Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lhpj1 Structured version   Visualization version   GIF version

Theorem lhpj1 37038
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 763 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) → 𝐾 ∈ HL)
2 simpr 485 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) → 𝑋𝐵)
3 lhpj1.b . . . . . 6 𝐵 = (Base‘𝐾)
4 lhpj1.h . . . . . 6 𝐻 = (LHyp‘𝐾)
53, 4lhpbase 37014 . . . . 5 (𝑊𝐻𝑊𝐵)
65ad2antlr 723 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) → 𝑊𝐵)
7 lhpj1.l . . . . 5 = (le‘𝐾)
8 eqid 2818 . . . . 5 (Atoms‘𝐾) = (Atoms‘𝐾)
93, 7, 8hlrelat2 36419 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑊𝐵) → (¬ 𝑋 𝑊 ↔ ∃𝑝 ∈ (Atoms‘𝐾)(𝑝 𝑋 ∧ ¬ 𝑝 𝑊)))
101, 2, 6, 9syl3anc 1363 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) → (¬ 𝑋 𝑊 ↔ ∃𝑝 ∈ (Atoms‘𝐾)(𝑝 𝑋 ∧ ¬ 𝑝 𝑊)))
11 simp1l 1189 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
12 simp2 1129 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → 𝑝 ∈ (Atoms‘𝐾))
13 simp3r 1194 . . . . . . 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 37036 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝 𝑊)) → (𝑊 𝑝) = 1 )
1711, 12, 13, 16syl12anc 832 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → (𝑊 𝑝) = 1 )
18 simp3l 1193 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → 𝑝 𝑋)
19 simp1ll 1228 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → 𝐾 ∈ HL)
2019hllatd 36380 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → 𝐾 ∈ Lat)
213, 8atbase 36305 . . . . . . . . 9 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
22213ad2ant2 1126 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → 𝑝𝐵)
23 simp1r 1190 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → 𝑋𝐵)
2463ad2ant1 1125 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → 𝑊𝐵)
253, 7, 14latjlej2 17664 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑝𝐵𝑋𝐵𝑊𝐵)) → (𝑝 𝑋 → (𝑊 𝑝) (𝑊 𝑋)))
2620, 22, 23, 24, 25syl13anc 1364 . . . . . . 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 5081 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → 1 (𝑊 𝑋))
29 hlop 36378 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ OP)
3019, 29syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → 𝐾 ∈ OP)
313, 14latjcl 17649 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑊𝐵𝑋𝐵) → (𝑊 𝑋) ∈ 𝐵)
3220, 24, 23, 31syl3anc 1363 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → (𝑊 𝑋) ∈ 𝐵)
333, 7, 15op1le 36208 . . . . . 6 ((𝐾 ∈ OP ∧ (𝑊 𝑋) ∈ 𝐵) → ( 1 (𝑊 𝑋) ↔ (𝑊 𝑋) = 1 ))
3430, 32, 33syl2anc 584 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → ( 1 (𝑊 𝑋) ↔ (𝑊 𝑋) = 1 ))
3528, 34mpbid 233 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → (𝑊 𝑋) = 1 )
3635rexlimdv3a 3283 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) → (∃𝑝 ∈ (Atoms‘𝐾)(𝑝 𝑋 ∧ ¬ 𝑝 𝑊) → (𝑊 𝑋) = 1 ))
3710, 36sylbid 241 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) → (¬ 𝑋 𝑊 → (𝑊 𝑋) = 1 ))
3837impr 455 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → (𝑊 𝑋) = 1 )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wrex 3136   class class class wbr 5057  cfv 6348  (class class class)co 7145  Basecbs 16471  lecple 16560  joincjn 17542  1.cp1 17636  Latclat 17643  OPcops 36188  Atomscatm 36279  HLchlt 36366  LHypclh 37000
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-p1 17638  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-lhyp 37004
This theorem is referenced by:  lhpmcvr  37039  cdleme30a  37394
  Copyright terms: Public domain W3C validator