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Theorem lhpm0atN 37284
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 1193 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → (𝑋 𝑊) = 0 )
2 simpl 486 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3 simpr1 1191 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → 𝑋𝐵)
4 simpr2 1192 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → 𝑋0 )
5 hllat 36618 . . . . . . . . . . 11 (𝐾 ∈ HL → 𝐾 ∈ Lat)
65ad2antrr 725 . . . . . . . . . 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 37253 . . . . . . . . . . 11 (𝑊𝐻𝑊𝐵)
109ad2antlr 726 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → 𝑊𝐵)
11 eqid 2822 . . . . . . . . . . 11 (le‘𝐾) = (le‘𝐾)
12 lhpm0at.m . . . . . . . . . . 11 = (meet‘𝐾)
137, 11, 12latleeqm1 17680 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋(le‘𝐾)𝑊 ↔ (𝑋 𝑊) = 𝑋))
146, 3, 10, 13syl3anc 1368 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → (𝑋(le‘𝐾)𝑊 ↔ (𝑋 𝑊) = 𝑋))
1514biimpa 480 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) ∧ 𝑋(le‘𝐾)𝑊) → (𝑋 𝑊) = 𝑋)
16 simplr3 1214 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) ∧ 𝑋(le‘𝐾)𝑊) → (𝑋 𝑊) = 0 )
1715, 16eqtr3d 2859 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) ∧ 𝑋(le‘𝐾)𝑊) → 𝑋 = 0 )
1817ex 416 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → (𝑋(le‘𝐾)𝑊𝑋 = 0 ))
1918necon3ad 3024 . . . . 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 2822 . . . . 5 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
227, 11, 12, 21, 8lhpmcvr 37278 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋(le‘𝐾)𝑊)) → (𝑋 𝑊)( ⋖ ‘𝐾)𝑋)
232, 3, 20, 22syl12anc 835 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → (𝑋 𝑊)( ⋖ ‘𝐾)𝑋)
241, 23eqbrtrrd 5066 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → 0 ( ⋖ ‘𝐾)𝑋)
25 simpll 766 . . 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 36542 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑋𝐴0 ( ⋖ ‘𝐾)𝑋))
2925, 3, 28syl2anc 587 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → (𝑋𝐴0 ( ⋖ ‘𝐾)𝑋))
3024, 29mpbird 260 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → 𝑋𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2114  wne 3011   class class class wbr 5042  cfv 6334  (class class class)co 7140  Basecbs 16474  lecple 16563  meetcmee 17546  0.cp0 17638  Latclat 17646  ccvr 36517  Atomscatm 36518  HLchlt 36605  LHypclh 37239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-proset 17529  df-poset 17547  df-plt 17559  df-lub 17575  df-glb 17576  df-join 17577  df-meet 17578  df-p0 17640  df-p1 17641  df-lat 17647  df-clat 17709  df-oposet 36431  df-ol 36433  df-oml 36434  df-covers 36521  df-ats 36522  df-atl 36553  df-cvlat 36577  df-hlat 36606  df-lhyp 37243
This theorem is referenced by: (None)
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