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Theorem atnlej1 35186
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
atnlej1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑃 (𝑄 𝑅)) → 𝑃𝑄)

Proof of Theorem atnlej1
StepHypRef Expression
1 hllat 35170 . . 3 (𝐾 ∈ HL → 𝐾 ∈ Lat)
213ad2ant1 1127 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑃 (𝑄 𝑅)) → 𝐾 ∈ Lat)
3 simp21 1248 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑃 (𝑄 𝑅)) → 𝑃𝐴)
4 eqid 2771 . . . 4 (Base‘𝐾) = (Base‘𝐾)
5 atnlej.a . . . 4 𝐴 = (Atoms‘𝐾)
64, 5atbase 35096 . . 3 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
73, 6syl 17 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑃 (𝑄 𝑅)) → 𝑃 ∈ (Base‘𝐾))
8 simp22 1249 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑃 (𝑄 𝑅)) → 𝑄𝐴)
94, 5atbase 35096 . . 3 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
108, 9syl 17 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑃 (𝑄 𝑅)) → 𝑄 ∈ (Base‘𝐾))
11 simp23 1250 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑃 (𝑄 𝑅)) → 𝑅𝐴)
124, 5atbase 35096 . . 3 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
1311, 12syl 17 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑃 (𝑄 𝑅)) → 𝑅 ∈ (Base‘𝐾))
14 simp3 1132 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑃 (𝑄 𝑅)) → ¬ 𝑃 (𝑄 𝑅))
15 atnlej.l . . 3 = (le‘𝐾)
16 atnlej.j . . 3 = (join‘𝐾)
174, 15, 16latnlej1l 17277 . 2 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) ∧ ¬ 𝑃 (𝑄 𝑅)) → 𝑃𝑄)
182, 7, 10, 13, 14, 17syl131anc 1489 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑃 (𝑄 𝑅)) → 𝑃𝑄)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  w3a 1071   = wceq 1631  wcel 2145  wne 2943   class class class wbr 4787  cfv 6030  (class class class)co 6796  Basecbs 16064  lecple 16156  joincjn 17152  Latclat 17253  Atomscatm 35070  HLchlt 35157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-lub 17182  df-join 17184  df-lat 17254  df-ats 35074  df-atl 35105  df-cvlat 35129  df-hlat 35158
This theorem is referenced by:  4atlem0be  35402  dalem5  35474  dalem-cly  35478  4atexlemex6  35881  cdleme00a  36017  cdleme21a  36133  cdleme21b  36134  cdleme21c  36135
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