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Theorem lhpm0atN 33506
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 1013 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  ( X  ./\  W )  =  .0.  )
2 simpl 458 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
3 simpr1 1011 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  X  e.  B )
4 simpr2 1012 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  X  =/=  .0.  )
5 hllat 32841 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  Lat )
65ad2antrr 730 . . . . . . . . . 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 33475 . . . . . . . . . . 11  |-  ( W  e.  H  ->  W  e.  B )
109ad2antlr 731 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  W  e.  B )
11 eqid 2428 . . . . . . . . . . 11  |-  ( le
`  K )  =  ( le `  K
)
12 lhpm0at.m . . . . . . . . . . 11  |-  ./\  =  ( meet `  K )
137, 11, 12latleeqm1 16268 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X ( le
`  K ) W  <-> 
( X  ./\  W
)  =  X ) )
146, 3, 10, 13syl3anc 1264 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  ( X ( le `  K ) W  <->  ( X  ./\ 
W )  =  X ) )
1514biimpa 486 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/=  .0.  /\  ( X  ./\  W )  =  .0.  ) )  /\  X ( le `  K ) W )  ->  ( X  ./\  W )  =  X )
16 simplr3 1049 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/=  .0.  /\  ( X  ./\  W )  =  .0.  ) )  /\  X ( le `  K ) W )  ->  ( X  ./\  W )  =  .0.  )
1715, 16eqtr3d 2464 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/=  .0.  /\  ( X  ./\  W )  =  .0.  ) )  /\  X ( le `  K ) W )  ->  X  =  .0.  )
1817ex 435 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  ( X ( le `  K ) W  ->  X  =  .0.  )
)
1918necon3ad 2614 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  ( X  =/=  .0.  ->  -.  X ( le `  K ) W ) )
204, 19mpd 15 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  -.  X ( le `  K ) W )
21 eqid 2428 . . . . 5  |-  (  <o  `  K )  =  ( 
<o  `  K )
227, 11, 12, 21, 8lhpmcvr 33500 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X
( le `  K
) W ) )  ->  ( X  ./\  W ) (  <o  `  K
) X )
232, 3, 20, 22syl12anc 1262 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  ( X  ./\  W ) ( 
<o  `  K ) X )
241, 23eqbrtrrd 4389 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  .0.  (  <o  `  K ) X )
25 simpll 758 . . 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 32765 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( X  e.  A  <->  .0.  (  <o  `  K ) X ) )
2925, 3, 28syl2anc 665 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  ( X  e.  A  <->  .0.  (  <o  `  K ) X ) )
3024, 29mpbird 235 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  X  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2599   class class class wbr 4366   ` cfv 5544  (class class class)co 6249   Basecbs 15064   lecple 15140   meetcmee 16133   0.cp0 16226   Latclat 16234    <o ccvr 32740   Atomscatm 32741   HLchlt 32828   LHypclh 33461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-preset 16116  df-poset 16134  df-plt 16147  df-lub 16163  df-glb 16164  df-join 16165  df-meet 16166  df-p0 16228  df-p1 16229  df-lat 16235  df-clat 16297  df-oposet 32654  df-ol 32656  df-oml 32657  df-covers 32744  df-ats 32745  df-atl 32776  df-cvlat 32800  df-hlat 32829  df-lhyp 33465
This theorem is referenced by: (None)
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