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Theorem 2atmat 34313
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 1089 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝐾 ∈ HL)
2 hllat 34116 . . . . . 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 2626 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
5 2atmat.j . . . . . . 7 = (join‘𝐾)
6 2atmat.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
74, 5, 6hlatjcl 34119 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
873ad2ant1 1080 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → (𝑃 𝑄) ∈ (Base‘𝐾))
9 simp21 1092 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑅𝐴)
104, 6atbase 34042 . . . . . 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 1093 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑆𝐴)
134, 6atbase 34042 . . . . . 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 17011 . . . . 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 1325 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → (((𝑃 𝑄) 𝑅) 𝑆) = ((𝑃 𝑄) (𝑅 𝑆)))
17 simp33 1097 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑆 ((𝑃 𝑄) 𝑅))
184, 5latjcl 16967 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾))
193, 8, 11, 18syl3anc 1323 . . . . . 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 16980 . . . . . 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 1323 . . . . 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 2662 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → ((𝑃 𝑄) (𝑅 𝑆)) = ((𝑃 𝑄) 𝑅))
25 simp23 1094 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑃𝑄)
26 simp32 1096 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → ¬ 𝑅 (𝑃 𝑄))
27 simp12 1090 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑃𝐴)
28 simp13 1091 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑄𝐴)
29 eqid 2626 . . . . . 6 (LPlanes‘𝐾) = (LPlanes‘𝐾)
3020, 5, 6, 29islpln2a 34300 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾) ↔ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))))
311, 27, 28, 9, 30syl13anc 1325 . . . 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 956 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → ((𝑃 𝑄) 𝑅) ∈ (LPlanes‘𝐾))
3324, 32eqeltrd 2704 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → ((𝑃 𝑄) (𝑅 𝑆)) ∈ (LPlanes‘𝐾))
34 eqid 2626 . . . . 5 (LLines‘𝐾) = (LLines‘𝐾)
355, 6, 34llni2 34264 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → (𝑃 𝑄) ∈ (LLines‘𝐾))
361, 27, 28, 25, 35syl31anc 1326 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → (𝑃 𝑄) ∈ (LLines‘𝐾))
37 simp31 1095 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑃𝑄) ∧ (𝑅𝑆 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑅𝑆)
385, 6, 34llni2 34264 . . . 4 (((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) ∧ 𝑅𝑆) → (𝑅 𝑆) ∈ (LLines‘𝐾))
391, 9, 12, 37, 38syl31anc 1326 . . 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 34312 . . 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 1323 . 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 384  w3a 1036   = wceq 1480  wcel 1992  wne 2796   class class class wbr 4618  cfv 5850  (class class class)co 6605  Basecbs 15776  lecple 15864  joincjn 16860  meetcmee 16861  Latclat 16961  Atomscatm 34016  HLchlt 34103  LLinesclln 34243  LPlanesclpl 34244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-preset 16844  df-poset 16862  df-plt 16874  df-lub 16890  df-glb 16891  df-join 16892  df-meet 16893  df-p0 16955  df-lat 16962  df-clat 17024  df-oposet 33929  df-ol 33931  df-oml 33932  df-covers 34019  df-ats 34020  df-atl 34051  df-cvlat 34075  df-hlat 34104  df-llines 34250  df-lplanes 34251
This theorem is referenced by:  4atexlemc  34821
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