632
This theorem is referenced by:  lhpmcvr2  33668  lhpm0atN  33673
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