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Theorem lplni2 39236
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 1134 . . 3 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → (𝑄𝐴𝑅𝐴𝑆𝐴))
2 simp3l 1198 . . 3 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → 𝑄𝑅)
3 simp3r 1199 . . 3 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → ¬ 𝑆 (𝑄 𝑅))
4 eqidd 2727 . . 3 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → ((𝑄 𝑅) 𝑆) = ((𝑄 𝑅) 𝑆))
5 neeq1 2993 . . . . 5 (𝑞 = 𝑄 → (𝑞𝑟𝑄𝑟))
6 oveq1 7431 . . . . . . 7 (𝑞 = 𝑄 → (𝑞 𝑟) = (𝑄 𝑟))
76breq2d 5165 . . . . . 6 (𝑞 = 𝑄 → (𝑠 (𝑞 𝑟) ↔ 𝑠 (𝑄 𝑟)))
87notbid 317 . . . . 5 (𝑞 = 𝑄 → (¬ 𝑠 (𝑞 𝑟) ↔ ¬ 𝑠 (𝑄 𝑟)))
96oveq1d 7439 . . . . . 6 (𝑞 = 𝑄 → ((𝑞 𝑟) 𝑠) = ((𝑄 𝑟) 𝑠))
109eqeq2d 2737 . . . . 5 (𝑞 = 𝑄 → (((𝑄 𝑅) 𝑆) = ((𝑞 𝑟) 𝑠) ↔ ((𝑄 𝑅) 𝑆) = ((𝑄 𝑟) 𝑠)))
115, 8, 103anbi123d 1433 . . . 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 2994 . . . . 5 (𝑟 = 𝑅 → (𝑄𝑟𝑄𝑅))
13 oveq2 7432 . . . . . . 7 (𝑟 = 𝑅 → (𝑄 𝑟) = (𝑄 𝑅))
1413breq2d 5165 . . . . . 6 (𝑟 = 𝑅 → (𝑠 (𝑄 𝑟) ↔ 𝑠 (𝑄 𝑅)))
1514notbid 317 . . . . 5 (𝑟 = 𝑅 → (¬ 𝑠 (𝑄 𝑟) ↔ ¬ 𝑠 (𝑄 𝑅)))
1613oveq1d 7439 . . . . . 6 (𝑟 = 𝑅 → ((𝑄 𝑟) 𝑠) = ((𝑄 𝑅) 𝑠))
1716eqeq2d 2737 . . . . 5 (𝑟 = 𝑅 → (((𝑄 𝑅) 𝑆) = ((𝑄 𝑟) 𝑠) ↔ ((𝑄 𝑅) 𝑆) = ((𝑄 𝑅) 𝑠)))
1812, 15, 173anbi123d 1433 . . . 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 5156 . . . . . 6 (𝑠 = 𝑆 → (𝑠 (𝑄 𝑅) ↔ 𝑆 (𝑄 𝑅)))
2019notbid 317 . . . . 5 (𝑠 = 𝑆 → (¬ 𝑠 (𝑄 𝑅) ↔ ¬ 𝑆 (𝑄 𝑅)))
21 oveq2 7432 . . . . . 6 (𝑠 = 𝑆 → ((𝑄 𝑅) 𝑠) = ((𝑄 𝑅) 𝑆))
2221eqeq2d 2737 . . . . 5 (𝑠 = 𝑆 → (((𝑄 𝑅) 𝑆) = ((𝑄 𝑅) 𝑠) ↔ ((𝑄 𝑅) 𝑆) = ((𝑄 𝑅) 𝑆)))
2320, 223anbi23d 1436 . . . 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 3625 . . 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 1369 . 2 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → ∃𝑞𝐴𝑟𝐴𝑠𝐴 (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ ((𝑄 𝑅) 𝑆) = ((𝑞 𝑟) 𝑠)))
26 simp1 1133 . . 3 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → 𝐾 ∈ HL)
27 hllat 39061 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
28273ad2ant1 1130 . . . 4 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → 𝐾 ∈ Lat)
29 simp21 1203 . . . . 5 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → 𝑄𝐴)
30 simp22 1204 . . . . 5 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → 𝑅𝐴)
31 eqid 2726 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
32 lplni2.j . . . . . 6 = (join‘𝐾)
33 lplni2.a . . . . . 6 𝐴 = (Atoms‘𝐾)
3431, 32, 33hlatjcl 39065 . . . . 5 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) ∈ (Base‘𝐾))
3526, 29, 30, 34syl3anc 1368 . . . 4 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → (𝑄 𝑅) ∈ (Base‘𝐾))
36 simp23 1205 . . . . 5 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → 𝑆𝐴)
3731, 33atbase 38987 . . . . 5 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
3836, 37syl 17 . . . 4 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → 𝑆 ∈ (Base‘𝐾))
3931, 32latjcl 18464 . . . 4 ((𝐾 ∈ Lat ∧ (𝑄 𝑅) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → ((𝑄 𝑅) 𝑆) ∈ (Base‘𝐾))
4028, 35, 38, 39syl3anc 1368 . . 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 39234 . . 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 582 . 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 256 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 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  wne 2930  wrex 3060   class class class wbr 5153  cfv 6554  (class class class)co 7424  Basecbs 17213  lecple 17273  joincjn 18336  Latclat 18456  Atomscatm 38961  HLchlt 39048  LPlanesclpl 39191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-proset 18320  df-poset 18338  df-plt 18355  df-lub 18371  df-glb 18372  df-join 18373  df-meet 18374  df-p0 18450  df-lat 18457  df-clat 18524  df-oposet 38874  df-ol 38876  df-oml 38877  df-covers 38964  df-ats 38965  df-atl 38996  df-cvlat 39020  df-hlat 39049  df-llines 39197  df-lplanes 39198
This theorem is referenced by:  islpln2a  39247  2llnjaN  39265  lvolnle3at  39281  dalem42  39413  cdleme16aN  39958
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