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Theorem lhpm0atN 34333
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 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → 𝑋𝐴)

Proof of Theorem lhpm0atN
StepHypRef Expression
1 simpr3 1062 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → (𝑋 𝑊) = 0 )
2 simpl 472 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3 simpr1 1060 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → 𝑋𝐵)
4 simpr2 1061 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → 𝑋0 )
5 hllat 33668 . . . . . . . . . . 11 (𝐾 ∈ HL → 𝐾 ∈ Lat)
65ad2antrr 758 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → 𝐾 ∈ Lat)
7 lhpm0at.b . . . . . . . . . . . 12 𝐵 = (Base‘𝐾)
8 lhpm0at.h . . . . . . . . . . . 12 𝐻 = (LHyp‘𝐾)
97, 8lhpbase 34302 . . . . . . . . . . 11 (𝑊𝐻𝑊𝐵)
109ad2antlr 759 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → 𝑊𝐵)
11 eqid 2610 . . . . . . . . . . 11 (le‘𝐾) = (le‘𝐾)
12 lhpm0at.m . . . . . . . . . . 11 = (meet‘𝐾)
137, 11, 12latleeqm1 16902 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋(le‘𝐾)𝑊 ↔ (𝑋 𝑊) = 𝑋))
146, 3, 10, 13syl3anc 1318 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → (𝑋(le‘𝐾)𝑊 ↔ (𝑋 𝑊) = 𝑋))
1514biimpa 500 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) ∧ 𝑋(le‘𝐾)𝑊) → (𝑋 𝑊) = 𝑋)
16 simplr3 1098 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) ∧ 𝑋(le‘𝐾)𝑊) → (𝑋 𝑊) = 0 )
1715, 16eqtr3d 2646 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) ∧ 𝑋(le‘𝐾)𝑊) → 𝑋 = 0 )
1817ex 449 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → (𝑋(le‘𝐾)𝑊𝑋 = 0 ))
1918necon3ad 2795 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → (𝑋0 → ¬ 𝑋(le‘𝐾)𝑊))
204, 19mpd 15 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → ¬ 𝑋(le‘𝐾)𝑊)
21 eqid 2610 . . . . 5 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
227, 11, 12, 21, 8lhpmcvr 34327 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋(le‘𝐾)𝑊)) → (𝑋 𝑊)( ⋖ ‘𝐾)𝑋)
232, 3, 20, 22syl12anc 1316 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → (𝑋 𝑊)( ⋖ ‘𝐾)𝑋)
241, 23eqbrtrrd 4607 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → 0 ( ⋖ ‘𝐾)𝑋)
25 simpll 786 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → 𝐾 ∈ HL)
26 lhpm0at.o . . . 4 0 = (0.‘𝐾)
27 lhpm0at.a . . . 4 𝐴 = (Atoms‘𝐾)
287, 26, 21, 27isat2 33592 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑋𝐴0 ( ⋖ ‘𝐾)𝑋))
2925, 3, 28syl2anc 691 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → (𝑋𝐴0 ( ⋖ ‘𝐾)𝑋))
3024, 29mpbird 246 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → 𝑋𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780   class class class wbr 4583  cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  meetcmee 16768  0.cp0 16860  Latclat 16868  ccvr 33567  Atomscatm 33568  HLchlt 33655  LHypclh 34288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-p1 16863  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-lhyp 34292
This theorem is referenced by: (None)
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