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Theorem lplni2 36553
Description: The join of 3 different atoms is a lattice plane. (Contributed by NM, 4-Jul-2012.)
Hypotheses
Ref Expression
lplni2.l = (le‘𝐾)
lplni2.j = (join‘𝐾)
lplni2.a 𝐴 = (Atoms‘𝐾)
lplni2.p 𝑃 = (LPlanes‘𝐾)
Assertion
Ref Expression
lplni2 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → ((𝑄 𝑅) 𝑆) ∈ 𝑃)

Proof of Theorem lplni2
Dummy variables 𝑟 𝑞 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 1129 . . 3 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → (𝑄𝐴𝑅𝐴𝑆𝐴))
2 simp3l 1193 . . 3 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → 𝑄𝑅)
3 simp3r 1194 . . 3 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → ¬ 𝑆 (𝑄 𝑅))
4 eqidd 2819 . . 3 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → ((𝑄 𝑅) 𝑆) = ((𝑄 𝑅) 𝑆))
5 neeq1 3075 . . . . 5 (𝑞 = 𝑄 → (𝑞𝑟𝑄𝑟))
6 oveq1 7152 . . . . . . 7 (𝑞 = 𝑄 → (𝑞 𝑟) = (𝑄 𝑟))
76breq2d 5069 . . . . . 6 (𝑞 = 𝑄 → (𝑠 (𝑞 𝑟) ↔ 𝑠 (𝑄 𝑟)))
87notbid 319 . . . . 5 (𝑞 = 𝑄 → (¬ 𝑠 (𝑞 𝑟) ↔ ¬ 𝑠 (𝑄 𝑟)))
96oveq1d 7160 . . . . . 6 (𝑞 = 𝑄 → ((𝑞 𝑟) 𝑠) = ((𝑄 𝑟) 𝑠))
109eqeq2d 2829 . . . . 5 (𝑞 = 𝑄 → (((𝑄 𝑅) 𝑆) = ((𝑞 𝑟) 𝑠) ↔ ((𝑄 𝑅) 𝑆) = ((𝑄 𝑟) 𝑠)))
115, 8, 103anbi123d 1427 . . . 4 (𝑞 = 𝑄 → ((𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ ((𝑄 𝑅) 𝑆) = ((𝑞 𝑟) 𝑠)) ↔ (𝑄𝑟 ∧ ¬ 𝑠 (𝑄 𝑟) ∧ ((𝑄 𝑅) 𝑆) = ((𝑄 𝑟) 𝑠))))
12 neeq2 3076 . . . . 5 (𝑟 = 𝑅 → (𝑄𝑟𝑄𝑅))
13 oveq2 7153 . . . . . . 7 (𝑟 = 𝑅 → (𝑄 𝑟) = (𝑄 𝑅))
1413breq2d 5069 . . . . . 6 (𝑟 = 𝑅 → (𝑠 (𝑄 𝑟) ↔ 𝑠 (𝑄 𝑅)))
1514notbid 319 . . . . 5 (𝑟 = 𝑅 → (¬ 𝑠 (𝑄 𝑟) ↔ ¬ 𝑠 (𝑄 𝑅)))
1613oveq1d 7160 . . . . . 6 (𝑟 = 𝑅 → ((𝑄 𝑟) 𝑠) = ((𝑄 𝑅) 𝑠))
1716eqeq2d 2829 . . . . 5 (𝑟 = 𝑅 → (((𝑄 𝑅) 𝑆) = ((𝑄 𝑟) 𝑠) ↔ ((𝑄 𝑅) 𝑆) = ((𝑄 𝑅) 𝑠)))
1812, 15, 173anbi123d 1427 . . . 4 (𝑟 = 𝑅 → ((𝑄𝑟 ∧ ¬ 𝑠 (𝑄 𝑟) ∧ ((𝑄 𝑅) 𝑆) = ((𝑄 𝑟) 𝑠)) ↔ (𝑄𝑅 ∧ ¬ 𝑠 (𝑄 𝑅) ∧ ((𝑄 𝑅) 𝑆) = ((𝑄 𝑅) 𝑠))))
19 breq1 5060 . . . . . 6 (𝑠 = 𝑆 → (𝑠 (𝑄 𝑅) ↔ 𝑆 (𝑄 𝑅)))
2019notbid 319 . . . . 5 (𝑠 = 𝑆 → (¬ 𝑠 (𝑄 𝑅) ↔ ¬ 𝑆 (𝑄 𝑅)))
21 oveq2 7153 . . . . . 6 (𝑠 = 𝑆 → ((𝑄 𝑅) 𝑠) = ((𝑄 𝑅) 𝑆))
2221eqeq2d 2829 . . . . 5 (𝑠 = 𝑆 → (((𝑄 𝑅) 𝑆) = ((𝑄 𝑅) 𝑠) ↔ ((𝑄 𝑅) 𝑆) = ((𝑄 𝑅) 𝑆)))
2320, 223anbi23d 1430 . . . 4 (𝑠 = 𝑆 → ((𝑄𝑅 ∧ ¬ 𝑠 (𝑄 𝑅) ∧ ((𝑄 𝑅) 𝑆) = ((𝑄 𝑅) 𝑠)) ↔ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅) ∧ ((𝑄 𝑅) 𝑆) = ((𝑄 𝑅) 𝑆))))
2411, 18, 23rspc3ev 3634 . . 3 (((𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅) ∧ ((𝑄 𝑅) 𝑆) = ((𝑄 𝑅) 𝑆))) → ∃𝑞𝐴𝑟𝐴𝑠𝐴 (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ ((𝑄 𝑅) 𝑆) = ((𝑞 𝑟) 𝑠)))
251, 2, 3, 4, 24syl13anc 1364 . 2 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → ∃𝑞𝐴𝑟𝐴𝑠𝐴 (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ ((𝑄 𝑅) 𝑆) = ((𝑞 𝑟) 𝑠)))
