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Theorem atnlej2 35181
Description: If an atom is not less than or equal to the join of two others, it is not equal to either. (This also holds for non-atoms, but in this form it is convenient.) (Contributed by NM, 8-Jan-2012.)
Hypotheses
Ref Expression
atnlej.l = (le‘𝐾)
atnlej.j = (join‘𝐾)
atnlej.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
atnlej2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑃 (𝑄 𝑅)) → 𝑃𝑅)

Proof of Theorem atnlej2
StepHypRef Expression
1 hllat 35165 . . 3 (𝐾 ∈ HL → 𝐾 ∈ Lat)
213ad2ant1 1126 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑃 (𝑄 𝑅)) → 𝐾 ∈ Lat)
3 simp21 1247 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑃 (𝑄 𝑅)) → 𝑃𝐴)
4 eqid 2770 . . . 4 (Base‘𝐾) = (Base‘𝐾)
5 atnlej.a . . . 4 𝐴 = (Atoms‘𝐾)
64, 5atbase 35091 . . 3 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
73, 6syl 17 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑃 (𝑄 𝑅)) → 𝑃 ∈ (Base‘𝐾))
8 simp22 1248 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑃 (𝑄 𝑅)) → 𝑄𝐴)
94, 5atbase 35091 . . 3 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
108, 9syl 17 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑃 (𝑄 𝑅)) → 𝑄 ∈ (Base‘𝐾))
11 simp23 1249 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑃 (𝑄 𝑅)) → 𝑅𝐴)
124, 5atbase 35091 . . 3 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
1311, 12syl 17 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑃 (𝑄 𝑅)) → 𝑅 ∈ (Base‘𝐾))
14 simp3 1131 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑃 (𝑄 𝑅)) → ¬ 𝑃 (𝑄 𝑅))
15 atnlej.l . . 3 = (le‘𝐾)
16 atnlej.j . . 3 = (join‘𝐾)
174, 15, 16latnlej1r 17277 . 2 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) ∧ ¬ 𝑃 (𝑄 𝑅)) → 𝑃𝑅)
182, 7, 10, 13, 14, 17syl131anc 1488 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑃 (𝑄 𝑅)) → 𝑃𝑅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  w3a 1070   = wceq 1630  wcel 2144  wne 2942   class class class wbr 4784  cfv 6031  (class class class)co 6792  Basecbs 16063  lecple 16155  joincjn 17151  Latclat 17252  Atomscatm 35065  HLchlt 35152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-lub 17181  df-join 17183  df-lat 17253  df-ats 35069  df-atl 35100  df-cvlat 35124  df-hlat 35153
This theorem is referenced by:  lplnri2N  35355  lplnri3N  35356  lplnexllnN  35365  dalem41  35514  paddasslem2  35622  4atexlemc  35870
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