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Theorem lhprelat3N 34841
Description: The Hilbert lattice is relatively atomic with respect to co-atoms (lattice hyperplanes). Dual version of hlrelat3 34213. (Contributed by NM, 20-Jun-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
lhprelat3.b 𝐵 = (Base‘𝐾)
lhprelat3.l = (le‘𝐾)
lhprelat3.s < = (lt‘𝐾)
lhprelat3.m = (meet‘𝐾)
lhprelat3.c 𝐶 = ( ⋖ ‘𝐾)
lhprelat3.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
lhprelat3N (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ∃𝑤𝐻 (𝑋 (𝑌 𝑤) ∧ (𝑌 𝑤)𝐶𝑌))
Distinct variable groups:   𝑤,𝐶   𝑤,𝐻   𝑤,𝐾   𝑤,   𝑤,   𝑤,𝑋   𝑤,𝑌
Allowed substitution hints:   𝐵(𝑤)   < (𝑤)

Proof of Theorem lhprelat3N
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 simpr 477 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝 ∈ (Atoms‘𝐾))
2 simpll1 1098 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ HL)
3 lhprelat3.b . . . . . . . 8 𝐵 = (Base‘𝐾)
4 eqid 2621 . . . . . . . 8 (Atoms‘𝐾) = (Atoms‘𝐾)
53, 4atbase 34091 . . . . . . 7 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
65adantl 482 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝𝐵)
7 eqid 2621 . . . . . . 7 (oc‘𝐾) = (oc‘𝐾)
8 lhprelat3.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
93, 7, 4, 8lhpoc2N 34816 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑝𝐵) → (𝑝 ∈ (Atoms‘𝐾) ↔ ((oc‘𝐾)‘𝑝) ∈ 𝐻))
102, 6, 9syl2anc 692 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑝 ∈ (Atoms‘𝐾) ↔ ((oc‘𝐾)‘𝑝) ∈ 𝐻))
111, 10mpbid 222 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((oc‘𝐾)‘𝑝) ∈ 𝐻)
1211adantr 481 . . 3 (((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) ∧ (((oc‘𝐾)‘𝑌)𝐶(((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ∧ (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ((oc‘𝐾)‘𝑋))) → ((oc‘𝐾)‘𝑝) ∈ 𝐻)
13 hlop 34164 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ OP)
142, 13syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ OP)
15 hllat 34165 . . . . . . . . . 10 (𝐾 ∈ HL → 𝐾 ∈ Lat)
162, 15syl 17 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ Lat)
17 simpll3 1100 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑌𝐵)
183, 7opoccl 33996 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑝𝐵) → ((oc‘𝐾)‘𝑝) ∈ 𝐵)
1914, 6, 18syl2anc 692 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((oc‘𝐾)‘𝑝) ∈ 𝐵)
20 lhprelat3.m . . . . . . . . . 10 = (meet‘𝐾)
213, 20latmcl 16984 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ ((oc‘𝐾)‘𝑝) ∈ 𝐵) → (𝑌 ((oc‘𝐾)‘𝑝)) ∈ 𝐵)
2216, 17, 19, 21syl3anc 1323 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑌 ((oc‘𝐾)‘𝑝)) ∈ 𝐵)
23 lhprelat3.c . . . . . . . . 9 𝐶 = ( ⋖ ‘𝐾)
243, 7, 23cvrcon3b 34079 . . . . . . . 8 ((𝐾 ∈ OP ∧ (𝑌 ((oc‘𝐾)‘𝑝)) ∈ 𝐵𝑌𝐵) → ((𝑌 ((oc‘𝐾)‘𝑝))𝐶𝑌 ↔ ((oc‘𝐾)‘𝑌)𝐶((oc‘𝐾)‘(𝑌 ((oc‘𝐾)‘𝑝)))))
2514, 22, 17, 24syl3anc 1323 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((𝑌 ((oc‘𝐾)‘𝑝))𝐶𝑌 ↔ ((oc‘𝐾)‘𝑌)𝐶((oc‘𝐾)‘(𝑌 ((oc‘𝐾)‘𝑝)))))
26 hlol 34163 . . . . . . . . . 10 (𝐾 ∈ HL → 𝐾 ∈ OL)
272, 26syl 17 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ OL)
28 eqid 2621 . . . . . . . . . 10 (join‘𝐾) = (join‘𝐾)
293, 28, 20, 7oldmm3N 34021 . . . . . . . . 9 ((𝐾 ∈ OL ∧ 𝑌𝐵𝑝𝐵) → ((oc‘𝐾)‘(𝑌 ((oc‘𝐾)‘𝑝))) = (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝))
3027, 17, 6, 29syl3anc 1323 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((oc‘𝐾)‘(𝑌 ((oc‘𝐾)‘𝑝))) = (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝))
3130breq2d 4630 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (((oc‘𝐾)‘𝑌)𝐶((oc‘𝐾)‘(𝑌 ((oc‘𝐾)‘𝑝))) ↔ ((oc‘𝐾)‘𝑌)𝐶(((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝)))
3225, 31bitr2d 269 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (((oc‘𝐾)‘𝑌)𝐶(((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ↔ (𝑌 ((oc‘𝐾)‘𝑝))𝐶𝑌))
33 simpll2 1099 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑋𝐵)
34 lhprelat3.l . . . . . . . . 9 = (le‘𝐾)
353, 34, 7oplecon3b 34002 . . . . . . . 8 ((𝐾 ∈ OP ∧ 𝑋𝐵 ∧ (𝑌 ((oc‘𝐾)‘𝑝)) ∈ 𝐵) → (𝑋 (𝑌 ((oc‘𝐾)‘𝑝)) ↔ ((oc‘𝐾)‘(𝑌 ((oc‘𝐾)‘𝑝))) ((oc‘𝐾)‘𝑋)))
