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Theorem lhpm0atN 36150
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 1002 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  ( X  ./\  W )  =  .0.  )
2 simpl 455 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
3 simpr1 1000 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  X  e.  B )
4 simpr2 1001 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  X  =/=  .0.  )
5 hllat 35485 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  Lat )
65ad2antrr 723 . . . . . . . . . 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 36119 . . . . . . . . . . 11  |-  ( W  e.  H  ->  W  e.  B )
109ad2antlr 724 . . . . . . . . . 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 15908 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X ( le
`  K ) W  <-> 
( X  ./\  W
)  =  X ) )
146, 3, 10, 13syl3anc 1226 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  ( X ( le `  K ) W  <->  ( X  ./\ 
W )  =  X ) )
1514biimpa 482 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/=  .0.  /\  ( X  ./\  W )  =  .0.  ) )  /\  X ( le `  K ) W )  ->  ( X  ./\  W )  =  X )
16 simplr3 1038 . . . . . . . 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 432 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  ( X ( le `  K ) W  ->  X  =  .0.  )
)
1918necon3ad 2664 . . . . 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 36144 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X
( le `  K
) W ) )  ->  ( X  ./\  W ) (  <o  `  K
) X )
232, 3, 20, 22syl12anc 1224 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  ( X  ./\  W ) ( 
<o  `  K ) X )
241, 23eqbrtrrd 4461 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/= 
.0.  /\  ( X  ./\ 
W )  =  .0.  ) )  ->  .0.  (  <o  `  K ) X )
25 simpll 751 . . 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 35409 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( X  e.  A  <->  .0.  (  <o  `  K ) X ) )
2925, 3, 28syl2anc 659 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   lecple 14791   meetcmee 15773   0.cp0 15866   Latclat 15874    <o ccvr 35384   Atomscatm 35385   HLchlt 35472   LHypclh 36105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-preset 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-p1 15869  df-lat 15875  df-clat 15937  df-oposet 35298  df-ol 35300  df-oml 35301  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473  df-lhyp 36109
This theorem is referenced by: (None)
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