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Theorem lhprelat3N 35798
Description: The Hilbert lattice is relatively atomic with respect to co-atoms (lattice hyperplanes). Dual version of hlrelat3 35170. (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 479 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝 ∈ (Atoms‘𝐾))
2 simpll1 1231 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ HL)
3 lhprelat3.b . . . . . . . 8 𝐵 = (Base‘𝐾)
4 eqid 2748 . . . . . . . 8 (Atoms‘𝐾) = (Atoms‘𝐾)
53, 4atbase 35048 . . . . . . 7 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
65adantl 473 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝𝐵)
7 eqid 2748 . . . . . . 7 (oc‘𝐾) = (oc‘𝐾)
8 lhprelat3.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
93, 7, 4, 8lhpoc2N 35773 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑝𝐵) → (𝑝 ∈ (Atoms‘𝐾) ↔ ((oc‘𝐾)‘𝑝) ∈ 𝐻))
102, 6, 9syl2anc 696 . . . . 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 472 . . 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 35121 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ OP)
142, 13syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ OP)
15 hllat 35122 . . . . . . . . . 10 (𝐾 ∈ HL → 𝐾 ∈ Lat)
162, 15syl 17 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ Lat)
17 simpll3 1235 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑌𝐵)
183, 7opoccl 34953 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑝𝐵) → ((oc‘𝐾)‘𝑝) ∈ 𝐵)
1914, 6, 18syl2anc 696 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((oc‘𝐾)‘𝑝) ∈ 𝐵)
20 lhprelat3.m . . . . . . . . . 10 = (meet‘𝐾)
213, 20latmcl 17224 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ ((oc‘𝐾)‘𝑝) ∈ 𝐵) → (𝑌 ((oc‘𝐾)‘𝑝)) ∈ 𝐵)
2216, 17, 19, 21syl3anc 1463 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑌 ((oc‘𝐾)‘𝑝)) ∈ 𝐵)
23 lhprelat3.c . . . . . . . . 9 𝐶 = ( ⋖ ‘𝐾)
243, 7, 23cvrcon3b 35036 . . . . . . . 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 1463 . . . . . . 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 35120 . . . . . . . . . 10 (𝐾 ∈ HL → 𝐾 ∈ OL)
272, 26syl 17 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ OL)
28 eqid 2748 . . . . . . . . . 10 (join‘𝐾) = (join‘𝐾)
293, 28, 20, 7oldmm3N 34978 . . . . . . . . 9 ((𝐾 ∈ OL ∧ 𝑌𝐵𝑝𝐵) → ((oc‘𝐾)‘(𝑌 ((oc‘𝐾)‘𝑝))) = (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝))
3027, 17, 6, 29syl3anc 1463 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((oc‘𝐾)‘(𝑌 ((oc‘𝐾)‘𝑝))) = (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝))
3130breq2d 4804 . . . . . . 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 1233 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑋𝐵)
34 lhprelat3.l . . . . . . . . 9 = (le‘𝐾)
353, 34, 7oplecon3b 34959 . . . . . . . 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 1463 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑋 (𝑌 ((oc‘𝐾)‘𝑝)) ↔ ((oc‘𝐾)‘(𝑌 ((oc‘𝐾)‘𝑝))) ((oc‘𝐾)‘𝑋)))
3730breq1d 4802 . . . . . . 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 749 . . . . 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 502 . . . 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 466 . . 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 6809 . . . . . 6 (𝑤 = ((oc‘𝐾)‘𝑝) → (𝑌 𝑤) = (𝑌 ((oc‘𝐾)‘𝑝)))
4342breq2d 4804 . . . . 5 (𝑤 = ((oc‘𝐾)‘𝑝) → (𝑋 (𝑌 𝑤) ↔ 𝑋 (𝑌 ((oc‘𝐾)‘𝑝))))
4442breq1d 4802 . . . . 5 (𝑤 = ((oc‘𝐾)‘𝑝) → ((𝑌 𝑤)𝐶𝑌 ↔ (𝑌 ((oc‘𝐾)‘𝑝))𝐶𝑌))
4543, 44anbi12d 749 . . . 4 (𝑤 = ((oc‘𝐾)‘𝑝) → ((𝑋 (𝑌 𝑤) ∧ (𝑌 𝑤)𝐶𝑌) ↔ (𝑋 (𝑌 ((oc‘𝐾)‘𝑝)) ∧ (𝑌 ((oc‘𝐾)‘𝑝))𝐶𝑌)))
4645rspcev 3437 . . 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 696 . 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 1204 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝐾 ∈ HL)
4948, 13syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝐾 ∈ OP)
50 simpl3 1208 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝑌𝐵)
513, 7opoccl 34953 . . . 4 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
5249, 50, 51syl2anc 696 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
53 simpl2 1206 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝑋𝐵)
543, 7opoccl 34953 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
5549, 53, 54syl2anc 696 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
56 simpr 479 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝑋 < 𝑌)
57 lhprelat3.s . . . . . 6 < = (lt‘𝐾)
583, 57, 7opltcon3b 34963 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ ((oc‘𝐾)‘𝑌) < ((oc‘𝐾)‘𝑋)))
5949, 53, 50, 58syl3anc 1463 . . . 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 35170 . . 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 1466 . 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 3204 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 383  w3a 1072   = wceq 1620  wcel 2127  wrex 3039   class class class wbr 4792  cfv 6037  (class class class)co 6801  Basecbs 16030  lecple 16121  occoc 16122  ltcplt 17113  joincjn 17116  meetcmee 17117  Latclat 17217  OPcops 34931  OLcol 34933  ccvr 35021  Atomscatm 35022  HLchlt 35109  LHypclh 35742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-riota 6762  df-ov 6804  df-oprab 6805  df-preset 17100  df-poset 17118  df-plt 17130  df-lub 17146  df-glb 17147  df-join 17148  df-meet 17149  df-p0 17211  df-p1 17212  df-lat 17218  df-clat 17280  df-oposet 34935  df-ol 34937  df-oml 34938  df-covers 35025  df-ats 35026  df-atl 35057  df-cvlat 35081  df-hlat 35110  df-lhyp 35746
This theorem is referenced by: (None)
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