3614, 33, 22, 35syl3anc 1323 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑋 (𝑌 ((oc‘𝐾)‘𝑝)) ↔ ((oc‘𝐾)‘(𝑌 ((oc‘𝐾)‘𝑝))) ((oc‘𝐾)‘𝑋)))
3730breq1d 4628 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (((oc‘𝐾)‘(𝑌 ((oc‘𝐾)‘𝑝))) ((oc‘𝐾)‘𝑋) ↔ (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ((oc‘𝐾)‘𝑋)))
3836, 37bitr2d 269 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ((oc‘𝐾)‘𝑋) ↔ 𝑋 (𝑌 ((oc‘𝐾)‘𝑝))))
3932, 38anbi12d 746 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((((oc‘𝐾)‘𝑌)𝐶(((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ∧ (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ((oc‘𝐾)‘𝑋)) ↔ ((𝑌 ((oc‘𝐾)‘𝑝))𝐶𝑌𝑋 (𝑌 ((oc‘𝐾)‘𝑝)))))
4039biimpa 501 . . . 4 (((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) ∧ (((oc‘𝐾)‘𝑌)𝐶(((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ∧ (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ((oc‘𝐾)‘𝑋))) → ((𝑌 ((oc‘𝐾)‘𝑝))𝐶𝑌𝑋 (𝑌 ((oc‘𝐾)‘𝑝))))
4140ancomd 467 . . 3 (((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) ∧ (((oc‘𝐾)‘𝑌)𝐶(((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ∧ (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ((oc‘𝐾)‘𝑋))) → (𝑋 (𝑌 ((oc‘𝐾)‘𝑝)) ∧ (𝑌 ((oc‘𝐾)‘𝑝))𝐶𝑌))
42 oveq2 6618 . . . . . 6 (𝑤 = ((oc‘𝐾)‘𝑝) → (𝑌 𝑤) = (𝑌 ((oc‘𝐾)‘𝑝)))
4342breq2d 4630 . . . . 5 (𝑤 = ((oc‘𝐾)‘𝑝) → (𝑋 (𝑌 𝑤) ↔ 𝑋 (𝑌 ((oc‘𝐾)‘𝑝))))
4442breq1d 4628 . . . . 5 (𝑤 = ((oc‘𝐾)‘𝑝) → ((𝑌 𝑤)𝐶𝑌 ↔ (𝑌 ((oc‘𝐾)‘𝑝))𝐶𝑌))
4543, 44anbi12d 746 . . . 4 (𝑤 = ((oc‘𝐾)‘𝑝) → ((𝑋 (𝑌 𝑤) ∧ (𝑌 𝑤)𝐶𝑌) ↔ (𝑋 (𝑌 ((oc‘𝐾)‘𝑝)) ∧ (𝑌 ((oc‘𝐾)‘𝑝))𝐶𝑌)))
4645rspcev 3298 . . 3 ((((oc‘𝐾)‘𝑝) ∈ 𝐻 ∧ (𝑋 (𝑌 ((oc‘𝐾)‘𝑝)) ∧ (𝑌 ((oc‘𝐾)‘𝑝))𝐶𝑌)) → ∃𝑤𝐻 (𝑋 (𝑌 𝑤) ∧ (𝑌 𝑤)𝐶𝑌))
4712, 41, 46syl2anc 692 . 2 (((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) ∧ (((oc‘𝐾)‘𝑌)𝐶(((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ∧ (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ((oc‘𝐾)‘𝑋))) → ∃𝑤𝐻 (𝑋 (𝑌 𝑤) ∧ (𝑌 𝑤)𝐶𝑌))
48 simpl1 1062 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝐾 ∈ HL)
4948, 13syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝐾 ∈ OP)
50 simpl3 1064 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝑌𝐵)
513, 7opoccl 33996 . . . 4 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
5249, 50, 51syl2anc 692 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
53 simpl2 1063 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝑋𝐵)
543, 7opoccl 33996 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
5549, 53, 54syl2anc 692 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
56 simpr 477 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝑋 < 𝑌)
57 lhprelat3.s . . . . . 6 < = (lt‘𝐾)
583, 57, 7opltcon3b 34006 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ ((oc‘𝐾)‘𝑌) < ((oc‘𝐾)‘𝑋)))
5949, 53, 50, 58syl3anc 1323 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → (𝑋 < 𝑌 ↔ ((oc‘𝐾)‘𝑌) < ((oc‘𝐾)‘𝑋)))
6056, 59mpbid 222 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ((oc‘𝐾)‘𝑌) < ((oc‘𝐾)‘𝑋))
613, 34, 57, 28, 23, 4hlrelat3 34213 . . 3 (((𝐾 ∈ HL ∧ ((oc‘𝐾)‘𝑌) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵) ∧ ((oc‘𝐾)‘𝑌) < ((oc‘𝐾)‘𝑋)) → ∃𝑝 ∈ (Atoms‘𝐾)(((oc‘𝐾)‘𝑌)𝐶(((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ∧ (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ((oc‘𝐾)‘𝑋)))
6248, 52, 55, 60, 61syl31anc 1326 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ∃𝑝 ∈ (Atoms‘𝐾)(((oc‘𝐾)‘𝑌)𝐶(((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ∧ (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ((oc‘𝐾)‘𝑋)))
6347, 62r19.29a 3072 1 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ∃𝑤𝐻 (𝑋 (𝑌 𝑤) ∧ (𝑌 𝑤)𝐶𝑌))
Colors of variables: wff setvar class
Syntax hints:  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  occoc 15881  ltcplt 16873  joincjn 16876  meetcmee 16877  Latclat 16977  OPcops 33974  OLcol 33976  ccvr 34064  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: (None)
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