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Theorem lhpmcvr 30030
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 29371 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
21ad2antrr 706 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  K  e.  Lat )
3 simprl 732 . . 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 30005 . . . 4  |-  ( W  e.  H  ->  W  e.  B )
76ad2antlr 707 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  W  e.  B )
8 lhpmcvr.m . . . 4  |-  ./\  =  ( meet `  K )
94, 8latmcom 14230 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  =  ( W 
./\  X ) )
102, 3, 7, 9syl3anc 1182 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( X  ./\  W
)  =  ( W 
./\  X ) )
11 eqid 2316 . . . . . 6  |-  ( 1.
`  K )  =  ( 1. `  K
)
12 lhpmcvr.c . . . . . 6  |-  C  =  (  <o  `  K )
1311, 12, 5lhp1cvr 30006 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W C ( 1.
`  K ) )
1413adantr 451 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  W C ( 1. `  K ) )
15 lhpmcvr.l . . . . 5  |-  .<_  =  ( le `  K )
16 eqid 2316 . . . . 5  |-  ( join `  K )  =  (
join `  K )
174, 15, 16, 11, 5lhpj1 30029 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( W ( join `  K ) X )  =  ( 1. `  K ) )
1814, 17breqtrrd 4086 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  W C ( W (
join `  K ) X ) )
19 simpll 730 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  K  e.  HL )
204, 16, 8, 12cvrexch 29427 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  B  /\  X  e.  B )  ->  ( ( W  ./\  X ) C X  <->  W C
( W ( join `  K ) X ) ) )
2119, 7, 3, 20syl3anc 1182 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( ( W  ./\  X ) C X  <->  W C
( W ( join `  K ) X ) ) )
2218, 21mpbird 223 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( W  ./\  X
) C X )
2310, 22eqbrtrd 4080 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 176    /\ wa 358    = wceq 1633    e. wcel 1701   class class class wbr 4060   ` cfv 5292  (class class class)co 5900   Basecbs 13195   lecple 13262   joincjn 14127   meetcmee 14128   1.cp1 14193   Latclat 14200    <o ccvr 29270   HLchlt 29358   LHypclh 29991
This theorem is referenced by:  lhpmcvr2  30031  lhpm0atN  30036
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-undef 6340  df-riota 6346  df-poset 14129  df-plt 14141  df-lub 14157  df-glb 14158  df-join 14159  df-meet 14160  df-p0 14194  df-p1 14195  df-lat 14201  df-clat 14263  df-oposet 29184  df-ol 29186  df-oml 29187  df-covers 29274  df-ats 29275  df-atl 29306  df-cvlat 29330  df-hlat 29359  df-lhyp 29995
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