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Theorem lplni2 35295
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 1220 . . 3 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → 𝑄𝑅)
3 simp3r 1221 . . 3 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → ¬ 𝑆 (𝑄 𝑅))
4 eqidd 2749 . . 3 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → ((𝑄 𝑅) 𝑆) = ((𝑄 𝑅) 𝑆))
5 neeq1 2982 . . . . 5 (𝑞 = 𝑄 → (𝑞𝑟𝑄𝑟))
6 oveq1 6808 . . . . . . 7 (𝑞 = 𝑄 → (𝑞 𝑟) = (𝑄 𝑟))
76breq2d 4804 . . . . . 6 (𝑞 = 𝑄 → (𝑠 (𝑞 𝑟) ↔ 𝑠 (𝑄 𝑟)))
87notbid 307 . . . . 5 (𝑞 = 𝑄 → (¬ 𝑠 (𝑞 𝑟) ↔ ¬ 𝑠 (𝑄 𝑟)))
96oveq1d 6816 . . . . . 6 (𝑞 = 𝑄 → ((𝑞 𝑟) 𝑠) = ((𝑄 𝑟) 𝑠))
109eqeq2d 2758 . . . . 5 (𝑞 = 𝑄 → (((𝑄 𝑅) 𝑆) = ((𝑞 𝑟) 𝑠) ↔ ((𝑄 𝑅) 𝑆) = ((𝑄 𝑟) 𝑠)))
115, 8, 103anbi123d 1536 . . . 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 2983 . . . . 5 (𝑟 = 𝑅 → (𝑄𝑟𝑄𝑅))
13 oveq2 6809 . . . . . . 7 (𝑟 = 𝑅 → (𝑄 𝑟) = (𝑄 𝑅))
1413breq2d 4804 . . . . . 6 (𝑟 = 𝑅 → (𝑠 (𝑄 𝑟) ↔ 𝑠 (𝑄 𝑅)))
1514notbid 307 . . . . 5 (𝑟 = 𝑅 → (¬ 𝑠 (𝑄 𝑟) ↔ ¬ 𝑠 (𝑄 𝑅)))
1613oveq1d 6816 . . . . . 6 (𝑟 = 𝑅 → ((𝑄 𝑟) 𝑠) = ((𝑄 𝑅) 𝑠))
1716eqeq2d 2758 . . . . 5 (𝑟 = 𝑅 → (((𝑄 𝑅) 𝑆) = ((𝑄 𝑟) 𝑠) ↔ ((𝑄 𝑅) 𝑆) = ((𝑄 𝑅) 𝑠)))
1812, 15, 173anbi123d 1536 . . . 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 4795 . . . . . 6 (𝑠 = 𝑆 → (𝑠 (𝑄 𝑅) ↔ 𝑆 (𝑄 𝑅)))
2019notbid 307 . . . . 5 (𝑠 = 𝑆 → (¬ 𝑠 (𝑄 𝑅) ↔ ¬ 𝑆 (𝑄 𝑅)))
21 oveq2 6809 . . . . . 6 (𝑠 = 𝑆 → ((𝑄 𝑅) 𝑠) = ((𝑄 𝑅) 𝑆))
2221eqeq2d 2758 . . . . 5 (𝑠 = 𝑆 → (((𝑄 𝑅) 𝑆) = ((𝑄 𝑅) 𝑠) ↔ ((𝑄 𝑅) 𝑆) = ((𝑄 𝑅) 𝑆)))
2320, 223anbi23d 1539 . . . 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 3453 . . 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 1465 . 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 35122 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
28273ad2ant1 1125 . . . 4 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → 𝐾 ∈ Lat)
29 simp21 1225 . . . . 5 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → 𝑄𝐴)
30 simp22 1226 . . . . 5 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → 𝑅𝐴)
31 eqid 2748 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
32 lplni2.j . . . . . 6 = (join‘𝐾)
33 lplni2.a . . . . . 6 𝐴 = (Atoms‘𝐾)
3431, 32, 33hlatjcl 35125 . . . . 5 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) ∈ (Base‘𝐾))
3526, 29, 30, 34syl3anc 1463 . . . 4 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → (𝑄 𝑅) ∈ (Base‘𝐾))
36 simp23 1227 . . . . 5 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → 𝑆𝐴)
3731, 33atbase 35048 . . . . 5 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
3836, 37syl 17 . . . 4 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → 𝑆 ∈ (Base‘𝐾))
3931, 32latjcl 17223 . . . 4 ((𝐾 ∈ Lat ∧ (𝑄 𝑅) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → ((𝑄 𝑅) 𝑆) ∈ (Base‘𝐾))
4028, 35, 38, 39syl3anc 1463 . . 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 35293 . . 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 696 . 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 247 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 196  wa 383  w3a 1072   = wceq 1620  wcel 2127  wne 2920  wrex 3039   class class class wbr 4792  cfv 6037  (class class class)co 6801  Basecbs 16030  lecple 16121  joincjn 17116  Latclat 17217  Atomscatm 35022  HLchlt 35109  LPlanesclpl 35250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-riota 6762  df-ov 6804  df-oprab 6805  df-preset 17100  df-poset 17118  df-plt 17130  df-lub 17146  df-glb 17147  df-join 17148  df-meet 17149  df-p0 17211  df-lat 17218  df-clat 17280  df-oposet 34935  df-ol 34937  df-oml 34938  df-covers 35025  df-ats 35026  df-atl 35057  df-cvlat 35081  df-hlat 35110  df-llines 35256  df-lplanes 35257
This theorem is referenced by:  islpln2a  35306  2llnjaN  35324  lvolnle3at  35340  dalem42  35472  cdleme16aN  36018
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