Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lhpm0atN Unicode version

Theorem lhpm0atN 30840
Description: If the meet of a lattice hyperplane with a nonzero element is zero, the element is an atom. (Contributed by NM, 28-Apr-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
lhpm0at.b  |-  B  =  ( Base `  K
)
lhpm0at.m  |-  ./\  =  ( meet `  K )
lhpm0at.o  |-  .0.  =  ( 0. `  K )
lhpm0at.a  |-  A  =  ( Atoms `  K )
lhpm0at.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
lhpm0atN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  X  e.  A )

Proof of Theorem lhpm0atN
StepHypRef Expression
1 simpr3 963 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  ( X  ./\  W )  =  .0.  )
2 simpl 443 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
3 simpr1 961 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  X  e.  B )
4 simpr2 962 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  X  =/=  .0.  )
5 hllat 30175 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  Lat )
65ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  K  e.  Lat )
7 lhpm0at.b . . . . . . . . . . . 12  |-  B  =  ( Base `  K
)
8 lhpm0at.h . . . . . . . . . . . 12  |-  H  =  ( LHyp `  K
)
97, 8lhpbase 30809 . . . . . . . . . . 11  |-  ( W  e.  H  ->  W  e.  B )
109ad2antlr 707 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  W  e.  B )
11 eqid 2296 . . . . . . . . . . 11  |-  ( le
`  K )  =  ( le `  K
)
12 lhpm0at.m . . . . . . . . . . 11  |-  ./\  =  ( meet `  K )
137, 11, 12latleeqm1 14201 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X ( le
`  K ) W  <-> 
( X  ./\  W
)  =  X ) )
146, 3, 10, 13syl3anc 1182 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  ( X ( le `  K ) W  <->  ( X  ./\ 
W )  =  X ) )
1514biimpa 470 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/=  .0.  /\  ( X  ./\  W )  =  .0.  ) )  /\  X ( le `  K ) W )  ->  ( X  ./\  W )  =  X )
16 simplr3 999 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/=  .0.  /\  ( X  ./\  W )  =  .0.  ) )  /\  X ( le `  K ) W )  ->  ( X  ./\  W )  =  .0.  )
1715, 16eqtr3d 2330 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/=  .0.  /\  ( X  ./\  W )  =  .0.  ) )  /\  X ( le `  K ) W )  ->  X  =  .0.  )
1817ex 423 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  ( X ( le `  K ) W  ->  X  =  .0.  )
)
1918necon3ad 2495 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  ( X  =/=  .0.  ->  -.  X ( le `  K ) W ) )
204, 19mpd 14 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  -.  X ( le `  K ) W )
21 eqid 2296 . . . . 5  |-  (  <o  `  K )  =  ( 
<o  `  K )
227, 11, 12, 21, 8lhpmcvr 30834 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X
( le `  K
) W ) )  ->  ( X  ./\  W ) (  <o  `  K
) X )
232, 3, 20, 22syl12anc 1180 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  ( X  ./\  W ) ( 
<o  `  K ) X )
241, 23eqbrtrrd 4061 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  .0.  (  <o  `  K ) X )
25 simpll 730 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  K  e.  HL )
26 lhpm0at.o . . . 4  |-  .0.  =  ( 0. `  K )
27 lhpm0at.a . . . 4  |-  A  =  ( Atoms `  K )
287, 26, 21, 27isat2 30099 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( X  e.  A  <->  .0.  (  <o  `  K ) X ) )
2925, 3, 28syl2anc 642 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  ( X  e.  A  <->  .0.  (  <o  `  K ) X ) )
3024, 29mpbird 223 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  X  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   meetcmee 14095   0.cp0 14159   Latclat 14167    <o ccvr 30074   Atomscatm 30075   HLchlt 30162   LHypclh 30795
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-lhyp 30799
  Copyright terms: Public domain W3C validator