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Theorem lhpmcvr 33053
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 32394 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
21ad2antrr 726 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  K  e.  Lat )
3 simprl 758 . . 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 33028 . . . 4  |-  ( W  e.  H  ->  W  e.  B )
76ad2antlr 727 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  W  e.  B )
8 lhpmcvr.m . . . 4  |-  ./\  =  ( meet `  K )
94, 8latmcom 16031 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  =  ( W 
./\  X ) )
102, 3, 7, 9syl3anc 1232 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( X  ./\  W
)  =  ( W 
./\  X ) )
11 eqid 2404 . . . . . 6  |-  ( 1.
`  K )  =  ( 1. `  K
)
12 lhpmcvr.c . . . . . 6  |-  C  =  (  <o  `  K )
1311, 12, 5lhp1cvr 33029 . . . . 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 2404 . . . . 5  |-  ( join `  K )  =  (
join `  K )
174, 15, 16, 11, 5lhpj1 33052 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( W ( join `  K ) X )  =  ( 1. `  K ) )
1814, 17breqtrrd 4423 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  W C ( W (
join `  K ) X ) )
19 simpll 754 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  K  e.  HL )
204, 16, 8, 12cvrexch 32450 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  B  /\  X  e.  B )  ->  ( ( W  ./\  X ) C X  <->  W C
( W ( join `  K ) X ) ) )
2119, 7, 3, 20syl3anc 1232 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( ( W  ./\  X ) C X  <->  W C
( W ( join `  K ) X ) ) )
2218, 21mpbird 234 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( W  ./\  X
) C X )
2310, 22eqbrtrd 4417 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 186    /\ wa 369    = wceq 1407    e. wcel 1844   class class class wbr 4397   ` cfv 5571  (class class class)co 6280   Basecbs 14843   lecple 14918   joincjn 15899   meetcmee 15900   1.cp1 15994   Latclat 16001    <o ccvr 32293   HLchlt 32381   LHypclh 33014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-preset 15883  df-poset 15901  df-plt 15914  df-lub 15930  df-glb 15931  df-join 15932  df-meet 15933  df-p0 15995  df-p1 15996  df-lat 16002  df-clat 16064  df-oposet 32207  df-ol 32209  df-oml 32210  df-covers 32297  df-ats 32298  df-atl 32329  df-cvlat 32353  df-hlat 32382  df-lhyp 33018
This theorem is referenced by:  lhpmcvr2  33054  lhpm0atN  33059
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