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Theorem lhpmcvr 34819
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 34160 . . . 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 34794 . . . 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 15558 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  =  ( W 
./\  X ) )
102, 3, 7, 9syl3anc 1228 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( X  ./\  W
)  =  ( W 
./\  X ) )
11 eqid 2467 . . . . . 6  |-  ( 1.
`  K )  =  ( 1. `  K
)
12 lhpmcvr.c . . . . . 6  |-  C  =  (  <o  `  K )
1311, 12, 5lhp1cvr 34795 . . . . 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 2467 . . . . 5  |-  ( join `  K )  =  (
join `  K )
174, 15, 16, 11, 5lhpj1 34818 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( W ( join `  K ) X )  =  ( 1. `  K ) )
1814, 17breqtrrd 4473 . . 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 34216 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  B  /\  X  e.  B )  ->  ( ( W  ./\  X ) C X  <->  W C
( W ( join `  K ) X ) ) )
2119, 7, 3, 20syl3anc 1228 . . 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 4467 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 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14486   lecple 14558   joincjn 15427   meetcmee 15428   1.cp1 15521   Latclat 15528    <o ccvr 34059   HLchlt 34147   LHypclh 34780
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-p1 15523  df-lat 15529  df-clat 15591  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-lhyp 34784
This theorem is referenced by:  lhpmcvr2  34820  lhpm0atN  34825
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