26 simp1 1128 . . 3 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → 𝐾 ∈ HL)
27 hllat 36379 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
28273ad2ant1 1125 . . . 4 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → 𝐾 ∈ Lat)
29 simp21 1198 . . . . 5 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → 𝑄𝐴)
30 simp22 1199 . . . . 5 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → 𝑅𝐴)
31 eqid 2818 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
32 lplni2.j . . . . . 6 = (join‘𝐾)
33 lplni2.a . . . . . 6 𝐴 = (Atoms‘𝐾)
3431, 32, 33hlatjcl 36383 . . . . 5 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) ∈ (Base‘𝐾))
3526, 29, 30, 34syl3anc 1363 . . . 4 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → (𝑄 𝑅) ∈ (Base‘𝐾))
36 simp23 1200 . . . . 5 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → 𝑆𝐴)
3731, 33atbase 36305 . . . . 5 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
3836, 37syl 17 . . . 4 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → 𝑆 ∈ (Base‘𝐾))
3931, 32latjcl 17649 . . . 4 ((𝐾 ∈ Lat ∧ (𝑄 𝑅) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → ((𝑄 𝑅) 𝑆) ∈ (Base‘𝐾))
4028, 35, 38, 39syl3anc 1363 . . 3 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → ((𝑄 𝑅) 𝑆) ∈ (Base‘𝐾))
41 lplni2.l . . . 4 = (le‘𝐾)
42 lplni2.p . . . 4 𝑃 = (LPlanes‘𝐾)
4331, 41, 32, 33, 42islpln5 36551 . . 3 ((𝐾 ∈ HL ∧ ((𝑄 𝑅) 𝑆) ∈ (Base‘𝐾)) → (((𝑄 𝑅) 𝑆) ∈ 𝑃 ↔ ∃𝑞𝐴𝑟𝐴𝑠𝐴 (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ ((𝑄 𝑅) 𝑆) = ((𝑞 𝑟) 𝑠))))
4426, 40, 43syl2anc 584 . 2 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → (((𝑄 𝑅) 𝑆) ∈ 𝑃 ↔ ∃𝑞𝐴𝑟𝐴𝑠𝐴 (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ ((𝑄 𝑅) 𝑆) = ((𝑞 𝑟) 𝑠))))
4525, 44mpbird 258 1 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → ((𝑄 𝑅) 𝑆) ∈ 𝑃)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wrex 3136   class class class wbr 5057  cfv 6348  (class class class)co 7145  Basecbs 16471  lecple 16560  joincjn 17542  Latclat 17643  Atomscatm 36279  HLchlt 36366  LPlanesclpl 36508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-proset 17526  df-poset 17544  df-plt 17556  df-lub 17572  df-glb 17573  df-join 17574  df-meet 17575  df-p0 17637  df-lat 17644  df-clat 17706  df-oposet 36192  df-ol 36194  df-oml 36195  df-covers 36282  df-ats 36283  df-atl 36314  df-cvlat 36338  df-hlat 36367  df-llines 36514  df-lplanes 36515
This theorem is referenced by:  islpln2a  36564  2llnjaN  36582  lvolnle3at  36598  dalem42  36730  cdleme16aN  37275
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