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Theorem lhpj1 34823
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 789 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) → 𝐾 ∈ HL)
2 simpr 477 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) → 𝑋𝐵)
3 lhpj1.b . . . . . 6 𝐵 = (Base‘𝐾)
4 lhpj1.h . . . . . 6 𝐻 = (LHyp‘𝐾)
53, 4lhpbase 34799 . . . . 5 (𝑊𝐻𝑊𝐵)
65ad2antlr 762 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) → 𝑊𝐵)
7 lhpj1.l . . . . 5 = (le‘𝐾)
8 eqid 2621 . . . . 5 (Atoms‘𝐾) = (Atoms‘𝐾)
93, 7, 8hlrelat2 34204 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑊𝐵) → (¬ 𝑋 𝑊 ↔ ∃𝑝 ∈ (Atoms‘𝐾)(𝑝 𝑋 ∧ ¬ 𝑝 𝑊)))
101, 2, 6, 9syl3anc 1323 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) → (¬ 𝑋 𝑊 ↔ ∃𝑝 ∈ (Atoms‘𝐾)(𝑝 𝑋 ∧ ¬ 𝑝 𝑊)))
11 simp1l 1083 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
12 simp2 1060 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → 𝑝 ∈ (Atoms‘𝐾))
13 simp3r 1088 . . . . . . 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 34821 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝 𝑊)) → (𝑊 𝑝) = 1 )
1711, 12, 13, 16syl12anc 1321 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → (𝑊 𝑝) = 1 )
18 simp3l 1087 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → 𝑝 𝑋)
19 simp1ll 1122 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → 𝐾 ∈ HL)
20 hllat 34165 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ Lat)
2119, 20syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → 𝐾 ∈ Lat)
223, 8atbase 34091 . . . . . . . . 9 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
23223ad2ant2 1081 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → 𝑝𝐵)
24 simp1r 1084 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → 𝑋𝐵)
2563ad2ant1 1080 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → 𝑊𝐵)
263, 7, 14latjlej2 16998 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑝𝐵𝑋𝐵𝑊𝐵)) → (𝑝 𝑋 → (𝑊 𝑝) (𝑊 𝑋)))
2721, 23, 24, 25, 26syl13anc 1325 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → (𝑝 𝑋 → (𝑊 𝑝) (𝑊 𝑋)))
2818, 27mpd 15 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → (𝑊 𝑝) (𝑊 𝑋))
2917, 28eqbrtrrd 4642 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → 1 (𝑊 𝑋))
30 hlop 34164 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ OP)
3119, 30syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → 𝐾 ∈ OP)
323, 14latjcl 16983 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑊𝐵𝑋𝐵) → (𝑊 𝑋) ∈ 𝐵)
3321, 25, 24, 32syl3anc 1323 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → (𝑊 𝑋) ∈ 𝐵)
343, 7, 15op1le 33994 . . . . . 6 ((𝐾 ∈ OP ∧ (𝑊 𝑋) ∈ 𝐵) → ( 1 (𝑊 𝑋) ↔ (𝑊 𝑋) = 1 ))
3531, 33, 34syl2anc 692 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → ( 1 (𝑊 𝑋) ↔ (𝑊 𝑋) = 1 ))
3629, 35mpbid 222 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 𝑋 ∧ ¬ 𝑝 𝑊)) → (𝑊 𝑋) = 1 )
3736rexlimdv3a 3027 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) → (∃𝑝 ∈ (Atoms‘𝐾)(𝑝 𝑋 ∧ ¬ 𝑝 𝑊) → (𝑊 𝑋) = 1 ))
3810, 37sylbid 230 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) → (¬ 𝑋 𝑊 → (𝑊 𝑋) = 1 ))
3938impr 648 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → (𝑊 𝑋) = 1 )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wrex 2908   class class class wbr 4618  cfv 5852  (class class class)co 6610  Basecbs 15792  lecple 15880  joincjn 16876  1.cp1 16970  Latclat 16977  OPcops 33974  Atomscatm 34065  HLchlt 34152  LHypclh 34785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-preset 16860  df-poset 16878  df-plt 16890  df-lub 16906  df-glb 16907  df-join 16908  df-meet 16909  df-p0 16971  df-p1 16972  df-lat 16978  df-clat 17040  df-oposet 33978  df-ol 33980  df-oml 33981  df-covers 34068  df-ats 34069  df-atl 34100  df-cvlat 34124  df-hlat 34153  df-lhyp 34789
This theorem is referenced by:  lhpmcvr  34824  cdleme30a  35181
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