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Theorem lhpm0atN 34012
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 996 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  ( X  ./\  W )  =  .0.  )
2 simpl 457 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
3 simpr1 994 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  X  e.  B )
4 simpr2 995 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  X  =/=  .0.  )
5 hllat 33347 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  Lat )
65ad2antrr 725 . . . . . . . . . 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 33981 . . . . . . . . . . 11  |-  ( W  e.  H  ->  W  e.  B )
109ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  W  e.  B )
11 eqid 2454 . . . . . . . . . . 11  |-  ( le
`  K )  =  ( le `  K
)
12 lhpm0at.m . . . . . . . . . . 11  |-  ./\  =  ( meet `  K )
137, 11, 12latleeqm1 15369 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X ( le
`  K ) W  <-> 
( X  ./\  W
)  =  X ) )
146, 3, 10, 13syl3anc 1219 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  ( X ( le `  K ) W  <->  ( X  ./\ 
W )  =  X ) )
1514biimpa 484 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/=  .0.  /\  ( X  ./\  W )  =  .0.  ) )  /\  X ( le `  K ) W )  ->  ( X  ./\  W )  =  X )
16 simplr3 1032 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/=  .0.  /\  ( X  ./\  W )  =  .0.  ) )  /\  X ( le `  K ) W )  ->  ( X  ./\  W )  =  .0.  )
1715, 16eqtr3d 2497 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/=  .0.  /\  ( X  ./\  W )  =  .0.  ) )  /\  X ( le `  K ) W )  ->  X  =  .0.  )
1817ex 434 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  ( X ( le `  K ) W  ->  X  =  .0.  )
)
1918necon3ad 2662 . . . . 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 2454 . . . . 5  |-  (  <o  `  K )  =  ( 
<o  `  K )
227, 11, 12, 21, 8lhpmcvr 34006 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X
( le `  K
) W ) )  ->  ( X  ./\  W ) (  <o  `  K
) X )
232, 3, 20, 22syl12anc 1217 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  ( X  ./\  W ) ( 
<o  `  K ) X )
241, 23eqbrtrrd 4423 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  .0.  (  <o  `  K ) X )
25 simpll 753 . . 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 33271 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( X  e.  A  <->  .0.  (  <o  `  K ) X ) )
2925, 3, 28syl2anc 661 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  ( X  e.  A  <->  .0.  (  <o  `  K ) X ) )
3024, 29mpbird 232 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 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   Basecbs 14293   lecple 14365   meetcmee 15235   0.cp0 15327   Latclat 15335    <o ccvr 33246   Atomscatm 33247   HLchlt 33334   LHypclh 33967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-poset 15236  df-plt 15248  df-lub 15264  df-glb 15265  df-join 15266  df-meet 15267  df-p0 15329  df-p1 15330  df-lat 15336  df-clat 15398  df-oposet 33160  df-ol 33162  df-oml 33163  df-covers 33250  df-ats 33251  df-atl 33282  df-cvlat 33306  df-hlat 33335  df-lhyp 33971
This theorem is referenced by: (None)
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