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Theorem lplni2 35558
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 1168 . . 3 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → (𝑄𝐴𝑅𝐴𝑆𝐴))
2 simp3l 1259 . . 3 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → 𝑄𝑅)
3 simp3r 1260 . . 3 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → ¬ 𝑆 (𝑄 𝑅))
4 eqidd 2800 . . 3 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → ((𝑄 𝑅) 𝑆) = ((𝑄 𝑅) 𝑆))
5 neeq1 3033 . . . . 5 (𝑞 = 𝑄 → (𝑞𝑟𝑄𝑟))
6 oveq1 6885 . . . . . . 7 (𝑞 = 𝑄 → (𝑞 𝑟) = (𝑄 𝑟))
76breq2d 4855 . . . . . 6 (𝑞 = 𝑄 → (𝑠 (𝑞 𝑟) ↔ 𝑠 (𝑄 𝑟)))
87notbid 310 . . . . 5 (𝑞 = 𝑄 → (¬ 𝑠 (𝑞 𝑟) ↔ ¬ 𝑠 (𝑄 𝑟)))
96oveq1d 6893 . . . . . 6 (𝑞 = 𝑄 → ((𝑞 𝑟) 𝑠) = ((𝑄 𝑟) 𝑠))
109eqeq2d 2809 . . . . 5 (𝑞 = 𝑄 → (((𝑄 𝑅) 𝑆) = ((𝑞 𝑟) 𝑠) ↔ ((𝑄 𝑅) 𝑆) = ((𝑄 𝑟) 𝑠)))
115, 8, 103anbi123d 1561 . . . 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 3034 . . . . 5 (𝑟 = 𝑅 → (𝑄𝑟𝑄𝑅))
13 oveq2 6886 . . . . . . 7 (𝑟 = 𝑅 → (𝑄 𝑟) = (𝑄 𝑅))
1413breq2d 4855 . . . . . 6 (𝑟 = 𝑅 → (𝑠 (𝑄 𝑟) ↔ 𝑠 (𝑄 𝑅)))
1514notbid 310 . . . . 5 (𝑟 = 𝑅 → (¬ 𝑠 (𝑄 𝑟) ↔ ¬ 𝑠 (𝑄 𝑅)))
1613oveq1d 6893 . . . . . 6 (𝑟 = 𝑅 → ((𝑄 𝑟) 𝑠) = ((𝑄 𝑅) 𝑠))
1716eqeq2d 2809 . . . . 5 (𝑟 = 𝑅 → (((𝑄 𝑅) 𝑆) = ((𝑄 𝑟) 𝑠) ↔ ((𝑄 𝑅) 𝑆) = ((𝑄 𝑅) 𝑠)))
1812, 15, 173anbi123d 1561 . . . 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 4846 . . . . . 6 (𝑠 = 𝑆 → (𝑠 (𝑄 𝑅) ↔ 𝑆 (𝑄 𝑅)))
2019notbid 310 . . . . 5 (𝑠 = 𝑆 → (¬ 𝑠 (𝑄 𝑅) ↔ ¬ 𝑆 (𝑄 𝑅)))
21 oveq2 6886 . . . . . 6 (𝑠 = 𝑆 → ((𝑄 𝑅) 𝑠) = ((𝑄 𝑅) 𝑆))
2221eqeq2d 2809 . . . . 5 (𝑠 = 𝑆 → (((𝑄 𝑅) 𝑆) = ((𝑄 𝑅) 𝑠) ↔ ((𝑄 𝑅) 𝑆) = ((𝑄 𝑅) 𝑆)))
2320, 223anbi23d 1564 . . . 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 3514 . . 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 1492 . 2 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → ∃𝑞𝐴𝑟𝐴𝑠𝐴 (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ ((𝑄 𝑅) 𝑆) = ((𝑞 𝑟) 𝑠)))
26 simp1 1167 . . 3 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → 𝐾 ∈ HL)
27 hllat 35384 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
28273ad2ant1 1164 . . . 4 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → 𝐾 ∈ Lat)
29 simp21 1264 . . . . 5 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → 𝑄𝐴)
30 simp22 1265 . . . . 5 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → 𝑅𝐴)
31 eqid 2799 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
32 lplni2.j . . . . . 6 = (join‘𝐾)
33 lplni2.a . . . . . 6 𝐴 = (Atoms‘𝐾)
3431, 32, 33hlatjcl 35388 . . . . 5 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) ∈ (Base‘𝐾))
3526, 29, 30, 34syl3anc 1491 . . . 4 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → (𝑄 𝑅) ∈ (Base‘𝐾))
36 simp23 1266 . . . . 5 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → 𝑆𝐴)
3731, 33atbase 35310 . . . . 5 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
3836, 37syl 17 . . . 4 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅))) → 𝑆 ∈ (Base‘𝐾))
3931, 32latjcl 17366 . . . 4 ((𝐾 ∈ Lat ∧ (𝑄 𝑅) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → ((𝑄 𝑅) 𝑆) ∈ (Base‘𝐾))
4028, 35, 38, 39syl3anc 1491 . . 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 35556 . . 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 580 . 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 249 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 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wne 2971  wrex 3090   class class class wbr 4843  cfv 6101  (class class class)co 6878  Basecbs 16184  lecple 16274  joincjn 17259  Latclat 17360  Atomscatm 35284  HLchlt 35371  LPlanesclpl 35513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-proset 17243  df-poset 17261  df-plt 17273  df-lub 17289  df-glb 17290  df-join 17291  df-meet 17292  df-p0 17354  df-lat 17361  df-clat 17423  df-oposet 35197  df-ol 35199  df-oml 35200  df-covers 35287  df-ats 35288  df-atl 35319  df-cvlat 35343  df-hlat 35372  df-llines 35519  df-lplanes 35520
This theorem is referenced by:  islpln2a  35569  2llnjaN  35587  lvolnle3at  35603  dalem42  35735  cdleme16aN  36280
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