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Theorem lhpj1 37469
 Description: The join of a co-atom (hyperplane) and an element not under it is the lattice unit. (Contributed by NM, 7-Dec-2012.)
Hypotheses
Ref Expression
lhpj1.b 𝐵 = (Base‘𝐾)
lhpj1.l = (le‘𝐾)
lhpj1.j = (join‘𝐾)
lhpj1.u 1 = (1.‘𝐾)
lhpj1.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
lhpj1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → (𝑊 𝑋) = 1 )

Proof of Theorem lhpj1
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 simpll 766 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) → 𝐾 ∈ HL)
2 simpr 488 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) → 𝑋𝐵)
3 lhpj1.b . . . . . 6 𝐵 = (Base‘𝐾)
4 lhpj1.h . . . . . 6 𝐻 = (LHyp‘𝐾)
53, 4lhpbase 37445 . . . . 5 (𝑊𝐻𝑊𝐵)
65ad2antlr 726 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) → 𝑊𝐵)
7 lhpj1.l . . . . 5 = (le‘𝐾)
8 eqid 2798 . . . . 5 (Atoms‘𝐾) = (Atoms‘𝐾)
93, 7, 8hlrelat2 36850 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑊𝐵) → (¬ 𝑋 𝑊 ↔ ∃𝑝 ∈ (Atoms‘𝐾)(𝑝 𝑋 ∧ ¬ 𝑝 𝑊)))
101, 2, 6, 9syl3anc 1368 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) → (¬ 𝑋 𝑊 ↔ ∃𝑝 ∈ (Atoms‘𝐾)(𝑝 𝑋 ∧ ¬ 𝑝 𝑊)))
11 simp1l 1194 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
12 simp2 1134 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → 𝑝 ∈ (Atoms‘𝐾))
13 simp3r 1199 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → ¬ 𝑝 𝑊)
14 lhpj1.j . . . . . . . 8 = (join‘𝐾)
15 lhpj1.u . . . . . . . 8 1 = (1.‘𝐾)
167, 14, 15, 8, 4lhpjat1 37467 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝 𝑊)) → (𝑊 𝑝) = 1 )
1711, 12, 13, 16syl12anc 835 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → (𝑊 𝑝) = 1 )
18 simp3l 1198 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → 𝑝 𝑋)
19 simp1ll 1233 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → 𝐾 ∈ HL)
2019hllatd 36811 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → 𝐾 ∈ Lat)
213, 8atbase 36736 . . . . . . . . 9 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
22213ad2ant2 1131 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → 𝑝𝐵)
23 simp1r 1195 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → 𝑋𝐵)
2463ad2ant1 1130 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → 𝑊𝐵)
253, 7, 14latjlej2 17688 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑝𝐵𝑋𝐵𝑊𝐵)) → (𝑝 𝑋 → (𝑊 𝑝) (𝑊 𝑋)))
2620, 22, 23, 24, 25syl13anc 1369 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → (𝑝 𝑋 → (𝑊 𝑝) (𝑊 𝑋)))
2718, 26mpd 15 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → (𝑊 𝑝) (𝑊 𝑋))
2817, 27eqbrtrrd 5058 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → 1 (𝑊 𝑋))
29 hlop 36809 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ OP)
3019, 29syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → 𝐾 ∈ OP)
313, 14latjcl 17673 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑊𝐵𝑋𝐵) → (𝑊 𝑋) ∈ 𝐵)
3220, 24, 23, 31syl3anc 1368 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → (𝑊 𝑋) ∈ 𝐵)
333, 7, 15op1le 36639 . . . . . 6 ((𝐾 ∈ OP ∧ (𝑊 𝑋) ∈ 𝐵) → ( 1 (𝑊 𝑋) ↔ (𝑊 𝑋) = 1 ))
3430, 32, 33syl2anc 587 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → ( 1 (𝑊 𝑋) ↔ (𝑊 𝑋) = 1 ))
3528, 34mpbid 235 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → (𝑊 𝑋) = 1 )
3635rexlimdv3a 3246 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) → (∃𝑝 ∈ (Atoms‘𝐾)(𝑝 𝑋 ∧ ¬ 𝑝 𝑊) → (𝑊 𝑋) = 1 ))
3710, 36sylbid 243 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) → (¬ 𝑋 𝑊 → (𝑊 𝑋) = 1 ))
3837impr 458 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → (𝑊 𝑋) = 1 )
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∃wrex 3107   class class class wbr 5034  ‘cfv 6332  (class class class)co 7145  Basecbs 16495  lecple 16584  joincjn 17566  1.cp1 17660  Latclat 17667  OPcops 36619  Atomscatm 36710  HLchlt 36797  LHypclh 37431 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-riota 7103  df-ov 7148  df-oprab 7149  df-proset 17550  df-poset 17568  df-plt 17580  df-lub 17596  df-glb 17597  df-join 17598  df-meet 17599  df-p0 17661  df-p1 17662  df-lat 17668  df-clat 17730  df-oposet 36623  df-ol 36625  df-oml 36626  df-covers 36713  df-ats 36714  df-atl 36745  df-cvlat 36769  df-hlat 36798  df-lhyp 37435 This theorem is referenced by:  lhpmcvr  37470  cdleme30a  37825
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