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Theorem 2atmat 35362
Description: The meet of two intersecting lines (expressed as joins of atoms) is an atom. (Contributed by NM, 21-Nov-2012.)
Hypotheses
Ref Expression
2atmat.l = (le‘𝐾)
2atmat.j = (join‘𝐾)
2atmat.m = (meet‘𝐾)
2atmat.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
2atmat (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → ((𝑃 𝑄) (𝑅 𝑆)) ∈ 𝐴)

Proof of Theorem 2atmat
StepHypRef Expression
1 simp11 1244 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝐾 ∈ HL)
2 hllat 35165 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
31, 2syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝐾 ∈ Lat)
4 eqid 2770 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
5 2atmat.j . . . . . . 7 = (join‘𝐾)
6 2atmat.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
74, 5, 6hlatjcl 35168 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
873ad2ant1 1126 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → (𝑃 𝑄) ∈ (Base‘𝐾))
9 simp21 1247 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑅𝐴)
104, 6atbase 35091 . . . . . 6 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
119, 10syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑅 ∈ (Base‘𝐾))
12 simp22 1248 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑆𝐴)
134, 6atbase 35091 . . . . . 6 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
1412, 13syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑆 ∈ (Base‘𝐾))
154, 5latjass 17302 . . . . 5 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾))) → (((𝑃 𝑄) 𝑅) 𝑆) = ((𝑃 𝑄) (𝑅 𝑆)))
163, 8, 11, 14, 15syl13anc 1477 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → (((𝑃 𝑄) 𝑅) 𝑆) = ((𝑃 𝑄) (𝑅 𝑆)))
17 simp33 1252 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑆 ((𝑃 𝑄) 𝑅))
184, 5latjcl 17258 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾))
193, 8, 11, 18syl3anc 1475 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾))
20 2atmat.l . . . . . . 7 = (le‘𝐾)
214, 20, 5latleeqj2 17271 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑆 ∈ (Base‘𝐾) ∧ ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾)) → (𝑆 ((𝑃 𝑄) 𝑅) ↔ (((𝑃 𝑄) 𝑅) 𝑆) = ((𝑃 𝑄) 𝑅)))
223, 14, 19, 21syl3anc 1475 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → (𝑆 ((𝑃 𝑄) 𝑅) ↔ (((𝑃 𝑄) 𝑅) 𝑆) = ((𝑃 𝑄) 𝑅)))
2317, 22mpbid 222 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → (((𝑃 𝑄) 𝑅) 𝑆) = ((𝑃 𝑄) 𝑅))
2416, 23eqtr3d 2806 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → ((𝑃 𝑄) (𝑅 𝑆)) = ((𝑃 𝑄) 𝑅))
25 simp23 1249 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑃𝑄)
26 simp32 1251 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → ¬ 𝑅 (𝑃 𝑄))
27 simp12 1245 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑃𝐴)
28 simp13 1246 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑄𝐴)
29 eqid 2770 . . . . . 6 (LPlanes‘𝐾) = (LPlanes‘𝐾)
3020, 5, 6, 29islpln2a 35349 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾) ↔ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))))
311, 27, 28, 9, 30syl13anc 1477 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → (((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾) ↔ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))))
3225, 26, 31mpbir2and 684 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → ((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾))
3324, 32eqeltrd 2849 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → ((𝑃 𝑄) (𝑅 𝑆)) ∈ (LPlanes‘𝐾))
34 eqid 2770 . . . . 5 (LLines‘𝐾) = (LLines‘𝐾)
355, 6, 34llni2 35313 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → (𝑃 𝑄) ∈ (LLines‘𝐾))
361, 27, 28, 25, 35syl31anc 1478 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → (𝑃 𝑄) ∈ (LLines‘𝐾))
37 simp31 1250 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑅𝑆)
385, 6, 34llni2 35313 . . . 4 (((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) ∧ 𝑅𝑆) → (𝑅 𝑆) ∈ (LLines‘𝐾))
391, 9, 12, 37, 38syl31anc 1478 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → (𝑅 𝑆) ∈ (LLines‘𝐾))
40 2atmat.m . . . 4 = (meet‘𝐾)
415, 40, 6, 34, 292llnmj 35361 . . 3 ((𝐾 ∈ HL ∧ (𝑃 𝑄) ∈ (LLines‘𝐾) ∧ (𝑅 𝑆) ∈ (LLines‘𝐾)) → (((𝑃 𝑄) (𝑅 𝑆)) ∈ 𝐴 ↔ ((𝑃 𝑄) (𝑅 𝑆)) ∈ (LPlanes‘𝐾)))
421, 36, 39, 41syl3anc 1475 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → (((𝑃 𝑄) (𝑅 𝑆)) ∈ 𝐴 ↔ ((𝑃 𝑄) (𝑅 𝑆)) ∈ (LPlanes‘𝐾)))
4333, 42mpbird 247 1 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → ((𝑃 𝑄) (𝑅 𝑆)) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  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  meetcmee 17152  Latclat 17252  Atomscatm 35065  HLchlt 35152  LLinesclln 35292  LPlanesclpl 35293
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-preset 17135  df-poset 17153  df-plt 17165  df-lub 17181  df-glb 17182  df-join 17183  df-meet 17184  df-p0 17246  df-lat 17253  df-clat 17315  df-oposet 34978  df-ol 34980  df-oml 34981  df-covers 35068  df-ats 35069  df-atl 35100  df-cvlat 35124  df-hlat 35153  df-llines 35299  df-lplanes 35300
This theorem is referenced by:  4atexlemc  35870
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